Low-scale leptogenesis and dark matter in a three-loop radiative seesaw model

We show that three open questions in particle physics and cosmology: the origin of neutrino mass, the identity of dark matter, and the origin of the baryon asymmetry of the universe can be explained simultaneously in the three-loop seesaw model proposed by Krauss, Nasri, and Trodden. We discuss the difficulty of successful leptogenesis with three right-handed neutrinos, and we propose a scenario with four right-handed neutrinos that satisfies all observational constraints. This scenario predicts a sleptonlike particle as light as a few hundreds GeV that can be probed by future collider experiments.


INTRODUCTION
The observation of neutrino oscillation has confirmed that neutrinos have masses. The neutrino oscillation parameters [1] and the cosmological observations [2] indicate that neutrino masses are many orders of magnitude smaller than those of the other standard model (SM) fermions. We need some mechanism beyond the SM to explain such tiny masses.
When we consider cosmology, there are additional strong motivations to consider the new physics beyond the SM. One is that there is no appropriate candidate for dark matter (DM) in the SM. Another problem is that the baryogenesis does not work in the SM, as the electroweak baryogenesis requires a smaller Higgs boson mass than the observed one.
The canonical seesaw mechanism [3][4][5][6] is a favorable idea to address the origin of the neutrino masses, in which neutrino masses are suppressed by heavy right-handed (RH) neutrino masses. In such models, on the one hand, the thermal leptogenesis [7] works as a mechanism of the baryogenesis, where the CP -violating decay of the right-handed neutrino produces the lepton asymmetry, which is partially converted to the baryon asymmetry through the sphaleron process. The disadvantage of the thermal leptogenesis in the seesaw model is that it tends to require a right-handed neutrino to be as heavy as 10 9 GeV [8][9][10] and cannot be tested by experiments. In addition, further extensions would be necessary because the minimal seesaw model does not contain a suitable candidate for DM.
An alternative approach to explain the tininess of the neutrino masses is to utilize loop factors [11][12][13][14][15][16][17]. A class of models with right-handed neutrinos, so-called radiative seesaw models, is particularly attractive. In these models, a discrete symmetry under which the right-handed neutrinos and some extra scalars are odd is introduced to forbid the tree-level neutrino-mass generation. This symmetry can also stabilize the lightest odd-charged particle as the DM.
In this paper, we focus on the radiative seesaw model proposed by Krauss, Nasri, and Trodden [15] (often called the KNT model), where tiny neutrino masses are generated via three-loop diagrams. To forbid the tree-level contribution to neutrino masses, a Z 2 symmetry is introduced under which the right-handed neutrinos and a charged scalar S 2 is odd. Therefore, the lightest right-handed neutrino can be a candidate for the DM. The phenomenology of the KNT model has been studied in Refs. [18][19][20][21][22][23]. In our previous work [23], we have found that the KNT model is severely constrained for the inverted neutrino-mass ordering case by the experiments searching for the lepton flavor violation (LFV). A model proposed by Ma [16] (sometimes called the scotogenic model) also shares properties that neutrino masses are radiatively generated at the one-loop level and possess a dark matter candidate. From the viewpoint of the philosophy of radiative generation of neutrino mass, the KNT model would be more appealing than Ma's scotogenic model with a TeV scale mass of extra scalars that need a small scalar quartic coupling O(10 −5 ) besides the one-loop suppression factor that is not enough to reduce neutrino masses down to sub-eV scale. Phenomenologically, the DM in the KNT model is a Majorana right-handed neutrino, while that in the scotogenic model is the scalar in a heavier inert doublet Higgs doublet. 1 There are different dark matter phenomenology.
Baryogenesis in the context of the KNT model has been scarcely considered. The highscale leptogenesis in an extended KNT model has been studied [26] but not for the original KNT model. In the KNT model, the decay of the second lightest right-handed neutrino N 2 into a charged lepton ℓ ∓ R and S ± 2 can produce the lepton asymmetry. One difficulty of the leptogenesis in the KNT model is that the asymmetry generated by the N 2 decay is cancelled and washed out 2 after the decay S ± 2 → N 1 ℓ ± R . In this paper, we point out that this washout can be suppressed if S 2 is relatively light and its Boltzmann suppression is not strong at the sphaleron freeze-out time. Since the sphaleron freeze-out temperature T sph is about ≃ 130 GeV [27], S 2 must be lighter than a few hundreds GeV. The light S 2 behaves like a slepton and can be explored by direct searches at the collider experiments such as the International Linear Collider (ILC) [28,29], the Compact Linear Collider [30], the Future Circular Collider [31], or Circular Electron Positron Collider (CEPC) [32,33]. Another difficulty of the leptogenesis is due to the ∆L = 2 washout processes such as via exchange of right-handed neutrinos [34,35]. Those washout reaction rates are, in general, many order of magnitude larger than the cosmic expansion rate, hence, the generated lepton asymmetry hardly survive. Thus we may need an extension of the model to explain the baryon asymmetry via leptogenesis. A simple solution is introducing the fourth generation of the right-handed neutrino, and we adopt this possibility in this paper. We check that our scenario of leptogenesis is compatible with the observations such as the neutrino oscillation, the DM abundance, and LFV processes.
This paper is organized as follows. In Sec. 2, we briefly review the KNT model and discuss the constraints from the neutrino oscillation, the lepton flavor violations, the relic abundance of the dark matter, and direct searches of new particles. In Sec. 3, we discuss the possibility of leptogenesis, and we show that the observed baryon asymmetry can be explained in the case of four right-handed neutrinos. In Sec. 4, we summarize this paper and make concluding remarks.

