Some Phenomenologies of a Simple Scotogenic Inverse Seesaw Model

In this paper, we discuss and calculate the electroweak parameters $R_l$, $A_l$, and $N_{\nu}^l$ in a model that combine inverse seesaw with the scotogenic model. Dark matter relic density is also considered. Due to the stringent constraint from the ATLAS experimental data, it is difficult to detect the loop effect on $R_l$, $A_l$ in this model considering both the theoretical and future experimental uncertainties. However, $N_{\nu}^l$ can sometimes become large enough for the future experiments to verify.


I. INTRODUCTION
The Type I Seesaw mechanisms [1][2][3][4][5] are utilized to explain the smallness of the neutrino masses by introducing some extremely heavy right-handed neutrinos with the masses 10 8-12 TeV, which is far beyond the ability of any current or proposed collider facility. Suppressing the mass scales of the right-handed neutrinos below 1 TeV will also lead to tiny Yukawa couplings (∼ 10 (−7)-(−9) ), making it rather difficult to produce any experimental signals in reality.
The scotogenic model [6][7][8][9] and the inverse seesaw model [10][11][12][13] are the two different approaches toward the TeV-scale phenomenology corresponding to the neutrino sector. In the various versions of the scotogenic model, the active neutrinos acquire masses through loop corrections. In this case, the loop factor naturally suppresses the Majorana masses of the left-handed neutrinos. As for the inverse seesaw model, two groups of so-called "pseudo-Dirac" sterile neutrinos are introduced. The contributions from the large Yukawa couplings to the left-handed neutrino masses are nearly cancelled out, with a small remnant left over due to the small Majorana masses among the pseudo-Dirac sterile neutrinos which softly break the lepton number.
As far as we know about the literature, the combination of these two models can date back to Ref. [14], which appeared shortly after the Ref. [6]. There are also various papers in the literature, suggesting different variants or discussing the phenomenologies (For some examples, see Refs. [15][16][17][18], while Ref. [19] had discussed a similar linear seesaw model.). In this paper, we discuss about a simple version of such kind of models motivated by avoiding some tight restrictions on the Yukawa coupling orders. In the usual scotogenic models, Yukawa couplings are usually constrained by the leptonic flavour changing neutral current (FCNC) such as the µ → eγ bound. In the case of the inverse seesaw mechanisms, the invisible decay width of the Z boson also exert limits on the Yukawa couplings. This lead to the mixings between the active neutrinos and the sterile neutrinos and will result in the corrections to the Z → νν branching ratios on tree-level. Combining these two models can reach some relatively larger Yukawa couplings, while evading some constraints at the same time.

II. MODEL DESCRIPTIONS
The scotogenic model is based on the inert two Higgs doublet model (ITHDM). In this model, two SU(2) L Higgs doublets Φ 1 and Φ 2 are introduced. Let Φ 2 be Z 2 -odd, while Φ 1 together with other standard model (SM) fields be Z 2 -even, the potential for the Higgs sector is given by where Φ 1,2 are the two Higgs doublets with the hypercharge Y = 1 2 , λ 1-5 are the coupling constants, m 2 1 , m 2 2 are the mass parameters. In the ITHDM, only Φ 1 acquires the electroweak vacuum expectation value (VEV) v and the standard model (SM) Higgs h originates from this doublet. All the elements of the Φ 2 form the other scalar bosons H ± , H, A, and no mixing between the SM Higgs and the exotic bosons takes place. Therefore, Due to the Z 2 symmetry, all the fermions Q L , u R , d R , L L , e R only couple with the Φ 1 field where Y u,d,l are the 3 × 3 coupling constants.
The Z 2 -odd pseudo-Dirac sterile neutrinos N i = P L N Li + P R N Ri , (i = 1-3, P L,R = 1∓γ 5 2 ), together with the left-handed lepton doublets couple with the Φ 1 . The pseudo-Dirac 4spinors N i can be written in the form of where Y N is the 3 × 3 Yukawa coupling constant matrix, m N is the 3 × 3 Dirac mass matrix between the sterile neutrino pairs, µ is a 3 × 3 mass matrix which softly breaks the lepton the mass terms corresponding to µ 2 ij N c Li N Lj are generated and discussed [20][21][22]. In fact, it is easier to generate the correct light neutrino mass matrix pattern in a discrete symmetry and flavon-based model if the lepton flavour violation has only one source (It, in this paper, refers to N c Li N Lj .), though, in this paper, we discuss both the contribution from µ 1,2 for completion.

