Radiative Dirac Neutrino Mass with Dark Matter and it's implication to $0\nu 4\beta$ in the $U(1)_{B-L}$ extension of the Standard Model

The Standard Model gauge symmetry is extended by $U(1)_{B-L}$ which when spontaneously broken leads to residual $\mathbb{Z}_4$ symmetry. $U(1)_{B-L}$ gauge symmetry made anomaly free by introducing exotic SM singlets with corresponding $U(1)_{B-L}$ charges of $13$, $-14$, and $15$. $\mathbb{Z}_4$ symmetry ensures the Dirac nature of neutrinos, simultaneously stabilizing dark matter. Dirac neutrino mass is generated through scotogenic scenario. Dark matter, direct detection, cosmological constraints, and collider constraints analysis is performed. $\mathbb{Z}_4$ symmetry predicts the exact absence of neutrinoless double beta decay ($0\nu 2\beta$) and gives a prediction for an enhanced neutrinoless quadruple beta decay ($0\nu 4\beta$) via which this model can be tested. Model allows for Majorana dark matter as well as for long-lived dark matter candidates.

The Standard Model (SM) of strong and electroweak interactions has proven to be very successful so far with the last remaining piece experimentally discovered on July, 4'th 2012 [1,2].Nevertheless there are experimental observations that require new physics beyond Standard Model (BSM). One of these problems is the experimental observation of neutrino oscillations [3][4][5][6][7][8][9][10][11][12] back in 1990's. Theoretical explanation of neutrino masses requires addition of new particles BSM. The most minimalistic and simplest realizations of this are the seesaw mechanism of type I [13][14][15][16] which adds a fermion singlet to SM. Next would be seesaw of type II [17][18][19][20] which extends the SM by a scalar triplet. Last of this kind of realizations is seesaw of type III in which SM is extended by a fermionic electroweak triplet. All these tree level realizations of naturally small neutrino masses require either a small couplings or heavy new physics in order to explain the smallness of neutrino masses and they are lead to unique dimension-five effective operator known as Weinberg operator [21]. In order to avoid the requirement of heavy new physics or small couplings, for instance neutrino masses can be generated radiatively at one-loop order.
Examples of this realizations include [22] the Zee model from 1980, the canonical scotogenic model [23] (scotos from Greek meaning darkness) from 2006, and radiative inverse seesaw model [24]. Since neutrinos are neutral and colorless they can be of Dirac or Majorana type. Currently there is no experimental evidence toward any direction. But if neutrinos are Dirac in nature there must be a symmetry (conserved quantity) responsible for the absence of Majorana mass of neutrinos. This issue was systematically studied in [25]. Dirac neutrinos which obtain their masses radiatively via scotogenic scenario. Model naturally predicts neutrinoless quadruple beta decay whereas neutrinoless double beta decay is exactly absent. Furthermore, model allows for stable dark matter, fermionic or bosonic, and leptogenesis. Similar works done on U (1) B−L extension of SM are [27][28][29][30][31][32][33][34][35][36].
The paper is organized as follows: in Sec. II the model is introduced and the cancellation of chiral anomalies is explained; Sec. III demonstrates how radiative Dirac neutrino masses are generated; Secs. IV and V give the fermion and scalar mass spectrum, respectively; Sec. VI discusses dark matter candidates; in Sec. VII we go over the neutrinoless quadruple beta decay prediction; Sec. VIII presents the results and discusses relevant constraints for our model; and Sec. IX concludes.

II. MODEL
SM gauge symmetry is extended to The model is constructed as follows: ν R is introduced as the Dirac partner for Left-handed neutrinos, N L,R fermions, η, and χ scalars are introduced to complete the loop for radiative neutrino mass generation, i.e. scotogenic scenario, Ψ i are introduced for anomaly cancellation, and lastly S 4 is needed for spontaneous symmetry breaking (SSB) of U (1) B−L to residual Z 4 discrete symmetry in the leptonic sector. Here, the residual Z 4 symmetry is    N L,R and Ψ 2L in this case and fermions that transform as w and w * under Z 4 residual symmetry arrange themselves into Dirac pairs, ν L,R , Ψ 1,3L are among them. When electroweak symmetry is broken by H the η 0 and χ scalars mix through H, which is needed for the neutrino mass generation. Furthermore, when U (1) B−L symmetry is broken by S 4 vacuum expectation value (VEV), the mass eigenstates of (η 0 , χ )(call them ξ 1,2 ) obtain an effective operator ξ 4 i +H.c., which is invariant under Z 4 and generates the neutrinoless quadruple beta decay. Neutrinoless double beta decay is forbidden by Z 4 symmetry, therefore neutrinoless quadruple beta decay will be dominant. More on this in Secs. VII and V.

