Connecting Light Dirac Neutrinos to a Multi-component Dark Matter Scenario in Gauged $B-L$ Model

We propose a new gauged $B-L$ extension of the standard model where light neutrinos are of Dirac type, naturally acquiring sub-eV mass after electroweak symmetry breaking. This is realised by choosing a different $B-L$ charge for right handed neutrinos than the usual $-1$ so that the Dirac Yukawa coupling involves an additional neutrinophilic scalar doublet instead of the usual Higgs doublet. The model can be made anomaly free by considering four additional chiral fermions which give rise to two massive Dirac fermions by appropriate choice of singlet scalars. The choice of scalars not only helps in achieving the desired particle mass spectra via spontaneous symmetry breaking, but also leaves a remnant $Z_2 \times Z'_2$ symmetry to stabilise the two dark matter candidates. Apart from this interesting link between Dirac nature of light neutrinos and multi-component dark matter sector, we also find that the dark matter parameter space is constrained mostly by the cosmological upper limit on effective relativistic degrees of freedom $\Delta N_{\rm eff}$ which gets enhanced in this model due to the thermalisation of the light right handed neutrinos by virtue of their sizeable $B-L$ gauge interactions.


I. INTRODUCTION
In spite of convincing evidence for existence of light neutrino masses and their large mixing [1], the nature of light neutrinos is still unknown. While neutrino oscillation experiments (which have measured two mass squared differences and three mixing angles [2] Similarly, evidence from cosmology experiments like Planck suggests that a mysterious, non-luminous and nan-baryonic component of matter, known as dark matter (DM), gives rise to around 26% of the present universe's energy density. In terms of density parameter Ω DM and h = Hubble Parameter/(100 km s −1 Mpc −1 ), the present DM abundance is conventionally reported as [42]: Ω DM h 2 = 0.120 ± 0.001 at 68% CL. Apart from cosmological evidences, there are several astrophysical evidences too strongly supporting the presence of DM [43][44][45].
In spite of such convincing astrophysical and cosmological evidences, the particle nature of DM is not yet known. Since none of the SM particles can be a realistic DM candidate, several beyond standard model (BSM) proposals have been floated in the last few decades, the most popular of them being known as the weakly interacting massive particle (WIMP) paradigm. In this WIMP paradigm, a DM candidate typically with electroweak (EW) scale mass and interaction rate similar to EW interactions can give rise to the correct DM relic abundance, a remarkable coincidence often referred to as the WIMP Miracle [46]. However, the same electroweak type interactions could also give rise to DM-nucleon scattering at an observable rate which can, in principle, be observed at ongoing or future direct detection experiments like LUX [47], PandaX-II [48,49], XENON1T [50,51], LZ [52], XENONnT [53], DARWIN [54] and PandaX-30T [55]. However, there have been no observations of any DM signal yet in the experiments, putting stringent bounds on DM-nucleon scattering rates.
Motivated by growing interest in light Dirac neutrinos and multi-component DM scenario, here we propose a model where both of these can be accommodated naturally. Instead of choosing ad-hoc discrete symmetries to stabilise DM, here we consider gauged B − L symmetry where B and L correspond to baryon and lepton numbers respectively. While gauged B − L symmetric extension of the SM was proposed long ago [91][92][93][94][95], realising DM and light neutrino masses in the model require non-minimal field content or additional discrete symmetries. As far as we are aware of, there have been no proposals so far to accommodate light Dirac neutrinos in a gauged B − L model without any additional discrete or global symmetries. Here we not only show how this can be realised naturally, but also show that such a framework also predicts the existence of two fermion DM candidates in order to make the model anomaly free. Apart from constraining the model from experimental bounds related to neutrino mass, collider searches, DM relic and DM-nucleon scattering rates, we also apply other bounds like perturbativity of different dimensionless couplings, bounded from below criteria of the scalar potential. More importantly, due to the Dirac nature of light neutrinos having additional gauge interactions, additional light degrees of freedom can be thermalised in the early universe, which is severely constrained from big bang nucleosynthesis (BBN) and cosmic microwave background (CMB) data. We show that the corresponding CMB-BBN bounds on additional light degrees of freedom constrain the DM parameter space more strongly compared to other relevant bounds. This paper is organised as follows. In section II, we give a brief overview of gauged B − L models with different solutions to anomaly conditions including the one we choose to discuss in details in this work. In section III, we discuss our model in details followed by section IV where we mention different existing constraints on model parameters. In section V, we briefly discuss the relic abundance and direct detection of DM followed by discussion of our results in section VI. Finally we conclude in section VII.
II. GAUGED B − L SYMMETRY As pointed out above, the B − L gauge extension of the SM is a very natural and minimal possibility as the corresponding charges of all the SM fields under this new symmetry is well known. However, a U (1) B−L gauge symmetry with only the SM fermions is not anomaly free. This is because the triangle anomalies for both U (1) 3 B−L and the mixed U (1) B−L −(gravity) 2 diagrams are non-zero. These triangle anomalies for the SM fermion content turns out to be Interestingly, if three right handed neutrinos are added to the model, they contribute . These four chiral fermions constitute two Dirac fermion mass eigenstates, the lighter of which becomes the DM candidate having either thermal [103] or non-thermal origins [104]. The light neutrino mass in this model had its origin from a variant of type II seesaw mechanism and hence remained disconnected to the anomaly cancellation conditions. In a follow up work by the authors of [105], these fermions with fractional charges were also responsible for generating light neutrino masses at one loop level. One can have even more exotic right handed fermions with B − L charges −17/3, 6, −10/3 so that the triangle anomalies cancel [105].
