The scalar potential in the model is given by

2

The SM Higgs field is denoted by

3

and GeV is the vacuum expectation value (VEV) in the Higgs basis after the spontaneous symmetry breaking, z is absorbed by the neutral gauge boson of the SM Z, and is absorbed by the charged gauge boson of the SM . The charged scalar masses are given by

4

5

In the numerical analysis, we consider to be free parameters.

After the spontaneous electroweak symmetry breaking, the charged-lepton mass matrix is given by

6

Then, the charged-lepton mass matrix is diagonalized by a bi-unitary mixing matrix as . Therefore, , where is the mass eigenstate. These three parameters are used to fit the mass eigenvalues of charged-leptons by solving the following three relations:

7

8

9

For convenience in constructing the neutrino-mass matrix, we define , which is given by .

The heavier Majorana mass matrix is given by

10

where can be real without loss of generality. is diagonalized by ( ); therefore, . Here, is the mass eigenstate.

The active neutrino mass matrix is given at the three-loop level via the following Lagrangian in terms of the mass eigenstates:

11

where and . The Yukawa matrices y and h are as follows:

12

13

where are complex free parameters. The neutrino mass matrix is then given by

14

Here, F is a loop function via three loop diagram and it depends on the mass eigenvalues of . ^① Since the masses of contribute to the structure of neutrino mass matrix, there would be too many free parameters to get some predictions for the neutrino sector. Thus, we consider a special situation among the mass hierarchies of so that F is independent of the structure of neutrino mass matrix. When we assume , one finds that the dominant part of the loop-function F is a constant and can explicitly be given by . In detail, one finds Appendix A. Thus, we redefine the neutrino mass matrix as follows:

15

16

The dimensionless matrix is diagonalized by a unitary matrix as , where , and the Pontecorvo-Maki-Nakagawa-Sakata unitary matrix is defined by . Note here that the lightest neutrino mass is zero due to the two-matrix rank of the neutrino. Thus, the atmospheric mass squared difference is as follows:

17

18

where NH(IH) represents the normal(inverted) hierarchy. The solar mass squared difference is given by

19

20

The effective mass for neutrinoless double beta decay is given by

21

22

where the Majorana phase is defined by and we adopt the standard parametrization for the PMNS unitary matrix. The current KamLAND-Zen data [16] provide measured observables, and their upper bound is given by meV at a 90% confidence level. The minimal cosmological model ΛCDM provides an upper bound on 120 meV [17, 18]. Moreover, the recent combination of DESI and CMB data gives a more stringent upper bound on this bound; 72 meV [19].

process: First of all, let us consider the processes at one-loop level ^② . The formula for the branching ratio can generally be written as

23

where is the fine-structure constant, for ( ), GeV^-2 is the Fermi constant, and is given by

24

25

where

26

By assuming that , the formula can be simplified to

27

The formula for the muon can be written in terms of and and simplified as follows:

28

Notice here that this contribution to the muon is negative; however, it is negligible compared to the deviation in the experimental value [22].

Here, we employ only the results of lepton universality from a precursor work [23]; the results provide the upper bounds on coupling H in terms of and . We summarize these results in Table 3.

Relic density: Our DM is identified as the lightest Majorana fermion where we denote as X hereafter and its mass is . In order to analyze it simpler, we impose the following condition, , in order to evade an effect of co-annihilation interactions for the relic density of DM. ^③ Under the condition, the dominant contribution to the relic density arises from Y. Then, the non-relativistic cross section is expanded by relative velocity ; and found as follows:

29

where we have neglected the masses of charged leptons. The above cross section suggests that it is p-wave dominant. The relic density is then given by

30

where , , . In our numerical analysis, we use a rather relaxed experimental range because we simplify our analysis of the relic density.

First, we perform a χ square analysis adopting data from NuFit6.0 [15], where we use five reliable observables (three mixings, two mass square differences) for the analysis. The yellow points represent the interval of , and the red ones , where no solutions are obtained within . Our three input parameters are randomly selected within the following range:

31

where we work on the fundamental region of τ, and are complex.