Chiral anomalies
Model is chiral anomaly free and cancellation of anomalies per family is shown in Tab. III.
U (1) Y gravitational anomaly is cancelled like in SM and U (1) B−L gravitational anomaly is cancelled as follows:

Abelian kinetic mixing
Model Lagrangian must be augmented with renormalizable Abelian kinetic mixing(KM) 0 counter-term, since the one-loop corrections has singular contribution, same as in [38] where B and B are strength tensors for hypercharge(U (1) Y ) and B − L(U (1) B−L ) gauge groups. respectively. ε 0 represents the bare Abelian kinetic mixing counter-term which must be included to renormalize the divergent one-loop corrections [39,40] Fermions that contribute Abelian KM are Q, u c , d c , L, e c and scalar that contributes is η.
Their contributions to the divergence are given as S : Total 6 In order to regularize the divergence ε 0 must be given by where the tree level finite piece is denoted by ε finite 0 and the one-loop corrected finite contribution to Abelian KM is given by

III. NEUTRINO MASSES
Neutrino tree level mass is forbidden by U (1) B−L symmetry. This is the S symmetry from Ref. [25] and the neutrino mass is generated via first scenario of one-loop radiative case from Ref. [25]. Neutrino masses are obtained via a diagram shown in Fig It is actually the Z 4 plus the Lorentz symmetry that stabilizes the dark matter. Neutrino radiative mass is given as where F (x) is defined as and mixing angles of (η, χ) and (N L , N c R ) are given in Eqs. 29 and 17, respectively.

IV. FERMION SECTOR
The SM fermions generate their masses in a usual way. Since neutrinos transform as w * under residual Z 4 symmetry, their masses are of Dirac type and were given in Sec. III. N L,R transform as w 2 under Z 4 therefore they obtain Majorana masses through the seesaw-I texture matrix form In general Yukawas here can be complex but the Majorana phases of N L,R can be used to remove this phases, so they are not physical. If the (1, 1) component of the N c L ,N R mass matrix was nonzero this would not be the case. See App. A for more details on this. Eigenvalues and eigenvectors are given by In order for Ψ 1,2,3 to get their masses, for instance, a SM singlet scalar with U (1) B−L ∼ 28 (say S 28 ) can be introduced. When S 28 obtains a non-zero VEV (S 28 ∼ v 28 which is allowed by Z 4 symmetry, Ψ i masses would be generated through Lagrangian terms Ψ a 1L Y ab But in our case, we generate Ψ 1,2,3 masses through effective dimension-ten La-grangian terms given in Eq. 2. Λ scale is associated with mass scale involved in generation of m Ψ i .
Remark regarding the mixing of N L,R and ν L,R with Ψ 2i and Ψ 1,3 , respectively. Since there is no symmetry distinguishing N R,L from Ψ 2 and similarly for ν L,R and Ψ 1,3 , these will mix via dimension-6, 7, and 8 effective operators given bȳ On the other hand if S 28 is included, as was explained above, then Ψ 1,2,3 do generate their masses.
But since scalars with charge ∼ 20 and ∼ 12 under U (1) B−L are not included, there will be no mixing between Ψ sector and (N, ν) sector. Which means there is an inherent Z 2 symmetry induced under which Ψ sector is odd and all other particles are even(trivial). If this is the case, then lightest of Ψ eigenstates and Z 4 LSP can be DM candidates which would give a multi-component DM scenario.

V. SCALAR SECTOR
Most general scalar potential is given as Potential minimization conditions are Due to Z 4 residual symmetry mass eigenstates can be divided into three groups: trivially transforming under Z 4 (singlet representation), transforming as w or w * under Z 4 (complex irreducible representation), and transforming as w 2 under Z 4 (real irreducible representation). Scalars transforming trivially under Z 4 are those that obtain none-zero VEVs S 0 4 , H 0 and H 0 's charged multiplet partner H ± . Their mass matrices are given as and mixing angle is given by with residual discrete symmetry. The mass of η ± is given as transforming as w or w * under Z 4 also mix, their corresponding mass matrix is given by and mixing angles are given by Lastly, scalars transforming as w 2 under Z 4 consist only of S. Corresponding scalar and pseusoscalar mass eigenvalues are Here the mass splitting is due to µ S 4 term in Eq. 20 which is allowed by Z 4 residual symmetry,

