Pure Leptonic Gauge Symmetry, Neutrino Masses and Dark Matter

A possible extension of the Standard Model to include lepton number as local gauge symmetry is investigated. In such a model, anomalies are canceled by two extra fermions doublet. After leptonic gauge symmetry spontaneously broken, three active neutrinos may acquire non-zero Majorana masses through the modified Type-II seesaw mechanism. Constraints on the model from electro-weak precision measurements are studied. Due to the $Z_2$ discrete flavor symmetry, right-handed Majorana neutrinos can serve as cold dark matter candidate of the Universe. Constraint from dark matter relic abundance is calculated.


I. INTRODUCTION
The Standard Model (SM) is in spectacular agreement with all known experiments. However, it is almost certainly fundamentally incomplete. The solar [1], atmosphere [2], reactor [3] and accelerator [4] neutrino experiments have provided us very convincing evidence that neutrinos are massive and lepton flavors are mixed. Besides, precisely cosmological observations have confirmed the existence of non-baryonic cold dark matter: Ω D h 2 = 0.1123±0.0035 [5]. These two important discoveries can not be accommodated in the SM without introducing extra ingredients.
A possible extension of the SM is to add three right-handed heavy Majorana neutrinos N R so that light neutrino masses can be generated by the famous canonical seesaw mechanism (i.e., where M D is the Dirac mass matrix linking left-handed light neutrinos to right-handed heavy neutrinos and M R is the mass matrix of N R ). This is the so-called Type-I seesaw mechanism [6]. Actually, there are three types of tree-level seesaw mechanisms [6][7][8] and three types of loop-level seesaw mechanisms [9][10][11][12]. An advantage of the seesaw mechanism is that they can both explain neutrino masses and the baryon asymmetry of the Universe with the help of leptogenesis [13]. Putting some discrete flavor symmetry on the seesaw mechanism, heavy seesaw particles can also serve as cold dark matter (CDM) candidate [10,14,15]. This builds a bridge between the dark matter and neutrino physics.
We do not have enough information on the detailed nature of CDM, except for its relic density. There are major experimental efforts for direct and indirect detection of dark matter particle beside the gravitational effect that it has on the Universe, because they must have some connection to the SM particles. A direct detection probes the scattering of dark matter off nuclei in the dark matter detectors, while indirect detection investigate the SM final states from the annihilation of the dark matter by cosmic ray detectors.
In this paper, we consider the possible extension of the SM to include lepton number(L) as local gauge symmetry. Two reasons drive us to investigate this possibility: • Baryon number(B) and L are accidental global symmetries in the SM. B must be broken to explain the baryon asymmetry of the Universe. L should be broken to accommodate the Majorana masses of active neutrinos. Investigating the possibility of L as spontaneously broken local gauge symmetry would help us to study the origin of small neutrino masses and describe the seesaw scale.
• Recent results from PAMELA [16,17] and FERMI [18] suggest there should be pure leptonic interactions for dark matter to explain the e + e − excesses observed by these experiments. Inspired by Ref. [19], we investigate the case dark matter being charged under L, which is taken as local gauge symmetry.
We extend the SM with three right-handed Majorana neutrinos, two generation fermions doublet as well as pure leptonic gauge symmetry U(1) X and Z 2 discrete flavor symmetry.
In this model, all the quarks have zero U(1) X charge, while all the leptons (including righthanded neutrinos) have unit U(1) X charge. After U(1) X gauge symmetry spontaneously broken, right-handed neutrinos acquire heavy Majorana masses, while three active neutrinos may acquire non-zero Majorana masses through the modified Type-II seesaw mechanism.
Due to the Z 2 symmetry, right-handed Majorana neutrinos don't couple to the SM fermions, so that they can be cold dark matter candidate. We study constraints on our model from neutrino physics and electroweak precision measurements. We also investigate constraints on the leptonic gauge coupling constant and the mass of the lightest right-handed neutrino from the dark matter relic abundance.
The paper is organized as following: Section II is a brief introduction to the setup of the U(1) X gauge symmetry. In section III and section IV, we study constraints on our model from the neutrino physics and electroweak precision measurements. In section V, we investigate the possibility of taking the lightest right-handed neutrino as cold dark matter.
Some conclusions are drawn in section VI.