The KNT model 2.1 The Lagrangian and the neutrino-mass matrix
We consider the KNT model [15], which explains the tininess of neutrino masses by utilizing the loop factor. In the model, charged scalar fields S 1 and S 2 and RH neutrinos N I with I are the generation indices are introduced. Furthermore, a global Z 2 symmetry is introduced, under which S 1 and N I fields are assigned odd and S 2 is assigned even.
This symmetry is necessary to forbid the neutrino Yukawa couplings with the right-handed neutrinos, left-handed lepton doublets, and the Higgs doublet, which would provide too large Dirac neutrino-mass terms after the electroweak symmetry breaking. The Z 2 symmetry simultaneously guarantees the stability of the lightest Z 2 odd particle. If N 1 is the lightest, it can be a DM candidate. The Lagrangian terms added to the SM are where the superscript c denotes the charge conjugation, the Yukawa matrix (h ij ) is an antisymmetric, i.e., h ij = −h ji , g Ij are other Yukawa coupling constants, λ S is a complex coupling, and N I are in the mass basis. The scalar potential V includes four-point scalar coupling terms, With the Lagrangian, the neutrino masses are induced through the Feynman diagram in Fig. 1. A component of the neutrino-mass matrix is given by [19] Here, we use a simple expression for the loop function f I found in Ref. [23]: Figure 1: The diagram of the neutrino-mass generation in the KNT model.
As discussed later, since observations constrain m N 1 ≲ O(100) GeV, m S 1 ≳ O(10 4 ) GeV, and m S 2 ≃ O(10 2 ) GeV, the region of y ≫ x 1 and y ≫ 1 is of interest. In such a case, the loop function can be analytically estimated as where ζ is the Riemann zeta function. In Fig. 2, we plot the loop function in the case of y ≫ 1 and the analytic expression equation (5), and we find that the analytic expression provides a good approximation in the region m N 1 ≲ 0.1m S 2 .
To find a parameter set that reproduces the neutrino oscillation data by the mass matrix equation (3), we can use the relations found in Refs. [22,23]. In Eq. (3), there are terms suppressed by the small electron mass m e , which can be ignored. Under this approximation, we can extract conditions on h ij as

