Thermal Relic Right-Handed Neutrino Dark Matter

It is known that two heavy Majorana right-handed neutrinos are sufficient to generate the baryon asymmetry in the present universe. Thus, it is interesting to identify the third right-handed neutrino $N$ with the dark matter. We impose a new discrete symmetry $Z_2$ on this dark matter neutrino to stabilize it. However, the $U(1)_{B-L}$ gauge boson $A'$ couples to the right-handed neutrino $N$. If the $B-L$ breaking scale $V_{B-L}$ is sufficiently low, the dark matter neutrino $N$ can be in the thermal bath. We find that the thermal relic $N$ can explain the dark matter abundance for the $B-L$ breaking scale $ V_{B-L} \sim O(10)\,$TeV. After considering all the constraints from the existing experiments, a narrow mass region of the thermal produced right-handed neutrino dark matter $N$ is still surviving.

Introduction -The intriguing U (1) B−L extension of the Standard Model (SM) predicts three right-handed neutrinos (RHNs) to cancel the gauge anomalies.The presence of the heavy RHNs is a key point for the natural explanation of the observed small neutrino masses through the seesaw mechanism [1][2][3][4], as well as for the creation of the baryon asymmetry in the present Universe via the leptogenesis [5,6].It has been pointed out [7,8] that two heavy Majorana neutrinos N 1,2 are sufficient for generating the observed baryon asymmetry 1 .This fact has raised a new question, 'Why do we have three families?' A simple answer to this question is that one more RHN (called as N ) is required to explain the dark matter [10,11].
In the previous works, we have considered the nonthermal production of the RHN DM N [10,11], provided that the breaking scale of the U (1) B−L gauge symmetry is extremely high such as a GUT scale.Thus, it is very difficult to test such a model since the interaction between DM N and the SM particles is extremely weak.However, the sufficient baryon asymmetry can be produced at the Universe temperature of O (10) TeV if the two Majorana masses of N 1,2 are almost degenerate [12,13].Therefore, it is interesting to consider the U (1) B−L gauge symmetry is broken at a relatively low scale around O (10) TeV and the two Majorana masses for N 1,2 are also of the same scale.
In this paper, we consider the thermal production of the RHN DM N in the early Universe assuming the B − L breaking scale V B−L ≃ O (10) TeV to explain the observed DM density2 .After introducing the model, we first show that such scenario can successfully ex-plain the DM density in the present Universe.Then, the constraints from all existing experiments, including the fixed-target, collider, and DM direct detection experiments, are discussed.The combined constraints from the direct detection and collider exclude the mass parameter region m N ≳ 5 GeV.The direct N production experiments at the fixed targets give us very strong constraints on the lower mass range where m N < 1 GeV.Even though, a narrow parameter region for m N ⊂ (10 MeV, 5 GeV) still remains.Such a parameter space can be tested in the near future experiments.
The RHN DM Model -In this paper, we consider a simple extension of the SM which is based on the gauge symmetry SU (3) × SU (2) × U (1) Y × U (1) B−L .This simple extension is extremely attractive since we need to add three RHNs to cancel the gauge anomalies.The presence of RHNs is a key point of explaining the small neutrino masses via the seesaw mechanism [1][2][3][4]16] and of generating the observed baryon-number asymmetry in the present universe through the leptogenesis [5,6].
A new Higgs field Φ is introduced to break the gauged B − L symmetry spontaneously, and its couplings to RHNs N i (i = 1 − 3) generate large Majorana masses for them through 1  2 h i ΦN i N i .The related Lagrangian is, Here, L α , f , and H are the SM left-handed lepton doublets, fermions, and Higgs boson.The index i stands for the generation of RHN and α for all the species of leptons.The vacuum expectation value of Φ is the breaking scale of B − L gauge symmetry ⟨Φ⟩ = V B−L .Thus, the RHN mass is As explained in the introduction, one of the righthanded neutrinos N i (i = 1−3) is the DM.We can choose the third N 3 to be this RHN DM without losing the generality and hence λ 3α = 0. To guarantee the stability of DM, we impose a discrete Z 2 symmetry.Only the RHN DM N is odd under the Z 2 parity while others are even.First of all, we exclude the parameter region that, g B−L ∼ O(1).In such a region, the right-handed neutrino N is nothing but the so-called WIMP, and it is most likely excluded by the direct detection experiments.Therefore, we consider the small gauge coupling g B−L ≪ 1 region and hence the B − L gauge boson A ′ is relatively light with mass m A ′ ≪ V B−L = O(10) TeV.On the other hand, we assume the new Higgs boson Φ has a mass of the order of the B − L breaking scale.The effects of this Higgs boson at the freeze-out time of our DM N can be neglected.Thus, the physics at the decoupling time of the RHN DM is described by only lighter particles, the DM N , the U (1) B−L gauge boson A ′ , the SM particles, and their interactions.