After the numerical analysis, we find that the IH case is not favored in the model, where the minimal can be at most . Thus, we summarize our results using only the NH case in the next subsection. Note that the parameters are chosen to fit the observed charged-lepton masses, and are related to fix the scale of the neutrino mass via κ, defined in Eq. (15). Thus, the relative neutrino mass and three mixing angles are fitted using the remaining parameters corresponding to seven real parameters. Three of these real parameters are related to complex phases; therefore, fitting the neutrino data is nontriavial. In fact, we would not be able to obtain any solutions in the IH case. To improve the fitting further, such as for IH, we need to change the assignment of the modular weight to increase the number of free parameters.

In Fig. 1, we show the allowed region of τ, and find that the allowed region is concentrated at nearby and where the value is close to the fixed point . We also find a few points near the fixed point . ^④

In Fig. 2, we demonstrate the allowed regions for the absolute values (left) and argument ones (right) of and in NH. We show that the allowed region is at approximately and , where is preferred, and and can be any value with little correlation.

In Fig. 3, we display the allowed region for deg (left) and meV (right) in terms of meV. We show that most of the points are located at deg and few points are at approximately deg, and meV. The vertical magenta dotted line is the upper bound of the results of Planck+DESI [19] 72 meV, and the range of our model is meV, which is a trivial consequence of two nonzero mass eigenvalues of active neutrinos.

In Fig. 4, we show the allowed region for meV (left) and deg (right) in terms of deg in NH. We show that the allowed region of is concentrated at approximately deg with few points outside the region.

We show a benchmark point (BP) that has the minimum in Table 4 and this BP will be employed to analyze the LFV, , and DM in the next subsection.

Before our numerical analysis, we present some definitions. The neutrino-mass matrix does not depend on all the masses inside the loop, but the chi square analysis of the neutrino-oscillation data provides the value of κ. Their masses inside the loop determine the values of the LFVs, muon , and relic density of DM. Thus, we rewrite Eq. (15) as follows:

32

When , and are numerically fixed, is numerically determined. Then, we impose the perturbative limit in our numerical analysis to be

33

In addition, we restrict ourselves to the following conditions to forbid co-annihilation processes and obtain the mass-independent loop function of the neutrino-mass matrix:

34

35

where we have defined to be .

Our input parameters are randomly selected from the following range:

36

where are real and the other required parameters are employed by the BP in the previous section.

Our numerical analysis showed that Yukawa coupling exceeds the perturbative limit to obtain the observed relic density of DM while satisfying the constraints of LFVs and lepton universalities. The correct relic density requires for the NH case, applying allowed parameters that can fit the neutrino data. This implies that co-annihilations do not help to reduce the Yukawa couplings to the perturbative limit. We may move to one of the next minimum models by changing the modular weight of to –2 instead of 0 to obtain one more mass parameter. This provides a wider region of allowed parameters, where the other assignments are the same as our model. However, we would still encounter difficulty in realizing the correct relic density while keeping the perturbative limit for the Yukawa couplings. This is because the DM annihilation cross section, Eq. (29), is p-wave dominant and we need a relatively larger coupling constant than that of the s-wave case. In addition, neutrino data and LFV constraints require heavy DM and new scalars that also suppress the DM annihilation cross section. Thus, obtaining the correct relic density in our minimal setting is difficult, and some extension is necessary.

If we do not satisfy the observed relic density and we perform our numerical analysis under the perturbative limit, we obtain the tendencies for electron , muon , and LFVs, as shown in Fig. 5. These figures suggests that and are at most and , respectively. However, LFVs, especially the branching ratio, would be testable in the near future because its maximum value is close to the experimental limit.

We briefly illustrate one of the simplest solutions to explain the observed relic density without breaking our predictions for the neutrino sector, making use of a new interaction. We introduce a singlet scalar boson that leads to new interactions

37

where its modular weight is assigned to zero for simplicity, assuming it is a singlet under the symmetry, and we omit terms with . We then have a Higgs portal to the SM by mixing between and h induced by the last term of . Note that the addition of these interactions do not modify the neutrino mass, and the predictions in our analysis will not change.

As a result we have additional DM annihilation processes such as and . In particular, the s-channel cross section is useful for explaining the relic density because the annihilation cross section is enhanced nearby at , where is the mass of . The annihilation cross section of the process is approximately given by

38

where is the SM Yukawa coupling for fermion f and indicates the Higgs- mixing. The relic density of DM is estimated as , and we obtain

39

where we consider the top quark as f for simplicity. Thus, we can realize with TeV, , and , even if we do not have resonant enhancement. With the resonant effect, we can fit the relic density for the small Higgs-mixing case without conflicting constraints of direct detection searches [26].