III. NEUTRINO MASSES
The right panel of the Fig. 1 shows the diagram that induces the neutrino masses. Inside the loop there is a Majonara mass insertion term originated from the Eqn. (4). By principle, we can directly calculate through the this diagram, however, in this paper, we adopt another method. In fact, the pseudo-Dirac neutrinos can actually be regarded as a pair of nearly- In spite of the coupling constants, the kernel of the left panel of Fig. 1 is given by the Ref. [6].
where M k , m H,A are the mass of the Majorana sterile neutrino, and the masses of the CP-even and CP-odd neutral exotic Higgs bosons H and A.
In the Weyl basis, the mass terms in (4) can be written in the form of That is to say, in the N Li , N Ri basis, the blocking mass matrix is given by where m N = [m N ij ] and µ 1,2 = [µ 1,2 ij ] are the 3 × 3 submatrix. Without loss of generality, let m N be diagonalized with the eigenvalue m N i , i = 1, 2, 3, and regard µ 1,2 ij as the perturbation parameter, and diagonalize (7), the rotation matrix is given by where Replace each masses in (5) with V T M N V ii , and multiply the coupling constants Y N ik Y N jl , then sum over all the terms while drop higher orders of µ ij , we acquire where In this paper, we ignore all the CP phases for simplicity and adopt the central values [23,24] ∆m 2 21 = 7.37eV 2 , |∆m 2 | = |∆m 2 32 + ∆ where s ij = sin θ ij , c ij = cos θ ij , and θ ij 's are the mixing angles. The CP-phase angle δ,  [25,26]. Some papers (e.g., Ref. [27,28] ) also plot the allowed region constrained by the oblique parameters. From the formula and the figures in the literature, we can easily find that if m A ≈ m H ± , or if m H ≈ m H ± , the contributions from the exotic Higgs doublets will nearly disappear in the alignment limit. Therefore, in this paper, we discuss the following benchmark parameter spaces: Although m H,A > m H ± is also possible, however, this will make some parameters decouple and we aim at discussing as much phenomenology (allowed by the current constraints) as possible in this paper. In this paper,we do not discuss such parameter space here.
The leptonic flavour changing neutral current (FCNC) decays l 1 → l 2 + γ in Fig. 2 set constraints on the parameter space. In the original scotogenic model, Ref. [7] had pointed out that in the usual scotogenic model, the parameter space is quite constrained by the processed. We have calculated the CL. ratio according to the Ref. [37], and scanned in the m H ± -m N parameter space. We plot our results of 95% CL. exclusion limits in the Fig. 3.

V. CALCULATIONS OF SOME OBSERVABLES
In this section, we aim at calculating the following observables: • Relic density of the dark matter.
• Shiftings on the Z-resonance observables R l = Γ Z→hadrons • Shiftings on the Z →invisible parameter N l ν = The review on the R l , A l , and N l ν can be found in Ref. [24]. The relic density of the dark matter is calculated by micrOMEGAs 4.3.5 [38,39], with the our model file exported by FeynRules 2.3.28. The inert Two Higgs Doublet Model part of the model file is based on the Ref. [40,41].
The shiftings on the electroweak parameters δR l , δA l , and δN l ν are calculated according to the formulas and steps listed in Ref. [30], where the one-loop corrections to the Z-l-l coupling constants are computed and then replace their values to the expressions of δR l , δA l , and δN l ν which depend on them. The computing processes can be compared and checked with the Ref. [42]. Note that in the case of this paper, neither the tree-level correction to the muon's decay constant G F nor the tree-level mixings between the sterile neutrinos and the light neutrinos exists, therefore the computing procedures are much simpler than those in the Ref. [30].
In order to present our results, we only consider the sub parameter space of m H ± = 250, 350, 500 GeV. The Yukawa coupling constant [Y N ij ] = yI is adjusted in order for the relic density Ω DM h 2 to approach 0.1199 ± 0.0027 [43]. Combined with the two cases in the last section, we show six plots in the Fig. 4, 5 and 6.
As has been mentioned, we are only interested in the case when m N < m H,A , therefore the upper-left part of the plots are all left blank. The color boundary becomes a step-like shape due to the insufficient density of points on the horizontal axis and our limit on the computational resources.

