Dirac dark matter in $U(1)_{B-L}$ with Stueckelberg mechanism

We investigate a $U(1)_{B-L}$ gauge extension of the Standard Model (SM) where the gauge boson mass is generated by the Stueckelberg mechanism. Three right-handed neutrinos are added to cancel the gauge anomaly and hence the neutrino masses can be explained. A new Dirac fermion could be a WIMP dark matter whose interaction with the SM sector is mediated by the new gauge boson. Assuming the perturbativity of the gauge coupling up to the Planck scale, we find that only the resonance region is feasible for the dark matter abundance. After applying the $\Delta N_{eff}$ constraints from the current Planck experiment, the collider search constraints as well as the dark matter direct detection limits, we observe that the $B-L$ charge of dark matter satisfies $|Q_{\chi}|>0.11$. Such a scenario might be probed conclusively by the projected CMB-S4 experiment, assuming the right-handed neutrinos are thermalized with the SM sector in the early universe.


I. INTRODUCTION
The discovery of neutrino oscillations [1,2] indicates that neutrinos should have tiny but non-zero masses, which can not be explained in the framework of the Standard Model (SM). One of the compelling solutions to neutrino mass problem is to introduce three righthanded neutrinos which directly couple to the SM sector through Yukawa interactions. At the same time, the gauge sector can be extended with an additional anomaly free U (1) B−L .
Another fact demanding the presence of new physics is the existence of dark matter (DM) which constitutes about 27% of the global energy budget in the universe. Therefore, it is intriguing to explain these phenomena in a same framework 1 .
In this article, we investigate a Dirac fermionic dark matter in a B − L gauge extension of the SM where the new gauge boson Z obtains mass via the Stueckelberg mechanism.
Such a model is first proposed by [3] and here we will give a more comprehensive study. In this model the neutrinos are Dirac fermions and a vector-like Dirac particle χ charged under U (1) B−L is assumed to be a WIMP dark matter candidate. In the early universe, DM is in thermal equilibrium with the SM plasma by exchanging the Z boson and then freezes out when the expansion rate of the universe excesses its annihilation rate. Finally, the current DM relic abundance needs to be consistent with the Planck data [19].
On the other hand, since the right-handed neutrinos interact with the new gauge boson, they are also in the thermal equilibrium with the SM sector in the early universe. When the temperature goes much below the gauge boson mass, they decouple and become the hot relic.
Similar to the neutrinos and photons, they contribute to the radiation energy density which is usually described as the effective number of neutrino species N ef f , which is predicted to be 3.043 in the SM [20][21][22][23][24]. The radiation density can be probed by the observation of the anisotropies in the cosmic microwave background (CMB), which was proposed long time ago [25]. The recent result from the Planck satellite shows N ef f = 2.99 +0. 34 −0.33 , providing a strong constraint on the extensions of SM where the massless or light particles are present.
We will show that this value already gives a very strong limit on our scenario. Particularly, it is pointed out in a recent article [26] that the projected CMB-S4 experiment will provide serious constraints for almost all Dirac-neutrino models, especially those addressing the origin of small neutrino masses. This paper is organized as follows. In Section II, we introduce the U (1) B−L Stueckelberg extension of the SM. Next, in Section III, the calculation of the shift in the effective number of neutrino species, ∆N ef f , and the current and future experimental bounds are discussed.
In Section IV, we examine various constraints on the parameter space of the model. At last, we draw our conclusions in Section V.

II. THE MODEL
The gauge group of the model is The Stueckelberg mechanism as an alternative to the Higgs mechanism can give mass to abelian vector bosons without breaking gauge invariance [27][28][29][30][31][32][33][34][35]. The Lagrangian related to the Stueckelberg mechanism is given by which is invariant under the following transformation In the quantum theory, a gauge fixing term should be added to the total Lagrangian so that the new gauge boson becomes massive while the field σ decouples. Note that the scalar field σ can have Stueckelberg couplings to all abelian gauge bosons, including the hypercharge vector boson B in the SM [36][37][38][39][40].
However, in this work, we only focus on the pure Stueckelberg sector in the absence of the mass mixing of the gauge boson B with the U (1) B−L gauge boson Z for simplicity. Then the B − L vector current J coupling to the gauge boson Z is given as where g is the B − L gauge coupling and J comes from quarks, leptons and DM.
In the Stueckelberg scenario, neutrinos are Dirac-type and their masses can be generated by Yukawa interactions via the Higgs mechanism, For a sub-eV neutrino mass, the coupling Y ν should be generally smaller than 10 −12 . We can add a Dirac fermion χ which only takes the B − L charge and can be a dark matter candidate (its stability can be guaranteed if its B −L charge Q χ is not equal to ±1, otherwise it will mix with the right-handed neutrino and decay). The relevant Lagrangian for the DM is then written as The total Lagrangian in the model can be summarized as Based on the above Lagrangian, for M χ m f , DM can annihilate into

denotes three families of quarks and leptons) via the gauge boson Z . The non-relativistic form for these annihilation cross sections is
where υ is the relative velocity of the annihilating DM pair, N C f is the number of colors of the final state SM fermions, Q χ and Q f represent the B −L charges of DM and SM fermions, and Γ Z is the decay width of Z boson given by From Eq.(9) it can be seen that the ratio of the contribution of a quark to the total DM annihilation cross section and the contribution of a lepton is about 1 : 3. Besides, DM is also able to annihilate into two Z bosons when DM is heavier than Z . The annihilation cross section of this channel is