In the recent work on U (1) B−L gauge symmetry with two component DM [84], the authors considered two right handed neutrinos with B − L charge -1 each so that the model still remains anomalous. The remaining anomalies were cancelled by four chiral fermions with fractional B − L charges leading to two Dirac fermion mass eigenstates both of which are stable and hence DM candidates. The two right handed neutrinos with B − L charge -1 take part in generating light neutrino masses via type I seesaw mechanism resulting in massless lightest neutrino. In another recent work [89], while implementing type III seesaw in a These can be cancelled after introducing four chiral fermions χ L , χ R , ψ L , ψ R having B − L charges 13/9, 22/9, 1/9, 19/9 respectively. This can be seen as appropriately, we can realise light Dirac neutrinos along with two component fermion DM naturally without incorporating any additional discrete and ad-hoc symmetries. The fermion and scalar content of the model are shown in table I and II respectively. The necessity of the individual scalar fields will be discussed later.
The Lagrangian of this model can be written as Here, L SM represents the Lagrangian involving charged leptons, left handed neutrinos, quarks, gluons and electroweak gauge bosons. Second term denotes the kinetic term of new Where D H µ , D η µ and D φ µ denote the covariant derivatives for the scalar doublets H, η and scalar singlets φ i respectively and can be written as where g BL is the new gauge coupling and n η and n φ i are the charges under U(1) B−L for η and φ i respectively. After both B − L and electroweak symmetries get broken by the VEVs of H and φ i s the doublet and all three singlets are given by From equation (7), it is clear that the neutral component of the scalar doublet H and the scalar singlets φ i acquire non-zero vacuum expectation value (VEV) whereas the neutral component of η does not. This can be assured by suitably choosing the sign of bare mass squared term of η field to be positive definite (µ 2 η > 0). However, one crucial point to note here is that the neutral component of η will get a very tiny induced VEV after electroweak symmetry breaking because of the presence of trilinear term H † ηφ 1 in the Lagrangian (5). This can be realised by minimising the scalar potential with respect to η. This leads to To simplify the calculation we have assumed all two VEVs of singlet scalars are equal, i.e. u 1 = u 2 = u and also assumed the induced VEV to be negligible. After putting equation (7) in equation (5) we have found out the 4×4 mixing matrix for the real scalar fields in the The physical scalars 1 √ 2 (h s 1 s 2 η R ) can be obtained by diagonalising this real symmetric mass matrix and that can be done by the orthogonal matrix O S and the physical states can be expressed as In a similar manner, the 3×3 pseudo scalar mass matrix can be written as The physical pseudo-scalars and the Goldstone boson 1 √ 2 (A 1 A 2 η I ) can be obtained by diagonalising the above mass matrix and that can be done by the orthogonal matrix O P and the states can be expressed as After the analysing of the scalar potential and diagonalising the mass matrices there will be four independent quartic couplings left (λ η , λ Hη , λ ηφ 1 , λ ηφ 2 ). All the other couplings in the potential can be expressed in terms of the VEVs, scalar masses and the mixing angles.
In principle there should be nine different mixing angles present in the scalar sector out of which six will come from the real sector and three will come from the pseudo scalar sector.
Later we have shown that the result in the DM sector is almost independent of these mixing angles and through our discussion to simplify the numerical analysis we have assumed all of them to be equal to 0.1. The parametrisation of the orthogonal matrices O S , O P are shown in appendix A and B.
Lets discuss the fermionic sector of our model. We have three generations of right handed neutrinos and four chiral fermions and the corresponding interactions can be written as where L ν R is the interactions related to the right handed neutrinos can be expressed as The first term in the equation (14) represents the kinetic part of ν R and the second term is the Yukawa interaction between SM lepton doublet L , ν R , and η which is responsible for generating neutrino mass. As discussed above, the neutral component of η will get a small induced VEV v ν through the trilinear interaction present in the potential. This will generate a tiny Dirac neutrino mass as As can be seen from equation (8), a tiny induced VEV v ν ≈ O(eV) can be generated by appropriate tuning of the trilinear coupling µ Hη as well as bare mass squared term µ 2 η . Since u ∼ 10 TeV, v ∼ 100 GeV, we can have v ν ∼ 0.1 eV by choosing µ Hη /µ 2 η ∼ 10 −16 which can be ensured by choosing very large µ 2 η . This also ensures that the components of η decouple from the low energy particle spectra as well as their relevant phenomenology. The hierarchy between µ Hη can be reduced to around 10 −11 if we tune the Dirac Yukawa couplings to be as small as electron Yukawa coupling. Similar way of generating sub-eV Dirac neutrino mass from induced VEV of neutrinophilic Higgs was proposed earlier by the authors of [14,15,41].
The term L DM is the interactions correspond to the chiral fermions can be written as It is clear that, in the basis ξ 1 = χ L + χ R and ξ 2 = ψ L + ψ R , the Yukawa interactions in equation (17) are exactly diagonal. As discussed earlier, because of this reason, ξ 1 and ξ 2 is completely stable and will play the role of dark matter in this model. In the basis of ξ 1 and ξ 2 , the above Lagrangian (17) can be written as where P L,R = 1 ± γ 5 2 , left and right chiral projection operators. From the above Lagrangian (17) it is clear that DM particles will get mass after the breaking of B − L symmetry spontaneously by the VEV's of the singlet scalars (φs). ξ 1 and ξ 2 can annihilate to the SM particles through the interaction with Z BL and the singlet scalars.