VI. DISCUSSIONS
There are plans on the future experiments to measure R l , A l , N l ν [44][45][46][47]. Currently, the collider experiments have imposed very stringent bounds on the mass of m ± H . From Fig. 3, we can easily see that m ± H 325 GeV has been excluded in the case when m N → 0 and Br H ± →N l ± =100%. When Br H ± →N l ± <100%, bounds on m H ± can be somehow relaxed.
However, besides the leptonic channel, H ± can only decay into H/A + W ± , which requires  suppressed due to the constraints on m H ± . As has been discussed in the Ref. [30], taking both the experimental and theoretical uncertainties into account, |δR l | should be 0.001 and |δA l | should be 8 × 10 −5 in order to for the new physics effects to be observed on the future experiments that measure the electroweak parameters. When m H ± > 325 GeV, for example as in the Fig. 5 and 6, in some cases the predicted |δR l | or |δA l | might reluctantly reach this bound, however in this case the Yukawa coupling y approaches the perturbative constraint y < √ 4π ∼ 3. On the other hand, when m H ± 350 GeV, for example when m H ± = 250 GeV as shown in Fig. 4, the unconstrained parameter space usually refers to a too-small Yukawa coupling for the enough |δR l | or |δA l |. Therefore, it is difficult to detect the effects on δR l and δA l from this model on future experiments.
However, δN l ν can be large if m H or m A are relatively small. Ref. [47] has shown us that δN l ν can reach a statistical uncertainty of 0.00004 and a systematic uncertainty of 0.004 in its Table 1. However, The discussions in the section 7 mentioned that a desirable goal would be to reduce this uncertainty down to 0.001. In this case, the FCC-ee is enough to cover much of the parameter space in m H ± = 350, 500 GeV. However, as has been shown in Fig. 4, it is still difficult to detect δN l ν in its unconstrained parameter space when m H ± = 250 GeV. Interestingly, current LEP results N l ν = 2.984±0.008 show a 2-σ deviation from the standard model prediction. If this will be confirmed in the future collider experiments, it will become a circumstantial evidence to this model. GeV, y = 0.605, in this case µ ≃ 10 keV. Furthermore, if we again appoint m N = 10 GeV, and let y = 1.88 in order for a correct relic density, µ will become ≃ 100 keV. Therefore, In this model, the mass splitting δ can be large enough ( 100 keV) so that the dark matter can be regarded as a pure Majorana particle in some parameter space, while in some parameter space, the mass difference δ might become so small (∼ 100 eV) so that the dark matter can transfer between the two mass eigenstates during the collision with the nucleons. The latter case is more similar to the Dirac case discussed in the Ref. [48]. According to [48], The Z-portal spin-dependent cross section was calculated to be less than 10 −41 cm 2 in both the Dirac case and the Majorana case, while the spin-independent cross section in the Dirac case was calculated to be 10 −47 cm 2 when m N < 200 GeV. Although these are still below the experimental bounds [49][50][51][52], it is hopeful for the future experiments to cover some of our parameter space since the current bounds are not far from the predictions.

VII. CONCLUSIONS
We have discussed some phenomenologies of a simple inverse seesaw scotogenic model by calculating the electroweak parameters R l , A l . N l ν in the case of a correct dark matter relic density. The current ATLAS results have imposed stringent bounds on the parameter space, lowering the predicted R l and A l . Considering both the experimental and theoretical uncertainties, it is difficult to detect the effect from this model on R l , A l in the future measurements. However, δN l ν can become large enough, shedding lights on verifying or constrain this model in the future.