Lepton flavor violation
In the KNT model, the charged lepton flavor is not conserved. Thus, charged leptons ℓ i can decay into the lighter one ℓ j and a photon γ. The branching ratio of this process is estimated as [19,23] Br where A ij R and A ij L are given by α em is the fine structure constant, and G F is the Fermi constant. The loop function The most severe constraint comes from µ → eγ. The current upper bound on this process is given by Br(µ → eγ)< 4.2 × 10 −13 [37]. Even if A 21 R = 0 is satisfied by taking a specific form for g Ii , A 21 L cannot be taken as zero because of the antisymmetric structure of (h ij ). Since |A 21 R | 2 ≥ 0, the branching ratio satisfies Note that the factor h 13 /h 23 is determined by the elements of the neutrino-mass matrix, as shown in Eq. (7).
As for neutrino-mass parameters, we input the best-fit values of normal ordering with super-Kamiokande data in Ref. [1]. In the KNT model, the lightest active neutrino mass is zero because of the antisymmetric structure of (h ij ) [22]. Hence, the other active neutrino masses are determined by the observed mass-squared differences. In this case, the remaining free parameter in M ab is one Majorana phase. In the following, we set the Majorana phase to zero for simplicity. With a finite value of the Majorana phase, the analysis does not change much. If the neutrino-mass ordering is inverted, the factor |h 13 /h 23 | is larger than that of the normal ordering case [22]. The combination of the upper bound on Br(µ → eγ) and the perturbativity condition was studied in Ref. [23], and inverted ordering is severely constrained.
By inputting the neutrino oscillation data and the upper bound of Br(µ → eγ) to Eq. (11), we obtain the lower bound on m S 1 :

Slepton searches
In the KNT model, S 2 behaves like a purely right-handed slepton in supersymmetric models.
In the case considered below (g 11 = g 12 = 0), it is like a right-handed stauτ R . The mass bounds onτ R are obtained by the LEP experiments [38][39][40][41][42]. 5 The bounds come from the search for the stau decay (S 2 → τ + N 1 in our case) and they depend on m N 1 . In the exclusion plot in Ref. [44], the strongest bound is m S 2 ≳ 95.5 GeV (95% confidence level) at m N 1 ≃ 64 GeV. In the analysis below, we use a conservative bound m S 2 > 96 GeV (13) for all the m N 1 region.

Dark matter
The lightest right-handed neutrino N 1 is stabilized by the Z 2 symmetry and it can be the dark matter. We assume that N 1 was produced as thermal relics. The abundance of such dark matter is determined by the annihilation cross section. Since the dark matter mass m N 1 cannot be very large, we naively expect to have a significant contribution to the LFV via S 2 and N 1 exchange diagrams. To avoid such a contribution, it is preferred that N 1 couples to only one lepton flavor. Thus, we here make an ansatz that N 1 only couples to τ , i.e., g 11 = g 12 = 0 and g 13 ̸ = 0. With this ansatz, the cross section is calculated as [15,18,19] with x f ≃ 20 [45]. The relic abundance of the dark matter after the decoupling is approximated by The white region satisfies conditions to explain dark matter and leptogenesis. The blackdashed curve shows the simplified lower bound from LEP (m S2 > 96 GeV) [44] and reddashed curves show the simplified sensitivity up limit on m S2 of future experiments (CEPC: 113 GeV [33], ILC250: 123 GeV , ILC500: 245 GeV [29]). The dotted lines are contours of m S2 = 200 GeV, 300 GeV. These curves are obtained by using the DM condition equation (16). The right-bottom shaded corner is the region where the charged S 2 is lighter than N 1 . The horizontal-gray line shows the perturbativity up limit.
By comparing the observed dark matter abundance Ω N 1 h 2 = 0.120 ± 0.001 [2], we obtain where Fig. 3, we show the contour of m S 2 in the m N 1 -|g 13 | plane which can reproduce the thermal relic abundance of the DM. By the perturbativity condition |g 13 | ≤ 1, there is a lower and upper bound on the dark matter mass m N 1 : The upper bound gives As discussed later, m S 2 less than O(100) GeV is also preferred to explain the baryon asymmetry by the leptogenesis.