VI. DARK MATTER
Dark matter is stabilized by same Z 4 symmetry that ensures the Dirac nature of the neutrinos. Now in order to calculate the relic abundance of the particle dark matter which was in thermal equilibrium, we would need to calculate the Boltzmann equation where Y DM = n DM /s, n DM is the number density of the dark matter and s is the entropy density.
H is the Hubble expansion, z = M DM /T where T is the background temperature and σv is the thermally averaged cross-section of the dark matter annihilation process given as We can write the partial wave expansion σv = a + bv 2 . Now, the solution of the above Boltzmann equation in terms of this expansion can be given as where M pl = 2.4 × 10 18 GeV and g * is the number of relativistic degrees of freedom at the time of freeze-out. The freeze-out tempertaure can be calculated by the following expression which in turn derived from the equality condition of rate of expansion of the Universe H ≈ g 1/2 * T 2 /M pl . Now, since in our case we have additional particles with mass differences close the dark matter, then they can be thermally accessible during the freeze-out. This will eventually give rise to many additional channels through which the dark matter can co-annihilate and give Standard Model (SM) particles in the final states. The effective cross-section in this case would be as follows where And the thermally averaged cross-section is given as One remarkable thing here is that the symmetry that stabilizes DM is the same symmetry that makes neutrinos of Dirac type. The consequence of this is that neutrinos transform non-trivially under DM symmetry, Z 4 in this case. Therefore, any field that transforms as w 2 under Z 4 and is in tensor irrep of Poincare symmetry will always decay to pair of neutrinos. On the other hand, fields that transform as w 2 and are in spinor irrep of Poincare symmetry will not be able to decay to only neutrinos, therefore the lightest can be DM candidate.
Ψ i will not be considered for DM candidate since, as can be seen from eq. 2, they do not participate directly in neutrino mass and 0ν4β generations and will not lead to interesting phenomenology.
Main candidates to consider are ξ, N , S. ξ has a mixing with the neutral component of the η doublet, therefore it will have a direct detection channel mediated by Z SM gauge boson and is severely constrained [41][42][43].
Y M is the best DM candidate, since this N is naturally LSP as required by the smallness of neutrino mass and enhancement of 0ν4β. The only neutral Z 4 non-trivial particle that is lighter than N is neutrino, but N decay to neutrinos is forbidden by Z 4 and Poincare symmetry. Decay to the other Z 4 non-trivial particles is forbidden by U (1) em × Z 4 . The annihilation channels for N as a DM candiate are shown in fig.17. And since the dominant channel will be near resonance i.e (m Z = 2m N ), we have imposed the resonance condition while doing the analysis. The allowed parameter region to satisfy the relic is shown in fig. 2.
From the plot in Fig. 2 we infer that in order for N to be a plausible dark matter candidate the  0.02, Fig. 3. For the analysis we have implemented the model into SARAH 4 [44] and then we took the output to SPheno 3.1 [45] to calculate the mass spectrum. Finally for the dark matter analysis we used MicrOmega 4.3 [46], using the mass spectrum from SPheno 3.1.
Now we focus on S being DM candidate. For S to be a viable DM candidate we assume the following particle mass hierarchy: m Ψ i , m N j , m ξ k , m s4 , m η + > m s > m h > m w , m z > m e,u,d > m ν . Since S is a neutral scalar boson that transforms as w 2 under residual Z 4 symmetry, Z 4 , U (1) em , and Poincare symmetries allow S to decay only to ν's. Assuming all BSM Z 4 non-singlets are heavier than S, the decay of S to neutrino pair is radiative and shown in Fig. 4. The amplitude of the diagram in Fig. 4 is given by where µ x ij is given in eqs. 45 and 46, x, y are spinors in 2 component notation, s(c) N is the mixing angle of N states given in eq. 17, s(c) ξ is the mixing of (η 0 , χ * ) states given in eq. 29, and C 0 is given in eq. B1. Then the decay width is given by We assume the S R,I mass scale is above EW scale (v = 246GeV) but below U (1) B−L spontaneous breaking (v < m s < v 4 ), so at the moment of freeze-out of S R,I EW symmetry is conserved whereas U (1) B−L symmetry is broken to Z 4 . Then annihilation of S to SM particles will proceed through the Feynman diagrams shown in Fig. 5. The inelastic scatterring of (S R , (n, p) → S I , (n, p)) assuming   is much greater, τ > 10 25 , from the cosmic microwave background (CMB) constraint.
Important remark regarding Fig. 6 and S R,I being a viable DM candidate is that to make S R,I long-lived, τ s > 10 25 s, µ x must be tiny (≈ 10 −5 eV). As will be explained in sec. VII, in order to have enhanced 0ν4β µ x ∼ µ S ≈ 10 6 TeV is required. So, for S R,I to be a viable DM candidate means strongly suppressed 0ν4β. There are two ways to make µ x tiny: either √ 2µ s + λ 2 v 4 < 2 × 10 −15 TeV (strong fine-tuning), which will allow for observable 0ν4β via the other S component (S I ) or µ S , λ 2 v 4 < 10 −15 TeV in which can 0ν4β will be strongly suppressed.
We assume that the mass splitting ∆m 2 S between S R and S I is small, therefore both S R and S I freeze-out simultaneously (with S I decaying to S R for m S R < m S I ). Diagrams shown in Fig. 5 contribute to σ(S R,I S R,I → XX) S annihilation cross-section in order to get the correct relic abundance for S R,I , Ω S h 2 = 0.120 [49]. The contact diagram annihilation to Higgs pair is dominant since the Z s−channel diagram is suppressed due to large m Z > 4.2TeV(Sec. VIII). Even at the resonance, m Z ≈2m S , the Z s−channel diagram is sub-dominant due to g B−L < 0.127(Sec. VIII). Therefore, S R relic abundance and effective annihilation cross-section for S R,I as a function of DM mass(m S R ) and coupling λ HS with other parameters fixed is plotted in Fig. 7 As can be seen S is a viable long-lived DM candidate that also allows for correct neutrino masses to be satisfied but will simultaneously lead to highly suppressed 0ν4β signal.
In this case 0ν2β decay is forbidden by Dirac nature of neutrino masses, whereas 0ν4β signal is highly suppressed. As was shown above, the situation with N is quite different!

