New U(1) Gauge Model of Radiative Lepton Masses with Sterile Neutrino and Dark Matter

An anomaly-free U(1) gauge extension of the standard model (SM) is presented. Only one Higgs doublet with a nonzero vacuum expectation is required as in the SM. New fermions and scalars as well as all SM particles transform nontrivially under this U(1), resulting in a model of three active neutrinos and one sterile neutrino, all acquiring radiative masses. Charged-lepton masses are also radiative as well as the mixing between active and sterile neutrinos. At the same time, a residual $Z_2$ symmetry of the U(1) gauge symmetry remains exact, allowing for the existence of dark matter.

The notion that neutrino mass is connected to dark matter has motivated a large number of studies in recent years. The simplest realization is the one-loop "scotogenic" model [1], where the standard model (SM) of quarks and leptons is augmented with a second scalar doublet (η + , η 0 ) and three neutral singlet fermions N R as shown in Fig. 1. Under an exactly Figure 1: One-loop "scotogenic" neutrino mass.
conserved discrete Z 2 symmetry, (η + , η 0 ) and N are odd, allowing thus the existence of dark matter (DM). Whereas such models are viable phenomenologically, a deeper theoretical understanding of the origin of this connection is clearly desirable.
Another important input to this framework is the 2012 discovery of the 125 GeV particle [2,3] at the Large Hadron Collider (LHC) which looks very much like the one Higgs boson of the SM. This means that any extension of the SM should aim for a natural explanation of why electroweak symmetry breaking appears to be embodied completely in one Higgs scalar doublet and no more.
To these ends, we propose in this paper an anomaly-free U (1) X gauge extension of the SM with three active and one sterile neutrinos. Whereas there exist many studies on light sterile neutrino masses [4,5], we consider here for the first time the case where all masses and mixing of active and sterile neutrinos are generated in one loop through dark matter, which is stabilized by a residual Z 2 symmetry of the spontaneously broken U (1) X gauge symmetry. To maintain the hypothesis of only one electroweak symmetry breaking Higgs doublet (which couples directly only to quarks in this model), charged-lepton masses are also radiatively generated through dark matter.
The U (1) X gauge symmetry being considered is a variation of Model (C) of Ref. [6]. It has its origin from the observation [7,8,9,10] that replacing the neutral singlet fermion N of the Type I seesaw for neutrino mass with the fermion triplet (Σ + , Σ 0 , Σ − ) of the Type III seesaw also results in a possible U(1) gauge extension. The former is the well-known B − L, the latter is the model of Ref. [8], where there is one Σ for each of the three families of quarks and leptons. Here we consider a total of only two Σ's, in which case several N 's of different U (1) X charges must be added to render the model anomaly-free. Model (C) of Ref. [6] is the first such example with three N 's. It allows radiative neutrino masses with dark matter [11,12,13]. It may also accommodate a sterile neutrino with radiative mass [5], but then dark matter is lost. Here we choose to satisfy the anomaly-free conditions with three different N 's. In so doing, we obtain a model with dark matter as well as radiative masses and mixing for three active and one sterile neutrinos as described below.
have been added to allow for all fermions to acquire nonzero masses. An automatic residual Z 2 symmetry is obtained as U (1) X is spontaneously broken by χ 0 1,2 . The three neutral singlet fermions are relabelled N R and S 1R,2R .
The two heavy fermion triplets obtain masses from the Σ R Σ Rχ Note that Φ is the only scalar doublet with even Z 2 , corresponding to the one Higgs doublet of the SM, solely responsible for electroweak symmetry breaking. The three active neutrinos ν L and the one singlet "sterile" neutrino N R are massless at tree level. They acquire radiative masses in one loop as shown in Figs. 2 to 4. The requisite couplings areΣ 0  To evaluate the one-loop diagrams of Figs. 1 to 4, we note first that each is a sum of simple diagrams with one internal fermion line and one internal scalar line. Each contribution is infinite, but the sum is finite. In Fig. 1, it is given by [1] where . Let their mass eigenstates be ζ l with mass m l . There are 4 Majorana fermion fields, spanning Σ 0 1R , Σ 0 2R , S 1R , S 2R . Let their mass eigenstates be ψ k with mass M k .