III. THE EFFECTIVE NUMBER OF NEUTRINO SPECIES IN COSMOLOGY
Three additional right-handed neutrinos can be in thermal equilibrium with the SM plasma via the exchange of Z boson in the early universe so that they can contribute to the expansion rate of the universe. However, due to their weak interactions 2 , such particles decouple earlier from the plasma than the left-handed neutrinos and therefore their contribution to the energy density of the universe is suppressed compared to that of the left-handed neutrinos. The extra radiation energy density is usually expressed in terms of an effective number of neutrinos, N ef f = (8/7)(11/4) 4/3 ρ ν /ρ γ . The SM prediction of this value is 3.043.
In this scenario, the extra contribution of the right-handed neutrinos to the effective number of neutrino species is given as where N ν R represents the number of relativistic right-handed neutrinos, g * (T ) = g B (T ) + 7 8 g F (T ) with g B,F (T ) being the number of bosonic and fermionic relativistic degrees of freedom in equilibrium at the temperature T. The second equality is obtained from taking into account of the isentropic heating of the rest of the plasma between T dec ν R and T dec ν L decoupling temperatures. Taking three active neutrinos, e ± and photon into account, we have g * (T dec ν L ) = 43/4 at T dec ν L ≈ 2.3 MeV [41]. The effective number of neutrino species has a strong relation with the temperature at which the right-handed neutrinos decouple from the SM plasma, which can be decided by Here H(T ) is the Hubble expansion parameter which is estimated by 1.66g 1/2 * (T ) T 2 M P l where g * (T ) is the effective degree of freedom [42,43] including the contribution of right-handed neutrinos. Γ ν R (T ) is the right-handed neutrino interaction rate which can be calculated by In our paper we examine the bounds on the parameters of the model by using present and prospective experimental data. The current Planck CMB measurement gives the result N ef f = 2.99 +0. 34 −0.33 including baryon acoustic oscillation (BAO) data [19]. Combining with the data given above, N SM ef f = 3.043, we adopt a conservative limit ∆N ef f < 0.283 and then the bound with respect to the B − L gauge coupling g and Z boson mass M Z is given as M Z /g > 10.4 TeV. It gives a very strong limit on the parameter space, as we will show later.
Besides, there are several experiments with better sensitivities which are underway or projected. The South Pole Telescope (SPT-3G), which is a ground-based telescope in operation at present, will have a sensitivity of σ(∆N ef f ) = 0.058 [44]. The CMB Simons Observatory (SO), which will see first light in 2021 and start a five-year survey in 2022, is expected to reach a similar sensitivity in the range of σ(∆N ef f ) = 0.05 − 0.07 [45]. The CMB Stage IV (CMB-S4) experiment will have the potential to constrain ∆N ef f = 0.06 at 95% C.L. as a single parameter extension to ΛCDM [46]. Importantly, according to Eq.(12), the minimal shift in the effective number of neutrino species in our scenario can be evaluated to acquire ∆N ef f ≥ 0.141 when T dec ν R is high enough. Hence, the future CMB-S4 experiment will be able to probe this scenario for arbitrary decoupling temperatures conclusively as long as the right-handed neutrinos have a thermalization with the SM plasma in the early universe. In this work, we consider the constraints from the DM relic density Ω χ h 2 , the shift in the effective number of neutrino species ∆N ef f , the dark matter direct detection limits as well as the collider search limits for the Z . We also require the gauge coupling to keep perturbativity up to the Planck scale M pl .
The one-loop β function of the U (1) B−L gauge coupling is given by where i sums over all particles that carry B − L charge and β 0 is a function of Q χ . Assuming that the Landau pole does not occur below the Planck scale, then we get with the renormalization scale µ = M Z . 1T [49], LHC+LEP [50,51], ∆N ef f [19] and Landau pole, respectively. Black points satisfy the DM relic abundance but their gauge couplings are larger than 2 √ π. The remained blank region survives all these constraints.
In our numerical calculation, we use LanHEP 3.2.0 [47] to generate the Feynman rules of the model and apply MicrOMEGAs 5.0.9 [48] to compute the DM relic abundance and DM-nucleon scattering cross-section. In Fig. 1,  In Fig.2 we show the parameter space of M χ versus Q χ by setting M Z = 2M χ and changing g to meet the condition Ω χ h 2 = 0.12(note that we only scanned the positive Q χ ).