IV. CONSTRAINTS ON THE MODEL PARAMETERS
Before discussing our results, we first note down the existing constraints on the model parameters from both theory and experiments. We discuss them one by one in this section as follows.

A. Boundedness of Scalar Potential
The scalar potential of the model has to be bounded from below and that can be ensured by the following inequalities.

B. Perturbativity of Couplings
We have to also take care of the perturbative breakdown of the model and to to guarantee that all quartic, Yukawa and gauge couplings should obey the following conditions.

D. Cosmological Bound on Additional Light Degrees of Freedom
Another interesting way to constrain the model parameters is by calculating the additional relativistic degrees of freedom due to the presence of right handed neutrinos at sub-eV scale having sizeable gauge interactions. Through these gauge interactions, they will achieve the thermal equilibrium in the early universe and will contribute to the total relativistic degrees of freedom of the thermal plasma. However, the total effective degrees of freedom for neutrinos are already very much constrained from cosmological observations, more specifically from BBN. We have used this fact to constrain the parameter space of the model. Recent data from the CMB measurement by the Planck [42] suggests that the effective degrees of freedom for neutrinos as N eff = 2.99 +0.34 −0.33 (20) In this scenario the effective contribution from the right-handed neutrinos can be written as [117,118] where N ν R represents the number of relativistic right-handed neutrinos, g(T) corresponds to the relativistic degrees of freedom at temperature T, and T dec ν R , T dec ν L are the decoupling temperatures for ν R and ν L respectively. From equation (20) one can write where we have used N SM eff = 3.045 [119]. Now, to predict ∆N eff one needs to know the decoupling temperature of ν R which remains in thermal equilibrium until the interaction rate becomes smaller than the Hubble expansion of the universe.
Here the Hubble rate can be written as [118] where g ν R is the internal degrees of freedom for right-handed neutrinos. In this scenario, the interaction rate can be written as [118] Γ where f(ν R ) is the Fermi-Dirac distribution of right-handed neutrinos. As we have discussed earlier, ν R will achieve thermal equilibrium only through Z BL interactions and the crosssection can be written as where n f is is the charge of the SM fermions under U(1) B−L , N C f is the colour multiplicity of the fermions. Inserting the required input in equation (23) one can find out the decoupling temperature for right-handed neutrinos and using equations (21), (22) we can derive a bound on the unknown parameters of the model and in this case these are g BL and M Z BL . In fact, this is not a feature of this model but can be applicable to any gauge symmetric model with additional light degrees of freedom having sizeable gauge interactions. For example, in left-right symmetric models with light Dirac neutrinos or light right handed neutrinos one can derive similar bounds on additional gauge sector, as discussed by several earlier works including [21,24] and references therein.