Thermal Production -In this section, we discuss the thermal production of the RHN DM N and calculate its relic abundance.First of all, the B −L gauge boson A ′ can be in the thermal bath with the SM particles through its decay and inverse decay, Then, we consider possible thermal processes that make the RHN N in the thermal bath.The dominant processes are dependent of the mass difference ∆ between the B − L gauge boson A ′ and the RHN N , defined as ∆ ≡ (m A ′ −m N )/m N .We shall discuss it in the following three cases.I. m N > m A ′ (∆ < 0): In this parameter region, there are two processes rendering the RHN N in the thermal bath with the SM particles.First, the RHN DM can annihilate into gauge boson through N + N ↔ 2A ′ with a thermal averaged cross section, Second, it also annihilates into SM fermions f and antifermion f , N + N ↔ f + f , by exchanging the mediator A ′ .The thermal averaged cross section, is velocity dependent but with the same order of g 4 B−L .The thermal evolution of the RHN yield Y N ≡ n N /s (where n N is the number density of RHN and s = 2π 2 g * T 3 /45 is the entropy density of the Universe with degrees of freedom g * ) is then described by the Boltzmann equation, where ⟨σv⟩ ann is the total annihilation cross section containing both channels.As the number density decreases, the annihilation freezes out.The final yield can be solved analytically as, Here, x f is the freeze-out point.The DM relic density is related to Y f N as, where s 0 = 2891.2cm −3 is the entropy density today and ρ c = 1.05 × 10 −5 h 2 GeV/cm 3 is the critical density of the Universe.
The parameter space which explains the observed DM density today Ω N h 2 = 0.12 is shown as the gradient blue region of the lower panel in Fig. 1.The mass range m N < 10 GeV has a very serious constraint from the CMB observations [17,18].Furthermore, other parameter region is also excluded by the existing experimental data.More details shall be discussed in the next section.
The annihilation of RHN N into gauge boson A ′ is kinetically suppressed because the gauge boson becomes heavier.Thus, such a channel is neglected in Eq. ( 4), and the freeze-out is determined only by the annihilation channel N + N ↔ f + f .As shown in the upper panel of Fig. 1, since the total annihilation cross section does not change too much, the parameter space (gradient red region) giving the observed DM relic density is similar to the case that ∆ < 0. Although there is no CMB constraint here for the annihilation channel to gauge boson is forbidden, all the parameter space is still excluded by experimental constraints.III.m A ′ > 2m N (∆ > 1): Once the gauge boson mass is larger than two times of the RHN mass, the decay and inverse decay channel3 A ′ ↔ 2N dominants with a decay rate of order g 2 B−L .The Boltzmann equation in the early Universe becomes, The gauge boson A ′ keeps maintaining in equilibrium for its decay rate is always larger than the Hubble constant in the concerning energy scale.The freeze-out point x f can be obtained by the condition that the coefficient of the collision term becomes order of one, To match the observed DM density, the RHN mass has a relationship with the coupling g B−L and mass difference ∆ as [20], Here, we applied the DM number density, n N (x f ) ≃ π 2 g * T eq m 2 N /30x 2 f .The corresponding parameter space is shown in Fig. 2 as the gradient red padding area.The same as the standard freeze-out scenario, a larger DM mass requires a larger coupling.Besides, for a fixed DM mass, the B − L gauge boson mass m A ′ increases as the mass difference ∆ increase.As a result, the number density of the gauge boson becomes smaller and a larger coupling is needed to compensate for the decrease of the left-handed side of Eq. ( 9).One can see that the DM with a mass m N ≳ 5 GeV is excluded by DM direct detection and collider experiments.However, there is still some regions surviving for m N ⊂ (10 MeV, 5 GeV) with the vacuumexpectation value V B−L ⊂ (1 TeV, 5 TeV).