The minimal case
We here consider the minimal structure of (g Ii ) to explain both the neutrino mixing and the dark matter with satisfying the lepton flavor violation constraints.
To explain the neutrino oscillation data, we should reproduce M τ τ , M µµ , and M µτ . Once these elements are reproduced, the other elements can be tuned by h 12 /h 23 and h 13 /h 23 . To realize it, we need at least three independent g Ij . On the other hand, the elements g I1 are irrelevant to the neutrino-mass matrix because of the strong suppression by m e . Therefore, we need at least two RH neutrinos which have a significant size of the Yukawa coupling with µ and/or τ . If we use N 1 and N 3 for generating an appropriate neutrinomass matrix, 6 g 32 and g 33 should have significant size, as g 22 = 0 is taken in Sec. 2.4. In addition, g 31 should be small to avoid too large a contribution to µ → eγ through the S 2 exchange diagrams. Thus, the minimal setup of the Yukawa couplings with N c I ℓ Ri S + 2 for the neutrino mixing and the dark matter is given by With the setup, three components of the neutrino-mass matrix in Eq. (3) become and we can see that there are enough degrees of freedom to reproduce an appropriate neutrino matrix. Note that the inverted ordering case with the best-fit oscillation parameters is excluded by the µ → eγ constraint. 7 Thus, we consider the normal ordering case throughout this paper. Let us comment on the lepton flavor violation processes other than µ → eγ. If the constraint (12) is satisfied, the contribution of S 1 to the other decays τ → eγ and τ → µγ is the same order as µ → eγ and much weaker than the experimental constraints. In our case of g I1 = 0, S 2 does not contribute to µ → eγ and τ → eγ. On the other hand, the S 2 contribution to τ → µγ is given as Using the upper bound Br(τ → µγ)< 4.2 × 10 −8 [46] and Br(τ → µνν) = 0.1739 [47], we find 3 Leptogenesis

