The $B-L$ Scotogenic Models for Dirac Neutrino Masses

We construct the one-loop and two-loop scotogenic models for Dirac neutrino mass generation in the context of $U(1)_{B-L}$ extensions of standard model. It is indicated that the total number of intermediate fermion singlets is uniquely fixed by anomaly free condition and the new particles may have exotic $B-L$ charges so that the direct SM Yukawa mass term $\bar{\nu}_L\nu_R\overline{\phi^0}$ and the Majorana mass term $(m_N/2)\overline{\nu_R^C}\nu_R$ are naturally forbidden. After the spontaneous breaking of $U(1)_{B-L}$ symmetry, the discrete $Z_{2}$ or $Z_{3}$ symmetry appears as the residual symmetry and give rise to the stability of intermediated fields as DM candidate. Phenomenological aspects of lepton flavor violation, DM, leptogenesis and LHC signatures are discussed.


I. INTRODUCTION
The standard model(SM) needs extensions to incorporate two important missing pieces: the tiny neutrino masses and the cosmological dark matter (DM) candidates. The scotogenic model, proposed by Ma[1], has recently became an attractive and economical scenario to accommodate the above two issues in a unified framework. The main idea is based on the assumption that the DM  . In these models, the stability of DM is usually guaranteed by imposing the odd parity under ad hoc Z 2 or Z 3 symmetry. The origin of discrete symmetry is still unknown. An attractive scenario, known as Krauss-Wilczek mechanism [32], is that the discrete symmetry appears as the residual symmetry which originates from the spontaneous symmetry breaking(SSB) of a continuous gauge symmetry at high scale. The simplest and wellstudied gauge extension of SM is that of U (1) B−L , which was first realized within the framework of left-right symmetric models [33][34][35][36]. Following this spirit, several loop-induced Majorana neutrino mass models were constructed based on gauged U (1) B−L symmetry [37][38][39][40][41][42][43][44]. In these works, exotic B − L charges are assigned to new particles to satisfy the anomalies cancelation condition.
By taking appropriate charge assignment, the residual discrete Z 2 (Z 3 ) symmetry arises after the SSB of U (1) B−L symmetry. Then the lightest particles with odd Z 2 (Z 3 ) parity can not decay into SM ingredients, becoming a DM candidate.
On the other hand, the evidences establishing whether neutrinos are Majorana or Dirac fermion is still missing. If neutrinos are Dirac fermions, certain new physics beyond the SM should exist to account for the tiny neutrino mass. Several scotogenic models for Dirac neutrino masses were proposed in Refs. [45][46][47][48][49][50][51]. The generic one-loop topographies are discussed in Ref. [52] and subsequently, specific realizations with SU (2) L multiplets fields are presented in Ref. [53]. In these models, two ad hoc discrete symmetries were introduced, one is responsible for the absence of SM Yukawa couplingsν L ν R φ 0 and the other for the stability of intermediate fields as dark matter(DM).The symmetries could be discrete Z 2 [46,52,53], Z 3 [49,54], or Z 4 [55,56].
It is natural to ask if the B − L symmetry also shed light on Dirac neutrino mass generation and DM phenomena. Recently several efforts were made at tree level [54,[57][58][59], and a specific oneloop realization was also proposed based on left-right symmetry scheme [51]. In this brief article, we propose the U (1) B−L extensions of scotogenic Dirac neutrino mass models with intermediate Dirac fermion singlets. We will systematically discuss the one-and two-loop realizations for Dirac neutrino masses with typical topographies respectively. In these models, a singlet scalar σ is responsible for the SSB of gauged U (1) B−L symmetry as well as masses of the heavy intermediate Dirac fermions. To get the Dirac type neutrino mass term, we introduce three right-handed components ν R and assume that they share the same B − L charges. The intermediate Dirac fermions are SM singlets but carry B − L quantum numbers. This implies that the anomaly cancelations of Then the effective Dirac neutrino mass term m DνL ν R is induced by SSB of U (1) B−L . As we shall see, the discrete Z 2 or Z 3 symmetry could appear as a remnant symmetry of gauged U (1) B−L symmetry, naturally leading to DM candidates.
In Sec.II, we construct the one/two-loop diagrams for Dirac neutrino mass generation and discuss their validity under B − L anomaly free condition. We consider the phenomenology of the models in Sec.III. A summary is given in Sec. IV.

