Type III Neutrino Seesaw, Freeze-In Long-Lived Dark Matter, and the $W$ Mass Shift

In the framework of seesaw neutrino masses from heavy fermion triplets $(\Sigma^+,\Sigma^0,\Sigma^-)$, the addition of a light fermion singlet $N$ and a heavy scalar triplet $(\rho^+,\rho^0,\rho^-)$ has some important consequences. The new particles are assumed to be odd under a new $Z_2$ symmetry which is only broken softly, both explicitly and spontaneously. With $N-\Sigma^0$ mixing, freeze-in long-lived dark matter through Higgs decay becomes possible. At the same time, the $W$ mass is shifted slightly upward, as suggested by a recent precision measurement.

Introduction : The well-known seesaw mechanism for tiny Majorana neutrino masses has three simple tree-level realizations [1], depending on the heavy intermediary particles involved. Whereas Type I [heavy fermion singlet N ] and Type II [heavy scalar triplet (ξ ++ , ξ + , ξ 0 )] are routinely considered in numerous papers, Type III [heavy fermion triplet (Σ + , Σ 0 , Σ − )] is rather less studied [2,3]. In this paper, in addition to three copies of heavy Σ for obtaining three light Majorana neutrinos, a new Z 2 symmetry is assumed under which one new light singlet fermion N and one new heavy scalar triplet (ρ + , ρ 0 , ρ − ) are odd, and all other fields are even. Whereas all dimension-four terms of the resulting Lagrangian must respect Z 2 [which forbids N from coupling to lepton doublets through the usual Higgs doublet Φ of the standard model of quarks and leptons (SM)], this symmetry is broken explicitly by the dimension-three soft term Φ † ρΦ, resulting thus in a small nonzero vacuum expectation value v ρ for ρ 0 .
Three important consequences follow. (A) The W mass gets a slight upward shift [4].
(B) The φ 0 − ρ 0 and Σ 0 − N mixings allow the SM Higgs boson h to decay to N N . (C) N decays through its mixing with the heavy Σ 0 which couples to νφ 0 , converting thereby to νf f , where f is the heaviest fermion kinematically allowed. Hence N is possibly a long-lived dark-matter candidate, produced in a freeze-in scenario [5] through rare h decay [6].
Higgs Potential : The Higgs potential V of this proposal consists of the SM Higgs doublet Φ and the new real scalar triplet ρ which is odd under the assumed Z 2 , i.e.
where the µ 1 trilinear term breaks Z 2 softly. Let then v 0,1 are determined by For large and positive m 2 1 , the scalar seesaw solution [7] is The 2 × 2 mass-squared matrix spanning h and s is then with h − s mixing given by To explain the new precision measurement of the W mass [4], i.e.
which is several standard deviations above the prediction of the SM (v 1 = 0), a central value of v 1 3.68 GeV may be extracted from the analysis of Ref. [8].
Singlet-Triplet Fermion Mixing : Neutrinos obtain seesaw masses through the heavy fermion triplets from the Yukawa couplngs resulting in the Dirac mass m νΣ = f ν v 0 /2, and then the usual seesaw Majorana neutrino Since N is odd under Z 2 , it does not couple to ν through φ 0 . However, it does couple to Σ through ρ, i.e.
The N − Σ 0 mixing is then f N v 1 /m Σ and the s coupling to N N is Higgs Decay to N N : Since N is a singlet fermion, the Higgs boson h does not couple to N N directly. It does so first through h − s mixing, then through N − Σ 0 mixing, as shown in Fig. 1.
The effective coupling is then The decay rate of h → N N +NN is [9] Γ where r = m N /m h . Now N is assumed light and a candidate for long-lived dark matter.
The correct relic abundance is obtained [10] if f h ∼ 10 −12 r −1/2 , provided that the reheat temperature of the Universe is above m h but well below m ρ and m Σ .
Long-Lived Dark Matter : The singlet fermion N is assumed light and decays only through its mixing with Σ 0 which couples to νh. Through the virtual Higgs, its coupling to νf f is then given by where f is a fermion allowed kinematically in the decay, as shown in Fig. 2. The decay rate is analogous to that of muon decay, i.e.
N Σ × h f ν f for m ν = 0.1 eV. The N lifetime is then many orders of magnitude greater than the age of the Universe and satisfies bounds from all cosmological considerations [11].
Conclusion : In this paper, a first example of long-lived freeze-in dark matter is presented in the context of Type III seesaw neutrino masses using heavy fermion triplets Σ. The key is the addition of a light fermion singlet N and a real scalar triplet ρ, both odd under a softly broken Z 2 symmetry. The rare decay of the SM Higgs to N N accounts for the dark matter relic abundance of the Universe, with N having a lifetime many orders of magnitude greater than the age of the Universe. This is accomplished with m N = 0.1 GeV, m Σ ∼ 10 9 GeV, and ρ 0 = 3.68 GeV, which also explains the shift in the W boson mass, observed recently.