V. DARK MATTER: RELIC DENSITY AND DIRECT DETECTION
Relic abundance of two component DM in our model χ 1,2 can be found by numerically solving the corresponding Boltzmann equations. Let n 2 = n ξ 2 + nξ 2 and n 1 = n ξ 1 + nξ 1 are the total number densities of two dark matter candidates respectively. Assuming there is no asymmetry in number densities of ξ i andξ i , the two coupled Boltzmann equations in terms of n 2 and n 1 are given below [89], where, n eq i is the equilibrium number density of dark matter species i and H denotes the Hubble parameter, defined earlier. For further details of these Boltzmann equations for two component Dirac fermion DM and their annihilation channels (ξ iξi → XX, X being all particles where DM can annihilate into) contributing to σv , please refer to [89] where a similar scenario was discussed recently. We have solved these two coupled Boltzmann equations using micrOMEGAs [120] where the model information has been supplied to micrOMEGAs using FeynRules [121]. All the relevant annihilation cross sections of dark matter number changing processes required to solve the coupled equations are calculated using CalcHEP [122]. The most important DM annihilation channels are the ones mediated by Z BL and the singlet scalars. Since the two DM candidates are stabilised by two separate and accidental Z 2 symmetries, there is no coannihilation between them. On the other hand a pair of one DM can annihilate into a pair of the other, if kinematically allowed, as shown by the last terms on the right hand side of above two equations.
Just like the new gauge boson and singlet scalars mediate DM annihilation into SM particles, similarly, they can also mediate spin independent DM-nucleon scatterings. The Feynman diagrams corresponding to such direct detection (DD) processes are shown in the figure 1. Different ongoing experiments like Xenon1T [50,51], LUX [47], PandaX-II [48,49]  To compare the result of our model with Xenon1T bound, we have multiplied the elastic scattering cross-section by the relative number density of each DM candidate and used the following conditions For details regarding direct detection of multi component DM, please refer to [123,124].

VI. RESULTS AND DISCUSSION
Since we have two stable DM candidates i.e. ξ 1 and ξ 2 in this model, the total relic abundance can be expressed as the sum of the individual candidates, Equation (17) clearly shows that ξ 1 and ξ 2 have interactions with Z BL and the new singlet scalars φ 1 and φ 2 . Through these interactions they will achieve the thermal equilibrium in the early universe (unless the gauge and Yukawa couplings are extremely small) and eventually freeze-out as the universe expands. In figure 2   experimental bounds. Apart from the experimental bounds, we also apply the bounded from below criteria of the scalar potential as well as perturbativity of all dimensionless couplings.
One interesting point to note here is that the BBN-CMB bound on ∆N eff is putting much stronger bound in the high mass region of M Z BL compared to the other bounds like collider or direct detection. In figure 5 we have shown the allowed parameter space in M ξ 1 − M ξ 2 plane where the variation of g BL ( right panel )and M Z BL ( left panel ) have also been shown through colour coding. One can see that the region becomes broader as we go to the high mass region of both DM candidates whereas in the low DM mass region it becomes narrower.
This can be explained by noting the fact that in the low mass region only gauge coupling ( g BL ) is playing the role whereas as the mass increases the Yukawa coupling corresponding to each DM candidate also increases and starts to contribute significantly. As a result the region becomes broader as we go to the high mass region due to reduced dependence on the gauge boson mediated annihilation channels.