Experimental Constraints -The RHN DM N , as well as a new massive gauge boson A ′ , is predicted in our model.Both of them have interactions with SM particles and receive strong constraints from experiments.In this section, we shall analyze the results from fixed-target, collider, and DM direct detection experiments and show the surviving parameter space.All the constraints are shown in Fig. 2. I. Fixed-Target Experiments: The B − L gauge boson A ′ can be produced by the bremsstrahlung process in high-energy electron scattering off heavy nucleus, e − +Z → e − +Z +A ′ .Its subsequent invisible decay into a neutrino pair will cause missing energy events in fixedtarget experiments [21][22][23].With no signal events found, NA64 experiment data [24] gives the stringent constraints on the B −L gauge coupling, g B−L < (2×10 −5 , 10 −4 ) according to the mass range m A ′ ⊂ (10, 100) MeV as shown in the gray shaded region surrounded by solid curve.
The extra B −L gauge symmetry also generates a nonstandard ν − e interaction, which shall lead to some recoil excess in neutrino detectors.Thus, the strongest constraint for the mass region m A ′ ⊂ (0.1, 1) GeV comes from ν −e scattering experiments, CHARM II [25], which is shown as the gray shaded region surrounded by dashed curve.
II. Collider Search: Several searches for the B − L gauge boson produced in proton-proton collisions at LHC were also carried out by the ATLAS, CMS, LHCb and FASER collaborations [26][27][28][29].By searching for the new high-mass resonances phenomena in dielectron and dimuon final states, the ATLAS collaboration gives the constraint for g B−L < 10 −3 in m A ′ ⊂ (100, 1000) GeV [26] as shown by the green shaded region surrounded by dashed curve.An inclusive search for the gauge boson decay A ′ → µ + + µ − was performed by LHCb experiment [28,30].By searching for the displaced-vertex signature for long-lived A ′ in 2m µ < m A ′ < 350 MeV and prompt-like A ′ decay in 350 MeV < m A ′ < 70 GeV, the experiment constrains g B−L < 10 −4 as shown by the the green shaded region surrounded by solid curve.The FASER experiment is designed to search for longlived particles travelling in the far-forward direction of the proton beam.It excludes the parameter space where g B−L ⊂ (5×10 −6 , 2×10 −5 ) and m A ′ ⊂ (15, 40) MeV [29] as shown in the dark green shaded region surrounded by solid curve.III.Direct Detection: The RHN itself can scatter with SM fermions by exchanging a t-channel B − L gauge boson.Thus, it is also constrained from the DM direct detection experiments.The scattering is p-wave since the RHN DM N is Majorana type.Taking a general fermion target with mass m f , the leading order of the p-wave scattering cross section σ S is, Here, T N is the kinetic energy of initial DM.For the halo DM with velocity v ∼ 10 −3 , the kinetic energy is of order T N ∼ 10 −6 m N .In the non-relativistic limit, the interaction between Majorana RHN and SM fermion via a vector mediator contains both spin-independent and spindependent contributions [31].The spin-independent part plays a leading role and receives stronger constraints.
For the RHN DM with mass over O(1) GeV, the most sensitive constraints are from the Xenon [32], LZ [33], and PandaX [34] which are based on the DM-nuclear scattering.For the coherent scattering of Xenon nuclear, the target mass is m f = 131 GeV.As shown in the blue shaded region surrounded by solid lines, the nuclear recoil constraint is slightly weaker than collider ones.However, it fills the gap around (30,70) GeV so that the possibility of m N ≳ 5 GeV is totally excluded.On the other hand, one needs to consider the DM-electron scattering (m f = m e ) for the RHN DM with mass m N ≲ O(1) GeV because of the kinetic threshold.In this region, the most stringent constraint (blue shaded region surrounded by dashed curve) is from SENSEI [35], DAMIC [36] and Darkside [37].It is much weaker than that of fixed-target experiments.