Production and evolutions of the asymmetry
For baryogenesis, we consider a scenario that the CP -violating decay of N 2 to ℓ R and S 2 generates the lepton asymmetry, and the lepton asymmetry is converted to the baryon asym- 6 As discussed in Sec. 3, N2 will be necessary for the leptogenesis, and the mass mN 2 is required to be smaller than mN 3 . So that we here use N3 instead of N2.
7 It corresponds to the n eff = 1 case in Ref. [23]. Figure 4: The Feynman diagrams relevant to the CP asymmetric decay of N 2 → ℓ R S 2 at the one-loop level.
metry by the sphaleron process. The CP asymmetry in the decay N 2 → ℓ ± Ri S ∓ 2 is defined as which comes from the interference between the tree diagram and the loop diagrams shown in Fig. 4. For producing enough large lepton asymmetry, the decay width of N 2 should not be too large compared to the Hubble rate at around T = m N 2 . To examine this, we define K as the ratio of the total decay width of where g * = 110.5 is the effective relativistic degrees of freedom by taking into account S 2 , N 1 , and N 2 , and K ≲ 1 is required. To realize it, the Yukawa couplings g 2i should be too strongly suppressed for N 2 to contribute to the neutrino-mass matrix. Thus N 2 should be introduced in addition to N 1 and N 3 . We consider the chemical potentials before the sphaleron freeze-out to derive the relation between the baryon number and the lepton number in our model. We denote the chemical potentials by µ and subscripts q, u, d, L, ℓ R , ϕ indicate the SM quark doublet, right-handed up-type quark, right-handed down-type quark, lepton doublet, right-handed charged lepton, and the Higgs doublet. The chemical potential of the Majorana particles is zero µ N = 0. The SM Yukawa interactions and the sphaleron process yields 8 3µ q + µ L = 0.
The conditions from vanishing hypercharge in the Universe and the non-SM interaction are [26] 3 These equations can be solved as Now, the condition (39) becomes clear. If the condition is not satisfied, it ends up with µ S 2 = −µ ℓ R = 0. The relation of the baryon number B = 3 × 3 × (1/3) × (2µ q + µ u + µ d ) and the lepton number The sets of Boltzmann equations to be solved are those of N 2 , ℓ i and S ± 2 . Those are rewritten as the Boltzmann equations for the number density of N 2 and asymmetries of number densities of B/3 − L i and S + 2 − S − 2 , respectively, as (n X denotes the number density of the species X) where ⟨Γ⟩ and ⟨σv⟩ are thermal averaged decay (inverse decay) rates and thermal averaged scattering cross section times relative velocity, respectively.
It is a kind of thermal leptogenesis but has some remarkable features. First, the sum of B − L asymmetry and the asymmetry between S + 2 and S − 2 is always zero, i.e. i n B/3−L i + n S + 2 −S − 2 = 0, which is nothing but the electric charge neutrality of the Universe. If S 2 becomes nonrelativistic and decouples from the thermal equilibrium before the sphaleron freeze-out, the abundance of S 2 gets Boltzmann suppressed and the lepton asymmetry is washed out. However, after the sphaleron decouples from the thermal bath at the temperature T sph = 131.7 ± 2.3 GeV [27], the baryon asymmetry n B /s is not washed out anymore, while B − L asymmetry is decreasing. With Eq. (34), the final baryon asymmetry is given by We stress that the out-of-equilibrium condition for successful baryogenesis in this scenario is satisfied by the sphaleron decoupling. It is different from the situation in the canonical leptogenesis. As a consequence of the behavior, the mass of S 2 should not be much larger than T sph , so that Second, we discuss the strong washout effect of the scattering process. ∆L = 2 washout processes ℓ ± i S ∓ 2 ↔ ℓ ∓ j S ± 2 and ℓ ± i ℓ ± j ↔ S ± 2 S ± 2 via N I exchange are, in general, very strong [34]. Those cross sections are of the order of ⟨σv⟩ ∼ |g * Ii g Ij | 2 /T 2 . The interaction rate Γ ∼ T 3 ⟨σv⟩ is much larger than the Hubble parameter H ∼ T 2 /M P where M P = 1.22 × 10 19 GeV is the Planck mass. This means that the ∆L = 2 washout is very strong unless involved g are extremely small. Since the strength of ∆L = 2 washout are flavor dependent, we need to evaluate lepton asymmetry in each flavor. To reproduce the observed neutrino mixing in the KNT model, g 32 and g 33 cannot be small, so that the µ and τ asymmetries are strongly washed out. Producing e asymmetry by the CP -violating decay of N 2 → e ± R S ∓ 2 is only the possibility. The CP -violating parameter of N 2 → eS 2 decay is evaluated as In order to produce e asymmetry in the model with three RH neutrinos, a significant size of g 31 is required to provide the large CP violation in N 2 → e ± R S ∓ 2 decay. However, such a large g 31 is disfavored not only by the µ → eγ constraint but also by the strong washout effect via the flavor changing ∆L = 2 scattering e ± S ∓ 2 ↔ ℓ ∓ i S ± 2 (ℓ i = µ, τ ). In fact, we cannot find any point which reproduces enough large baryon asymmetry in the case with three RH neutrinos.
A simple extension to solve the above difficulty is introducing the fourth RH neutrino N 4 . The neutrino-mass matrix and the properties of the DM can be explained by N 1 and N 3 , while N 2 and N 4 play an important role in the leptogenesis. In the case that N 2 and N 4 only couple to an electron, ϵ 1 can be enhanced by large g 41 , while the contribution to µ → eγ via N 4 and N 2 exchange and the washout by the flavor changing ∆L = 2 scattering processes are absent.