A. One-loop Scotogenic Model
Consider first the one-loop scotogenic realization of Dirac neutrino masses. In the B − L extended scotogenic models, the particle content under SU (2) L × U (1) Y × U (1) B−L symmetry is listed as follow where several Dirac fermion singlets are added with their chiral components denoted as F Ri and F Li (i = 1 · · · n) respectively. In the scalar sector, we further add one doublet scalar η and one singlet scalar χ.
In the original Z 2 model [45,46], Z 2 odd parity is assigned to ν R and intermediated particle fields running in the loop. As a warm up, we start from the simplest U (1) B−L extension. We denote it as A 1 model with the corresponding Feynman diagrams illustrated as the first diagram in Fig. 1.
The relevant interactions for radiative Dirac neutrino mass generation are given as where L is the SM lepton doublet and we omit the summation indices. In terms of gauged U which, using relevant interactions given in Eq.
(2) , can be solved exactly as Given the interations in Eq.
(2), the charge assignments for other particles are listed in the A 1 row in Table. I. Therefore the total number of heavy fermions is fixed by the anomaly free conditions and the B − L charge assignments for all new particles are determined in terms of free parameter Q ν R . Let us now discuss precisely what values Q ν R can be taken. First, the condition Q ν R = −1 should also be imposed to forbid the SM direct Yukawa coupling termν L ν R φ 0 . Second, forbidding Majorana mass terms (m R )ν C R ν R , σν C R ν R and σ * ν C R ν R requires Q ν R = 0, −1/3 and 1 respectively (note that Q σ = Q ν R + 1 for A 1 model). Third, to generate a purely loop-induced neutrino mass term, Q σ and Q χ (= Q ν R − 1) appropriately assigned so that σ k χ and (σ * ) k χ(k = 1, 2, 3) terms, which cause the VEV of χ, are forbidden. This further requires Q ν R = 0, −1/3, −1/2, −2 and −3. Similarly, the (Φ † η)σ k and (Φ † η)(σ * ) k (k = 1, 2) should also be avoid to generate the VEV of η, leading to Q ν R = 0, −1/3, −3. Once an appropriate Q ν R is taken, the residual Z 2 symmetry appears in Eq.(2), under which the parity is odd for inert particles (η, χ, F L/R ) and even for all other particles.
We now consider other possible realizations. In the scalar sector, the interactions relevant to radiative neutrino mass generation are given by Taking appropriate charge assignment, at least one η − χ mixing term given in Eq. (5) should be selected to build the model. All the seven possible topological diagrams(denoted as A 1 − A 7 ) are depicted in Fig. 1, where we have already discussed the specific model A 1 above.
Figure 1. Possible one-loop topological diagrams that can generate the prototype model given in Ref. [46] after the SSB of U (1) B−L symmetry Under the gauged U (1) B−L symmetry, the quantum numbers of new particles are required to satisfy the anomaly free conditions. We summarize the B − L quantum number assignments for each diagram in Table I. We have checked that among the seven models, five of them (A 1 ,A 2 , A 4 , A 5 and A 6 ) are suitable for the gauged B − L extension. For each available model, the total number of intermediate fermions F R/L is uniquely determined by the anomaly free condition of The B − L quantum number of A 1 and A 2 model can not be uniquely fixed and we choose Q ν R as the variable. If χ linear terms are forbidden by appropriate Q ν R assignment, the residual Z 2 symmetry arises after the SSB of U (1) B−L . Thus the lightest particle with odd Z 2 parity can serve as a DM candidate.
Compared with A 1 and A 2 , for models A 4 , A 5 and A 6 , the B − L quantum numbers for new particles are fixed uniquely. This is due to the fact that the interaction χ 2 σ (χ 2 σ * ) contributes an  Table I. B − L charge assignments for new particles in each one-loop models. In A2 model, we set z ≡ (5x 2 − 6x + 5) 1/2 . The symbol "×" means that no appropriate charge assignment are available to meet the requirement of anomaly cancellation additional constraint on Q χ and Q σ , i.e., The existence of χ 2 σ(χ 2 σ * ) term has two-fold meanings: (i) that it automatically forbids the χ One recalls that in the prototype scotogenic Dirac model [46] with sizable Yukawa couplings, a relatively small coupling constants of η − χ mixing terms is required to reproduce the scale of neutrino masses. To rationalize such a unnaturally small coupling, an extra soften broken symmetry is added [46]. We emphasize that the fine tuning can be relaxed in A 4 − A 6 models with the help of double suppression from η − χ and χ R − χ I mixing interactions. Takeing A 5 model as an example, with scalar interactions λ(Φ † η)χ * σ and µ χ χ 2 σ * , the radiative neutrino mass is evaluated as where Λ ∼ m η , m R χ , m I χ denotes the scale of new physics, usually taken to be Λ ∼ σ ∼ O(1)TeV. Then for λ ∼ y 1 ∼ y 2 ∼ f ∼ 10 −2 and µ χ ∼ O(10)GeV, the neutrino mass scale (0.1 eV) can be reproduced.

