Parallel Seesaw Mechanisms for Neutrinos and Freeze-In Long-Lived Dark Matter

If dark matter is light, it may be due to a seesaw mechanism just as neutrinos are. It is postulated that both originate from the same type of heavy fermion anchors, either singlets or triplets. In the latter case, a shift of the $W$ mass is predicted, as suggested by the $CDF$ precision measurement. A spontaneously broken dark $U(1)$ gauge symmetry is assumed, resulting in freeze-in long-lived light dark matter.

Introduction : Neutrinos are elusive because they are light and weakly interacting.Dark matter is also elusive and yet to be observed directly.Perhaps the reason is the same, namely that dark matter is light and feebly interacting.As opposed to thermal freeze-out for heavy dark matter, it is produced through freeze-in [1,2,3,4,5] as the decay product of some other particle, such as [6] the Higgs boson of the Standard Model (SM).It may also be very long-lived and does not require an unbroken symmetry which would be necessary in the case of the usually assumed stable dark matter.
To understand why neutrinos are light, there is the well-known seesaw mechanism [7].If dark matter is also light, the reason may well be the same.To explore this idea in detail, two scenarios are studied where the anchors of both seesaw mechanisms are assumed to be of the same type.One is an extension of Type I seesaw with heavy neutrino singlets as the anchors, the other is its Type III analog, using heavy lepton triplets instead [8,9].In the latter case, a scalar triplet is also present, resulting in a shift [10,11,12,13,14,15] of the W boson mass from the prediction of the SM, as indicated by recent CDF data [16].
Dark U (1) D Gauge Symmetry : To implement the idea of a seesaw mass for dark matter, a dark U (1) D gauge symmetry is postulated.The relevant particles in the two proposed scenarios are listed in Tables 1 and 2. The model of Table 1 assumes the usual Type I seesaw mechanism where three heavy neutrino singlets N R act as anchors for imparting small masses to the three observed neutrinos ν L , through the Yukawa terms The neutrinos themselves have no invariant mass terms in the Lagrangian because they transform as doublets under the SU (2) L × U (1) Y gauge symmetry.In the dark sector, the U (1) D gauge symmetry is spontaneously broken by χ 0 , but S L has no invariant mass.It acquires a small mass from N ′ R as well through the Yukawa term (The question of anomaly cancellation will be discussed later.)For all three neutrinos and S L to acquire seesaw masses, there should be three N R copies and one N ′ R .They are distinguished by the dimension-four Yukawa terms of Eqs. ( 1) and ( 2), where ν, l, N may be chosen to be odd under lepton parity, and S, N ′ to be odd under dark parity.However, both N and N ′ do not transform under the gauge symmetries of the model, so they can mix.It is assumed here that they do so only through the dimension-three soft N N ′ Majorana mass terms, which will be assumed small in the following.The justification is that their absence would enhance the symmetries of the model, an argument attributed to 't Hooft and often used in many such discussions.
The model of Table 2 Neutrinos obtain small masses now in the Type III seesaw mechansim through the Yukawa terms In the dark sector, the scalar triplet ρ is added to allow the Yukawa terms Again, for all three neutrinos and S L to acquire seesaw masses, there should be three Σ R fermion/scalar SU (2 Table 2: Seesaw dark matter with heavy lepton triplet anchors.
copies and one Σ ′ R .A hybrid variation of the two models is clearly also possible with some N R , N ′ R and some Σ R , Σ ′ R adding up to four copies.
Anomaly Cancellation : Since S L is the only fermion transforming under U (1) D , the proposed dark gauge theory is anomalous.However, it is crucial that S L does not have a partner with opposite dark charge, or else it would not be massless to begin with, which is the first requirement for the seesaw mechanism.To allow for this special condition, the following particle content under dark U (1) D may be considered.Let there be one copy of a singlet with one unit of dark charge, i.e. S L , two copies with two units, three copies with minus three units, and one copy of four units.Then [17] 1(1) + 2(2) + 3(−3) + 1(4) = 0; 1(1) + 2(8) + 3(−27) + 1(64) = 0.
Thus anomaly cancellation is achieved.Given the charges q i of the 7 left-handed chiral fermions, and the unit charge of the scalar χ, the only possible fermion mass terms correspond would require (3), (−3) would require (2), and (4) would require (5).Hence S L is massless, which is exactly what is desired for this proposal.The situation is analogous to the neutrino case, where the absence of a Higgs triplet (ξ ++ , ξ + , ξ 0 ) keeps the neutrino massless.
In this scenario, the lightest dark Dirac fermion is also stable.This accidental symmetry is analogous to that of baryon number and lepton number in the SM and is due to the chosen particle content under the dark U (1) D gauge symmetry.For the purpose of this study, it is assumed that the reheat temperature of the Universe is well above the SM Higgs boson mass but not high enough to produce these dark fermions.More discussion on these possible dark matter candidates will be presented later.
Two Higgs Potentials : In the model of Table 1, the Higgs sector consists of the SM doublet Φ and the dark singlet χ.Its scalar potential is simply given by As ϕ 0 and χ 0 acquire nonzero vacuum expectation values v 0 and v 1 , the SM gauge symmetry breaks to electromagnetic U (1) and the dark U (1) D symmetry is completely broken.The minimum of V 1 is given by The 2 × 2 mass-squared matrix spanning √ 2Re(ϕ 0 ) and √ 2Re(χ 0 ) is then In the model of Table 2, a complex scalar triplet ρ = (ρ + , ρ 0 , ρ − ) is added so that Assuming that m 2 2 >> µ 2 0,1 , the vacuum expectation value ⟨ρ 0 ⟩ = v 2 is small [18], i.e.
Assuming the latter to be much heavier, this mixing is small, i.e.
Another mixing is that of S L with N ′ R coming from Eq. ( 2), i.e.
This is the analog of the ν − N mixing in the neutrino sector.The seesaw mass of S is The third mixing comes from the assumed small N N ′ mass terms.Consider for simplicity the 4 × 4 mass matrix spanning (ν, S, N, N ′ ).It is of the form The reduced seesaw 2 × 2 mass matrix is then Assuming that m S >> m ν , the ν − S mixing is then which is indeed very much suppressed.This mixing shows that S may be considered a sterile neutrino, but its mixing with the active neutrinos is naturally very small.
In the model of Table 2, h mixes with ρ 0 according to Another mixing is that of S L with Σ ′ 0 R coming from Eq. ( 4), i.e.
This is the analog of the ν − Σ 0 mixing in the neutrino sector.The seesaw mass of S is Now v 2 contributes to the W mass shift as measured by CDF [16], i.e.
The central value here crresponds to v 2 ≃ 3.68 GeV from the analysis of Ref. [15].
The third mixing is of course the ΣΣ ′ analog of N N ′ as shown above.
Rare Higgs Decay to Dark Matter : In the model of Table 1, h decays to SS + S S through θ ϕχ and θ SN ′ as shown in Fig. 1.