Benchmark inputs
We construct a benchmark scenario to demonstrate how the baryon asymmetry is produced in our leptogenesis scenario. With the four RH neutrinos, the Yukawa matrix g is a 4 × 3 matrix. The minimal structure for the successful leptogenesis is given by With this structure, there are nine complex and six real parameters: complex : λ S , h 12 , h 13 , h 23 , g 13 , g 21 , g 32 , g 33 , g 41 These parameters have to satisfy the neutrino-mass conditions, Eqs. (6), (7), (20), (21), and (22), and two more conditions, i.e., leptogenesis and the DM conditions. The four inequalities (12), (24), (13), and (39), and the perturbativity condition (couplings have to be smaller than the order of unity) must also be obeyed.
In our analysis, we scan m N 2 and m S 2 in the range of [100, 330] GeV, and we fix the other parameters to satisfy the conditions obtained from the DM relic abundance and neutrino oscillation data. First, we consider the DM relic abundance. Once |g 13 | and m S 2 are fixed, m N 1 is determined to reproduce the relic abundance of the DM as shown in Fig. 3. For example, m S 2 = 110 GeV and g 13 = 1.0 give m N 1 = 12 GeV.
Second, we take into account the neutrino oscillation data. For the neutrino-mass parameters, we input the best-fit neutrino oscillation parameters in the case with super-Kamiokande data [1]. Once we fix h 23 = 1, Eqs. (6) and (7) determine the rest of the h ij as We additionally fix λ S = |g 32 | = 1. By using the neutrino-mass equations. (20), (21), and (22) and the approximation formula (5) of f 1 , we obtain The branching ratios of the LFV decays in the benchmark point are evaluated as Br(τ → µγ) = 7.5 × 10 −15 , which are far below the current limits. Finally, we fix the rest of the parameters relevant to the leptogenesis. For optimizing the production of the lepton asymmetry, we tune the value of |g 21 | to satisfy K = 1 in Eq. (26) i.e., and we take arg(g 21 ) = π/4 which maximize the CP asymmetry ϵ 1 . We fix the m N 4 and g 41 as m N 4 = 1.0 × 10 8 GeV and g 41 = 0.1, respectively. Our benchmark inputs are summarized in Table 1.

Numerical analysis
We show, in Fig. 5, the evolution of absolute values of asymmetry yield Y ≡ |n i |/s of each leptons with bluish (dotted) dashed curves, that of S + 2 − S − 2 with the green curve, that of N 2 with the orange curve, in the case with m S 2 = 110 GeV and m N 2 = 250 GeV. The asymmetry of the N 2 decay rates ϵ 1 is [48]: which reflects the charge neutrality, as mentioned above. The total B − L asymmetry drawn with the black dashed curve always coincides with that of S + 2 − S − 2 asymmetry. We can see that both the total B − L asymmetry and the S ± 2 asymmetry decreases for a large m N 2 /T . At T = T sph , the baryon asymmetry is frozen out as Y B = Y B−L | T =T sph due to the sphaleron decoupling. Figure 5 shows that enough large baryon asymmetry Y B = O(10 −10 ) is obtained in our benchmark. In Fig. 6, we show an example of contour plots of the final baryon asymmetry for a set of Yukawa coupling constants. The range m S 2 < 310 GeV is preferred by the DM relic abundance as shown in Fig. 3 and it can be explored by future e + e − collider experiments. For example, the CEPC can probe it up to 113 GeV [33], and the ILC with the center of mass energy of 250 GeV can do up to 123 GeV [29]. The Compact Linear Collider with 380 GeV or the ILC with 500 GeV can explore our predicted mass range.   Table 1.

Summary
In this paper, we have proved that neutrino masses, dark matter, and leptogenesis can be explained in the KNT model with four right-handed neutrinos. We have shown explicit parameters that can realize these phenomena and satisfy the observational constraints, such as the LFV decays. To avoid the severe constraint from the LFV decay on the inverted mass ordering case [23], we considered the normal ordering case. In our scenario, there is a definite prediction that a charged scalar particle S 2 should be as light as m S 2 < O(100) GeV. The S 2 in our scenario behaves as a staulike particle, and it can be detected by future lepton collider experiments.