Combining all the constraints, one can clearly see that the parameter space for m A ′ < 2m N (∆ < 1), which prefers g B−L > 10 −3 , has already been excluded.The only remaining region is m N ⊂ (10 MeV, 5 GeV) for ∆ > 1 and g B−L ≲ 10 −4 except for a small region with m N < 20 MeV which is excluded by the FASER experiment.
Discussion and Conclusions -We have considered, in this paper, the B-L extension of the standard model, which requires three RHNs to cancel the gauge anomalies.The two of them should be used to produced lepton asymmetry that is converted to the baryon asymmetry in the president universe.Therefore, the third RHN is a natural candidate for the dark matter.In this paper we have explored this interesting possibility postulating a discrete symmetry Z 2 to stabilize the third RHN.
It has been pointed out [12,13] that we can have a successful low-energy scale leptogenesis if the two Majorana RHNs N 1,2 have almost degenerate masses M 1 ≃ M 2 .However, there are several dangerous processes which dilutes the produced lepton asymmetry 4 .In order to show the resonant leptogenesis works we adopt here the non-thermal leptogenesis [6].We assume the inflaton decay is sufficiently slow and the reheating temperature is T R ∼ 140 GeV without suppressing the sphaleron process [38,39].We see that all dangerous processes are smaller than the Hubble rate H ∼ T 2 R /M P L once V B−L ≳ 3 TeV due to the Boltzmann suppression, and never comes into equilibrium [6].Clearly, the most dangerous inverse decay process of l + H → N 1,2 is also suppressed.
Once the breaking scale is fixed, all the coupling strengths in our scenario are well-predicted and receive strong constraints, which is different from other DM models with intermediate vector boson that have arbitrary couplings [40][41][42].Motivated by this observation, we consider thermal production of the RHN DM N in this paper and find the allowed parameter space, m N ⊂ (10 MeV , 5 GeV) and g B−L ≃ 10 −4 , consistent with the present DM density and with all existing experiments 5 .We show that the higher mass region for m N ≳ 5 GeV is already excluded by the DM direct detection and the LHC experiments.The fixed-target experiments give us stringent constraints for the lighter mass region, m N < O(1) GeV.Therefore, we expect that the high-luminosity LHC experiments will provide us crucial tests of the present scenario for the RHN DM of the mass in the surviving range and/or the B − L gauge boson A ′ will be discovered in future high intensity electron beam frontiers such as LDMX-style missing momentum experiments [21][22][23] and Belle II if the mass is in fact m A ′ < O(1) GeV.
Note that the Ref. [43] shows our remaining parameter space has also been excluded by the constraint from the invisible decay B → K + A ′ .However, the definition of the B−L charge has an ambiguity due to the hypercharge gauge rotation.For Q B−L → Q B−L − 2Y , the charge for Q qα = 0.Then, the above decay channel is sufficiently suppressed even after the electroweak symmetry breaking, but the B −L charge of the right-handed neutrino N is unchanged.Therefore, the constraints in [43] are not applicable in our model with the new B − L charge and our conclusion in this paper remains the same.
the present model for all particles are shown in Tab.I.The q α , u R , d R and e R are the left-handed quark doublets, right-handed up-and down-type quarks, and the right-handed charged leptons, respectively.
where H(T ) = 1.66 √ g * T 2 /M P L is the hubble constant and M P L = 1.22 × 10 19 GeV the planck mass.Since m A ′ = g B−L × V B−L , this condition reads to g B−L ≥ 12πV B−L /M P L ≃ 10 −13 , which is easily satisfied.

FIG. 1 :
FIG.1:The gradient red (blue) region in the upper (lower) panel corresponds to the parameter space which gives the observed DM density for the mass difference 0 < ∆ < 1 (∆ < 0).The gray shaded region in the lower panel stands for the CMB constraint while other colored shaded regions are excluded by different experiments whose labels can be found in FIG.2.

FIG. 2 :
FIG.2:The parameter space (red padding region) which gives the observed DM density for the mass difference ∆ > 1.The colored (green, blue, black) shaded regions are excluded by different experiments.The blue lines (solid, dashed, dash-dotted) from above to below correspond to VB−L = 1, 5, 10 TeV.

TABLE I :
B − L charge for different species Species qα uR dR Lα eR Ni Φ H