VII. NEUTRINOLESS QUADRUPLE BETA DECAY
In our construction of the model, by design, due to Z 4 residual symmetry neutrinoless double beta decay (0ν2β) is exactly absent. Therefore the dominant multipole will be neutrinoless quadruple beta decay (0ν4β). Contribution to neutrinoless quadruple beta decay is shown in Fig. 9. There will be also a diagram mediated by ν R right-chiral neutrinos with N R replaced by N L in Fig. 9. But due to suppression with neutrino mass at every leg and seesaw suppressed N L = (cosθN 1 − sinθN 2 ) Majorana mass (Eq. 15), contribution mediated by ν R can be safely ignored.
Reference [50] is the first paper to study experimental side of 0ν4β with B − L breaking to Z 2n where n = 2 naturally leading to neutrinoless quadruple beta decay. Neutrinoless quadruple beta decay has been searched for and experimentally studied by NEMO−3 collaboration in Refs. [51,52]. Another study was performed using 150 N d [53] nuclei at Kimballton Underground Research Facility setting upper limit for half life-time for 0ν4β.
The diagram in Fig. 10 effectively gives the Z 4 invariant vertex The relation between interaction eigenstates (η 0 , χ) and mass eigenstates ξ i is given in Eq. 29 and µ R,I ij in the basis (ξ 1 , ξ 2 ) are given as where s and c stand for sinθ ξ and cosθ ξ , respectively, and θ ξ was defined in Eq. 29. 0ν4β can be calculated as two one-loop diagrams. Neutrinoless quadruple beta decay is given by where Q abcd represents quadruple strength, Λ is the new physics scale relevant for the neutrinoless quadruple beta decay. Q abcd /Λ 2 explicitly is given by where the sum over repeated indices is assumed and the Majorana N mass represents the Λ scale in Eq. 47. a, b, c, d, α, β, γ, δ are flavor indices and take values 1 − 3. v S and M S are given as θ ξ is mixing angle between η and χ scalars and was given in Eq. 29. F (x, y, z) is the loop function and is given by In Eq. 48, x ij and y ij are given by where m i is the mass eigenstate of ξ i given in Eq. 28 and m N i is the Majorana mass eigenstate of N i given in Eq. 15. s N and c N stand for the sine and cosine of the mixing angle of the N L,R fermions and are given in Eq. 17. Lastly, c 4 is the combinatorics factor and is given as Important remark regarding eq. 48 is the presence of the λ 2 v 4 cross term in the last line. If λ 2 was absent (forbidden) in the model then 0ν4β would be proportional to the splitting of S scalar and pseudo-scalar masses, which is controlled by µ S 4 term in eq. 20. Neutrino mass suppression factors like Y L , θ ξ , ∆m ξ , v 4 also suppress 0ν4β but µ S freedom can be used to control the enhancement of 0ν4β. In the case if µ S v 4 ∼ O(10 2−3 TeV) the µ 2 S term will dominate and 0ν4β will scale as Below numerical calculation of Q 0ν4β Λ 2 is performed using pySecDec [54] software tool. Diagrams that have dominant contribution to are the ones with ν L legs and are shown in Fig. 11.
There are also diagrams with ν L replaced by ν R but they are suppressed by a factor of mν pν for each ν L → ν R leg replacement. Diagrams in Fig. 11 produce loop integrals FIG. 11: 2 two-loop diagrams contributing to neutrino quadruple beta decay.
with i p i = 0 and Y L given below.
After using pySecDec python code to calculate these integrals numerically, we compare numerical results with analytically obtained results in eq. 48 and plot both in Figs. 12 and 13.
Q 0ν4β Λ 2 dependence on m ξ , v 4 , and µ S for the analytical result from eq. 48 is plotted in fig. 12 with the other parameters fixed. As can be seen from eq. 48, for λ 2 v 4 µ S Q 0ν4β Λ 2 ∝ µ 2 S and the µ S can be used to enhance the Q 0ν4β Λ 2 for possible detection in the upcoming 0ν4β experiments. Current half-life lower limit on Q 0ν4β Λ 2 is given as τ 0ν4β 1/2 > 3.2 × 10 21 years [51]. The relation between half-life and Q 0ν4β Λ 2 is given as where G 0ν4β is the four particle phase space factor and A 0ν4β is the matrix element for 0ν4β process.
where the last factor was inserted for dimensional matching. Using this estimate and half-life lower Fig. 13 shows the comparison of numerical results from pySecDec with approximate analytical expression from eq. 48. As can be seen from the plot, the coupled loop (Fig. 11b) is relevant at v 4 < 10 6 TeV scales, where it of the order of the decoupled loop (Fig. 11a) and can interfere destructively (10 3 TeV < v 4 < 10 6 TeV). For v 4 > 10 6 TeV coupled loop becomes irrelevant. Important thing to notice is that µ S plays crucial role at enhancing 0ν4β at v 4 < 10 6 TeV scales (for λ 2 = 1).
Detailed study of phenomenology of the U (1) B−L model is done in [36]. Now, in our model