II. THE LEPTONIC U(1) X GAUGE SYMMETRY
Now we consider the extension of the SM with three right-handed Majorana neutrinos N R , as well as new gauge symmetry U(1) X and Z 2 discrete flavor symmetry. To generate Majorana masses for right-handed neutrinos, one Higgs singlet φ with U(1) X charge −2k is added to the model. Due to the Z 2 discrete symmetry, right-handed neutrinos do not couple to the SM particles, such that they can serve as cold dark matter candidate. We also include one Higgs triplet ∆ with U(1) X charge 2k in our model, so that small but non-zero neutrino masses can be generated through the modified Type-II seesaw mechanism. Representations of particles under the symmetry, Now we investigate how to cancel anomalies of the model. The global SU(2) L anomaly [22] requires fermions doublet to be even. Considering the conditions for the absence of axialvector anomaly [23][24][25] in the presence of U(1) X and the absence of the gravitational-gauge anomaly [26][27][28], which requires the sum of the U(1) X charges to vanish, one has where a i is the weak hypercharge of Ψ i with a 1(2) = a(−a). We find that Eqs. (1), (4)- (6) and (7)-(9) hold automatically, while Eq. (2) and Eq. (3) are equivalent. As a result, the upper equations can be simplified to the following relation: Anomalies put no constraint on a. We set a = 1/2 in our paper. Four interesting scenarios can be derived from Eq. (10): • m = 0 and k = 1. All the quarks have zero U(1) X charge. As a result, U(1) X is a pure lepton number gauge symmetry.
• k = 0 and m = 1/3. All the leptons have zero U(1) X charge. Such that U(1) X is a pure baryon number gauge symmetry.
• b = 0 and k = −3m = −1. It corresponds to U(1) B−L gauge symmetry, the phenomenology of which has been well-studied.
In this paper, we only investigate the phenomenologies of the first scenario. We will study constraints on the model from neutrino physics, electroweak precision measurements and cosmological observations. The phenomenology of the second scenario, which is interesting but beyond the reach of this paper, will be shown in somewhere else. The following is the leptonic part of the full lagrangian with There are no Yukawa interactions between new fermions (Ψ i ) and SM particles because of their special U(1) X quantum numbers. Then the neutral component of Ψ is stable and can be dark matter candidate. However, Ψ is strongly coupled to the SM Higgs boson in our model, i.e., O(Y Ψ ) > 1(of course it should be smaller than 4π to satisfy the perturbativity limit [20]). Assuming m Ψ > 400 GeV and m H ∼ 200 GeV, We can estimate the relic density of Ψ: Ω Ψ h 2 < 2 × 10 −3 , which is quite small compared with the dark matter relic density, even smaller than its statistical error. Therefore the contribution of Ψ to the dark matter relic density is almost ignorable. For simplification, we will not discuss the phenomenology of Ψ in this paper.

III. NEUTRINO MASSES
We now investigate the possible origin of Majorana masses for three active neutrinos in our model. Conventional seesaw mechanisms explicitly break the lepton number, which is gauged and spontaneously broken in our model. To overcome this difficulty, we modify the Type-II seesaw mechanism slightly. The most general gauge invariant Higgs potential can be written as 1 Here H plays the role of the SM Higgs doublet. On the contrary to the conventional Type-II seesaw mechanism, the last term in Eq. (12) conserves the lepton number. When φ gets vacuum expectation value(VEV), U(1) X gauge symmetry is broken down and right-handed neutrinos acquire Majorana masses. We set H = v and φ = v ′ . After imposing the 1 Actually the full Higgs potential should also contain the following terms: Λ 1 (Tr(∆ † ∆)) 2 , Λ 2 Tr(∆ † ∆∆ † ∆), conditions of global minimum, one obtains The light neutrino mass matrix is then Present constraint on the neutrino mass matrix from neutrino oscillation experiments and cosmological observations is O(M ν ) ∼ 0.1eV [29]. By setting φ = 1 TeV and M ∆ = 10 8 GeV, one has Y ∆ λ 7 ∼ 10 −2 .