B. Two-loop Scotogenic Models
Now let us discuss the two-loop scotogenic realizations of Dirac neutrino masses. The simple model with Z 3 discrete symmetry was proposed recently [49] where two classes of Dirac fermion singlets are added. Here we denote the corresponding chiral components as F R,Li (i = 1, 2 · · · n) and S R,Lj (j = 1, 2 · · · m) respectively. In the scalar sector, we add one scalar doublet η, two scalar singlets χ and ξ. In order to accomplish the U (1) B−L extension, a scalar singlet σ is also added to play the role as B − L symmetry breaking. The particle content and quantum number assignments Similar as the one-loop cases, the two-loop model can be realized though various pathways. As an illustration, we start from a simple U (1) B−L extension (denoted as B 1 ) with topology depicted by the first diagram in Fig. 2. The relevant interactions are Under gauged U (1) B−L symmetry, the condition of cancelation for [U (1) B−L ] × [Gravity] 2 anomaly is given by Notice that Q ν R = −1 is required to forbid ν L ν R φ 0 term. From Eq.(10), one obtains Clearly, only (n, m) = (1, 2) and (2, 1) patterns are allowed for model B 1 . In this secnario, the rank of effective neutrino mass matrix is two, implying a vanishing neutrino mass eigenvalue. Hence the models with condition n + m = 3 are the minimal two-loop realizations allowed phenomenologically. The anomaly free condition of [U (1) B−L ] 3 is given by Taking the interaction terms in Eq.(9) into account and solving Eq.(11), (12), we find Subsequently, the B − L charges of other particles are obtained, which are shown explicitly in Table II.  Table II. Obviously, after B − L breaking, the residual Z 3 symmetry arises with (Φ † η)χ * σ,χ 3 σ * ,χξσ (Φ † η)χ * ,χ 3 σ,χξσ 2 Table II. B − L quantum number assignments and relevant scalar interactions for two-loop models with n + m = 3.

III. PHENOMENOLOGY: A CASE STUDY
In the following, we consider some phenomenological aspects of the gauged B − L scotogenic Dirac models. From Table. I, we can see that besides the B − L charge and some scalar interactions being different, all the one-loop models have same interactions as in Eq. 2. Therefore, we can concentrate on the simplest one, i.e., model A 1 . As for the two-loop models, phenomenon will be similar provided the additional ξ and S L,R are heavy enough.
In model A 1 , the B − L charges of all the additional particles are determined by B − L charge of right-handed neutrino Q ν R . To make sure a residual Z 2 symmetry after the breaking of B − L, we fix Q ν R = 1/6 in the following discussion. The complete gauge invariant scalar potential for For the Z 2 even scalars, φ 0 R and σ R mix into physical scalars h and H with mixing angle α. Here, we regard h as the discovered 125 GeV scalar at LHC [60][61][62]. In order to escape various direct and indirect searches for the scalar H [63], a small mixing angle sin α = 0.01 is assumed in this work.