IX. CONCLUSION
The U (1) B−L extension of the SM was presented which is then spontaneously broken to residual Z 4 symmetry. The Z 4 symmetry is both responsible for the Dirac nature of neutrinos as well as for the stability of DM, a unique feature for this type of construction. Neutrino masses are generated radiatively through scotogenic scenario. Since the neutrinos are of Dirac type the neutrinoless double beta decay is exactly absent, but the Z 4 symmetry allows for non-zero neutrinoless quadruple beta decay, which is despite being an experimentally tiny effect is the dominant of the neutrinoless multipole beta decays. If future experiments on 0ν2nβ see no positive results in 0ν2β but do observe non-zero 0ν4β, this will be a strong indication toward neutrinos of Dirac type while still violating lepton number by 4 units and will hint toward this type of model. Z 4 allows for several WIMP like DM candidates in our model: best DM candidate is Majorana N which allows for small neutrino masses of O(0.1eV) scale, enhanced 0ν4β decay, U (1) B−L breaking scale as low as O(10TeV), and DM masses of O(1TeV); other possible DM candidate is S real scalar field which has a radiative decay to neutrinos and is suitable long-lived DM candidate, making S long-lived also suppresses 0ν4β decay, so it predicts no observable 0ν4β in current or future 0ν2nβ experiments without finetuning. In many models like this one, 0ν4β might be predicted to be non-zero but even in that case it is expected to be well below the sensitivity of current and future experiments looking for 0ν2nβ decays. Model presented here allows for arbitrary enhanced 0ν4β decay which can be made as large as 10 16−19 . The prize we pay for this is the introduction of S field which gives us a freedom of the enhancement of 0ν4β without effecting neutrino mass generation and DM related processes (for the N DM case). We have also shown that the model can satisfy all required collider constraints. More where tan (2θ) = −2 |Y N D | |Y N M | , (A9) As can be seen, Majorana phases of N L,R fermion fields can be used to remove phases from the mass matrix.