In Fig. 2, let theΣ 0 1R ν i η 0 2 andΣ 0 2R ν i η 0 2 couplings be h (2) i1 and h (2) i2 , then its contribution to M ν is given by Fig. 3, let thē S 1R ν i η 0 1 coupling be h (1) i1 , then its contribution to M ν is given by where Fig. 4, let the S 2R N R χ 0 3 coupling be h 2 , then where In the above, the three active neutrinos ν 1,2,3 acquire masses through their couplings to three dark neutral fermions, i.e. Σ 0 1R , Σ 0 2R , S 1R , whereas the one sterile neutrino N acquires mass through its coupling to S 2R . However, since S 1R mixes with S 2R at tree level, there is also mixing between ν i and N as shown in Fig. 5, with where k z S 1k z S 2k = l y R 1l y R 3l = l y I 1l y I 3l = 0. Note that the structures of these one-loop formulas are all similar, and there is enough freedom in choosing the various parameters to obtain masses of order 0.1 eV for ν and 1 eV for N , as well as a sizeable ν − N mixing. Note also that the last term in each case corresponds to the cancellation among several scalars which allow the loops to be finite and should be naturally small. In Fig. 1, it is represented by the well-known (λ 5 /2)(Φ † η) 2 + H.c. term which splits Re(η 0 ) and Im(η 0 ) in mass. In our case for example, in Eqs. (4) and (5), let h ∼ 10 −1 , the z 2 M factor ∼ 1 TeV, the [(y R ) 2 − (y I ) 2 ]F factor ∼ 10 −9 (which means that theχ 1 χ 2 3 coupling is very small), then m ν ∼ 0.1 eV. In Eq. (6), let the [(y R ) 2 −(y I ) 2 ]F factor ∼ 10 −8 instead, then m N ∼ 1 eV. In Eq. (7), let the z 1 z 2 M factor be 100 GeV, and the [y R y R − y I y I ]F factor ∼ 10 −9 (which means that the η † 1 Φχ 1 χ 3 coupling is very small), then m νN ∼ 0.1 eV. This is thus a possible framework for accommodating three active neutrinos plus a fourth light sterile neutrino, in the 3+1 scheme [14], with best fit values ∆m 2 41 = 0.93 eV 2 , |U e4 | = 0.15, |U µ4 | = 0.17, albeit having a large χ 2 , to ease the long-standing tension between ν e appearance and ν µ disappearance experiments in the ∆m 2 ∼ few eV 2 range.
The neutral dark scalars ζ l have in general components which are not electroweak singlets (η 0 1,2 , ξ 0 ). As such, they are not good dark-matter candidates because their interactions with the Z gauge boson would result in too large a cross section for their direct detection in underground experiments. Hence one of the neutral dark fermions ψ k is a much better DM candidate. Note that whereas Σ 0 1R and Σ 0 2R are components of SU (2) L triplets, they do not couple to Z because they have I 3 = 0. Note also that they mix with S 1R and S 2R only in one loop. The case of Σ 0 as dark matter in the triplet fermion analog of the scotogenic model was discussed in Ref. [19]. Here the important change is that Σ 0 1R and Σ 0 2R have both SU (2) L and U (1) X interactions. On the other hand, suppose the lighter linear combination of S 1R and S 2R is dark matter, call it ψ 0 , then only U (1) X is involved. As shown recently in [12,13], the allowed region of parameter space from dark matter relic abundance, direct detection and collider constraints corresponds to the s-wave resonance region near m X ≈ 2m ψ 0 . The U (1) X gauge boson mass m X in our model comes from χ 1,2 = u 1,2 . If n 1 = n 4 = 1 is chosen in Table 1, then the SM Higgs does not transform under U (1) X and there is no X − Z mixing. In that case, The U (1) X charges of (u, d) L , u R , d R , (ν, l) L are all 1, and those of l R , N R , χ 1 , χ 2 are −1, −3, 1, 3. These particles are even under the residual Z 2 of U (1) X . The dark sector consists of fermions Σ 1R , Σ 2R , S 1R , S 2R , with U (1) X charges 3/2, 3/2, 1/2, 5/2, as well as scalars η 1 , η 2 , χ 3 , χ 4 , ξ, with U (1) X charges −1/2, 1/2, 1/2, −3/2, −1/2.
Instead of having a sterile neutrino of 1 eV, it is also possible in our model to make it a few keV, thus rendering N a warm dark-matter candidate. This may require h 2 in Eq. (6) to be much greater than the corresponding Yukawa couplings in Eqs. (4) and (5) for the active neutrinos. On the other hand, ν − N mixing has to be much more suppressed in order not to overclose the Universe or conflict with observed X-ray data. According to Ref. [20], these may be avoided if the mixing |U i4 | is less than 10 −4 . Such a small mixing also makes the keV sterile neutrino long-lived on cosmological time scales. It could also provide an explanation to the recently observed 3.55 keV X-ray line [21] after analysing the data taken by the XMM-Newton X-Ray telescope in the spectrum of 73 galaxy clusters. The same line also appears in the Chandra observations of the Perseus cluster [22] and the XMM-Newton observations of the Milky Way Centre [23]. In the absence of any astrophysical interpretation of the line due to some atomic transitions, the origin of this X-ray line can be explained naturally by sterile neutrino dark matter with mass approximately 7.1 keV decaying into a photon and a standard model neutrino. As reported in Ref. [22], the required mixing angle of sterile neutrino with active neutrino should be of the order of sin 2 2θ ≈ 10 −11 − 10 −10 in order to give rise to the observed X-Ray line flux. For such a tiny mixing angle, Fig. 5 must be strongly suppressed, implying thus almost zero χ 3 − η 1 mixing. This in turn will make Fig. 3 vanish, thus predicting one nearly massless active neutrino.
We have shown in this paper how a stabilizing Z 2 symmetry for dark matter may be derived from a new anomaly-free U (1) X extension of the standard model. Using just the one Higgs doublet of the SM, we have also shown how three charged leptons and active neutrinos plus a sterile neutrino acquire radiative masses through the dark sector. This explains why the sterile neutrino mass itself is also small. Apart from the possibilities of long lived sterile neutrino dark matter and cold dark matter separately as discussed above, our model is wellsuited for the much more interesting mixed-dark-matter scenario, i.e. the coexistence of both.
Such a scenario could be important from the point of view of large structure formation, as well as offering proofs in different indirect detection experiments ranging from gamma rays to X-rays. We leave such a complete analysis to future investigations. This work is supported in part by the U. S. Department of Energy under Grant No. de-sc0008541.