Secondly, we briefly discuss the phenomenology of dark matter (DM). In this paper, we mainly consider scalar DM candidate, since for the fermion singlet, M F = f σ is naturally around TeVscale and it is more interesting to realize successful leptogenesis. We emphasis that the (Φ † η) 2 term is not allowed in U (1) B−L extensions to generate a mass splitting between η 0 R and η 0 I , rendering the η dominated component H 0 2 unsuitable as a DM candidate to escape the direct detection bound. Therefore, we concentrate on the χ dominated component H 0 1 as the DM candidate. With heavy F and relatively small Yukawa couplings, i.e., |y 2 | 0.01, the contribution of F to H 0 1 annihilation is negligible. To generate the correct relic density, the possible annihilation channels are: 1) SM Higgs h portal; 2) scalar singlet H portal; 3) gauge boson Z portal. For case 1), the extensive researches imply that M H 0 1 M h /2 is the only allowed region under tight constraints from relic density and direct detection [69,70]. For case 2), M H 0 1 ∼ M H /2 is needed, and electroweak scale H 0 1 DM is allowed [71]. Notably, when M H ∼ 100 GeV thus M H 0 1 ∼ 50 GeV, the observed excess in gamma-ray flux by Fermi-LAT can be interpreted [72,73]. For case 3), it requires M H 0 1 ∼ M Z /2, and M H 0 1 is usually around TeV-scale [74]. In Fig 4, we show the relic density Ωh 2 as a function of M H 0 1 . The Higgs h/H portal could easily acquire the correct relic density, while the Z portal could not due to too small g BL . Note that the process H 0 1 H 0 * 1 → HH could also realise correct relic density provided M H 0 1 ∼ M H . Thirdly, we consider Dirac leptogenesis. It is well known that the leptogenesis can be accomplished in Dirac neutrino models [75,76]. In model A 1 , the heavy Fermion singlet F can decay into Lη and ν R χ to generate lepton asymmetry in the left-handed L and right-handed sector R . Due to the fact that the sphaleron processes do not have direct effect on right-handed fields, the lepton asymmetry in the left-handed sector can be converted into a net baryon asymmetry via sphaleron processes, as long as the one-loop induced effective Dirac Yukawa couplings are small enough to prevent the lepton asymmetry from equilibration before the electroweak phase transition [77]. Under the assumption y 1 = y 2 , the final lepton asymmetry is calculated as [45] Define the parameter K = Γ F 1 /H(T = M F 1 ), where Γ F 1 is the tree-level decay width of F 1 and H(T ) = 8π 3 g * /90 T 2 /M Pl wit g * 114 and M Pl = 1.2 × 10 19 GeV. As in our case K 1, the final baryon asymmetry is estimated as [77] In Fig. 5, we depict Y B as a function of M F 1 . It is clear that the BP in Eq. 17 could predict the correct value of Y B , as well as satisfy the out of equilibration condition Then we turn to the collider phenomenology. The DM candidate H 0 1 will contribute to invisible Higgs decay. The corresponding decay width for h → H 0 1 H 0 * 1 is calculated as where g hH 0 1 H 0 * 1 = λ Φχ v cos α + λ χσ v σ sin α is the effective trilinear hH 0 1 H 0 * 1 coupling and v = 246 GeV, v σ = M Z /(g BL Q σ ). So the invisible branching ratio is BR inv = Γ inv /(Γ inv + Γ SM ) with Γ SM = 4.07 MeV at M h = 125 GeV [78]. Our BP in Eq. 17 with λ Φχ = λ χσ = 0.001 predicts BR inv ∼ 0.01, which can escape the most stringent bound comes from fitting to visible Higgs decays, i.e., BR inv < 0.23 [79]. As for the light scalar H, the dominant visible decay is H → bb and invisible decay is H → H 0 1 H 0 * 1 . The possible promising signatures are e + e − → ZH at future lepton colliders [80]. Meanwhile, due to the doublet nature of H ± 2 and H 0 2 , they can be pair produced at LHC via Drell-Yan processes as pp In the case of light H 0 1 DM, the most promising signature is The LEP II data requires that [83] M Z g BL = Q σ v σ 6 ∼ 7 TeV.
And the direct searches for Z with SM-like gauge coupling in the dilepton final states have excluded M Z 4 TeV [84]. Recasting of these searches in gauged U (1) B−L has been performed in Ref. [74,85], where the exclusion region in the M Z − g BL is obtained. In this paper, we consider M Z = 4 TeV and g BL = 0.1 to respect these bounds. In the limit that masses of SM fermions f (f ≡ q, l, ν L,R ) are small compared with the Z mass, the decay width of Z into fermion pair f f is given by where C l,ν = 1, C q = 3. Then the branch ratios of Z decay into each final states take the ratios as where l = e, µ. Thus, the B − L nature of Z can be confirmed when BR(Z → bb)/BR(Z → µ + µ − ) = 1/3 is measured [39]. In addition, the decay width of Z into scalar pair SS * is given by new particles in all the possible models present in Table I and Table II,  By considering phenomenology on lepton flavor violation, dark matter, leptogenesis and LHC signatures, we consider the benchmark point in Eq. 17. In addition to generate tiny neutrino mass via scalar DM mediator, this BP can also interpret the gamma-ray excess from the galactic center, and realize successful leptogenesis. As for collider signatures, the scalar DM H 0 1 will contribute to invisible Higgs decay as h → H 0 1 H 0 * 1 . The scalar singlet H might be testable via e + e − → ZH with H → bb/H 0 1 H 0 * 1 at lepton colliders. Meanwhile, the promising signature at LHC is the trilepton signature as pp → H ± 2 H 0( * ) 2 → W ± Z + H 0 1 H 0 * 1 with leptonic decays of W/Z. The new B − L gauge boson is expected discovered via the dilepton signature pp → Z → l + l − at LHC [86].
And in principle, the constructed models in Table I and Table II can