IV. ELECTROWEAK PRECISION MEASUREMENT CONSTRAINTS
We now perform an analysis of the electroweak precision observable on our model. The most stringent restrictions come from the S and T parameters [21], which can be expressed by the following equations where s = sin θ W , c = cos θ W with θ W the Weinberg angle. Y α is the weak hypercharge of Ψ α with Y 1,2 = −1/2, 1/2. To obtain a better understanding of the importance of the S and T parameters in constraining our model, we show in Fig. 1  there is sizeable region in the parameter space lying within 90% Confidence Level contour.
We proceed to consider the constraint on the model from the muon g-2. There has been a long history in measuring and calculating the muon abnormal magnetic moment a µ . In particular the steadily improving precision of both the measurements and the predictions of g-2 Experiment"(E821) for a µ reads a exp µ = (11659208 ± 6) × 10 −10 [30], which deviates from the SM prediction by In our model, heavy neutral gauge boson Z ′ will contribute to ∆a µ , which is where M Z ′ is the mass of Z ′ . There are a lot of works on the heavy neutral gauge bosons. For a recent review, see Ref. [31], and recent studies Z ′ at Tevatron and LHC [32]. In our model, the mass of the new gauge boson Z ′ is given by M Z ′ = 2g X v ′ . To satisfy the experimental lower bound, M Z ′ /g X > 5 ∼ 10 TeV [33]. Given M Z ′ > 500GeV and g X < 0.1, we can find the biggest contribution to ∆a µ is 4 × 10 −12 , which is far below the discrepancy listed in Eq. (17). In short there is almost no strong constraint on the g X and M Z ′ from the muon g-2.

V. DARK MATTER
In our model, the lightest heavy Majorana neutrino serves as dark matter candidate.
Assuming M N < M Ψ , there are two dominant annihilation channels: N N → Z ′ → ℓℓ and N N → ϕ →qq(lℓ), where ϕ is the SM Higgs. The second channel is heavily suppressed by the mixing between φ and H, such that we only consider the first channel. Ignoring charged lepton masses, one can write down the thermal average of the interaction rate σv in non-relativistic limit where x f = M N /T f with T f the freeze-out temperature of the relic particle. In assumption The present density of dark matter is simply given by ρ N = M N s 0 Y ∞ , where s 0 = 2889.2 cm −3 is the present entropy density and Y ∞ is the asymptotic value of the ratio where g * accounts the number of relativistic degrees of freedom at the freeze-out temperature. The relic density can finally be expressed in terms of the critical density where h is the Hubble constant in units of 100 km/s · Mpc and M P l = 1.22 × 10 19 GeV is the Planck mass. The freeze-out temperature x f can be estimated through the iterative solution of the following equation [34] x f = ln c(c + 2) 45 8 where c is the constant of order one determined by matching the late-time and early-time solutions.
We set x f equals to 20, a typical value at freeze-out for weakly interacting particles. In where the annihilation cross section is resonantly enhanced. All the experimental constraints can be fulfilled near these region.

VI. CONCLUSIONS
We have investigated the possibility of taking the lepton number as local gauge symmetry.
In such a model, at least two fermions doublet are needed to cancel the anomaly. We have introduced a modified Type-II seesaw mechanism to generate Majorana masses for three active neutrinos. Constraints from electroweak precision measurements were studied. The result shows that there are adequate parameter space for our model. Taking heavy Majorana neutrinos as dark matter candidate, we have studied the constraint on the leptonic gauge coupling constant and the mass of the lightest right-handed neutrino from dark matter relic abundance.