The effective coupling is
The decay rate of h → SS + S S is [19] Γ where r = m S /m h .Now S is light and a candidate for long-lived dark matter.The correct relic abundance is obtained [20] if In the model of Table 2, the decay of h to SS also proceeds with χ replaced with ρ 0 and N ′ replaced with Σ ′ 0 in Fig. 1.Hence Combining this with Eq. ( 24), m S = 1 keV is uniquely determined.
Long-Lived Dark Matter : Since S is a singlet fermion, it may be considered a sterile neutrino, but its mixing with the active neutrinos, i.e. θ νS of Eq. ( 17), is suppressed by N N ′ or ΣΣ ′ mixing which breaks the separate conservation of lepton parity and dark parity as already pointed out.It may therefore decay radiatively [21] into a neutrino and a photon.
Its lifetime is then given by where τ U = 4.35 × 10 17 s is the age of the Universe.However, for decaying dark matter not to conflict with the observations of the Cosmic Microwave Background (CMB), τ γ > 10 25 s is required [22].Hence This shows that m S should not be much greater than 10 keV or θ νS would have to be chosen to be extremely small.
Unlike the canonical scenario with sterile neutrinos as warm dark matter, the production mechanism of S through Higgs decay does not depend on θ νS whereas its decay rate does, thereby evading the strong constraints in the case of sterile neutrinos where both depend on θ νS .Although the low-energy phenomenology is indistinguishable from just adding a light sterile neutrino to the SM, the current proposal offers a theoretical understanding to why the sterile neutrino is light, and the hope that at energies beyond a few TeV, the dark gauge sector may be observed.

or − 1 .
Hence only the combinations 2 + (−3) = −1 and 4 + (−3) = 1 are allowed.This means that there is a 3 × 3 mass matrix linking the 3 fermions of charge (−3) to the 2 fermions of charge (2) and the one of charge (4), resulting in 3 Dirac fermions.S L of charge (1) is left by itself.For it to have a Majorana mass, a Higgs scalar of charge(2) would be needed.With only χ of charge (1), S L also cannot couple to the other fermions:(2)

Table 1 :
Seesaw dark matter with heavy neutrino singlet anchors.