A Three-Loop Neutrino Model with Global $U(1)$ Symmetry

We study a three-loop induced neutrino model with a global $U(1)$ symmetry at TeV scale, in which we naturally accommodate a bosonic dark matter candidate. We discuss the allowed regions of masses and quartic couplings for charged scalar bosons as well as the dark matter mass on the analogy of the original Zee-Babu model, and show the difference between them. We also discuss the possibility of the collider searches, in which future like-sign electron liner collider could be promising.


I. INTRODUCTION
Even after the discovery of the Higgs boson, the large Yukawa hierarchy required by the observed values of the fermion masses remains to be one of the unnatural issues in the Standard Model (SM). The situations get to be more serious in the neutrino sector since their corresponding values are sub-eV, which means that we have to realize at least O(10 11 )magnitude hierarchy by hand when we adapt the Dirac-type mass terms for explanation.
An elegant way for alleviating the unnaturalness is making the situation that the neutrino masses are loop-induced as initiated by A. Zee at one-loop level in Ref. [1].
In such a setting, loop factors naturally reduce their mass values and we can explain the minuscule neutrino masses with less fine-tuned Yukawa couplings. This mechanism is fascinating and lots of works have been done in this direction [1]- [73]. As a naive expectation, higher-loop generated neutrino masses would be preferable because much more improvement could be expected due to a large amount of loop factors. Several three-loop models have been proposed already, e.g., in Refs. [4,10,33,48,55]. In higher-loop models, a dark matter (DM) candidate tends to propagate inside the loop, whose stability is naturally ensured by symmetries for prohibiting lower-level neutrino masses. Also, when a continuous global symmetry is used in such a model, we would predict a Nambu-Goldstone boson (NGB). This kind of particles could play a significant role in an early stage of the Universe [74].
In this paper, we propose a model as a simple extension of the Zee-Babu model [3] with two-loop induced neutrino mass terms, by adding an additional singly-charged gauge singlet scalar and DM to the original one, where the radiative neutrino mass terms turn out to appear at the three-loop level. Note that a doubly-charged scalar (k ±± ) and a singlycharged singlet scalar (h ± ) are introduced in the Zee-Babu model [3]. Our model overcomes a shortcoming in the Zee-Babu model of the absence of DM candidate. On the other hand, the structure of the internal loops within the radiative neutrino masses gets to be morphed. Therefore, expected mass ranges of the charged particles are affected from the original ones. This paper is organized as follows. In Sec. II, we explain the construction of our model and analyze the system with declaring brief prospects in collider-related issues. We summarize and conclude in Sec. III.

A. Model setup
We discuss a three-loop induced radiative neutrino model. The particle contents and their charges are shown in Tab. I. We add new bosons, which are, two SU(2) L singlet neutral bosons (Σ 0 , χ 0 ), two singly-charged singlet scalars (h + 1 , h + 2 ), and one SU(2) L singlet doubly-charged boson k ++ to the SM. We assume that only the SM-like Higgs Φ and Σ 0 have vacuum expectation values (VEVs), which are symbolized by Φ ≡ v/ √ 2 and respectively. x ( = 0) is an arbitrary number of the charge of the global U(1) symmetry 1 , and the assignments can realize our neutrino model at the three-loop level 2 . Notice here that one can identify the global B − L symmetry in case x = 1. The NGB in Σ 0 due to breaking of the U(1) global symmetry can also evade experimental searches or constraints due to its very weak interactions with matter fields as can be seen in [75], when this symmetry is identified as the L symmetry. The Z 2 symmetry assures the stability of DM χ 0 .
The relevant Lagrangian for Yukawa sector, mass terms, and scalar potential under these 1 This symmetry cannot be gauged because its anomaly cannot be canceled. 2 Notice here that one can realize our model by assigning zero global U (1) charge to χ 0 instead of −x, where χ 0 can be still a (real) DM candidate due to the Z 2 symmetry. Then the following two relevant terms h + 1 h − 2 χ 0 and (Σ 0 ) 2 (χ 0 ) 2 are respectively replaced by Σ * 0 h + 1 h − 2 χ 0 and (χ 0 ) 2 . Such a mechanism has been done by the authors in Ref. [19]. We would like to thank our referee to draw our attention. symmetries are given by where the first term of L Y generates the SM charged-lepton masses and y L (y R ) are threeby-three antisymmetric (symmetric) matrices, respectively. We assume µ 12 , and µ 22 to be positive real, but λ 0 to be negative real to identify χ 0R as the DM candidate (see Eq. (II.7)).
Here, we briefly mention the correspondence to the Zee-Babu model in the trilinear couplings among the charged scalars. In the Zee-Babu model, only one singly-charged scalar is introduced and we regenerate the forms by taking the limits in our model,

B. Mass matrices of bosons
The scalar fields can be parameterized as where v ≃ 246 GeV is the VEV of the Higgs doublet field, and w ± and z are respectively (would-be) NGB which are absorbed as the longitudinal components of W and Z bosons.
Inserting the tadpole conditions, ∂V/∂φ| v = 0 and ∂V/∂σ| v ′ = 0, the resultant mass matrix of the CP even bosons (φ, σ) is given by where h is the SM-like Higgs and H is an additional CP-even Higgs mass eigenstate. The mixing angle α is determined as The Higgs bosons φ and σ are rewritten in terms of the mass eigenstates h and H as An NGB appears due to the spontaneous symmetry breaking of the global U(1) symmetry.
The mass eigenvalues for the neutral bosons χ 0R , χ 0I , the singly-charged bosons h ± 1 , h ± 2 and the doubly-charged boson k ±± are respectively given as where these particles are not mixed due to the invariance of the system and thus they themselves are mass eigenstates, respectively.

C. Vacuum stability of electrically charged bosons
The vacuum stability has to be especially assured by the Higgs potential for electricallycharged bosons (h ± 1 , h ± 2 , k ±± ). However, our model has some loop contributions to the leading order of these quartic couplings. Here, we examine this issue at the one-loop level.
Let us define these quartic couplings as follows: where the upper indices denote the number of the order, and the one-loop contributions can be given as where each of m 1 and m 2 of F 0 represent a mass of propagating fields in the loops. We will include these constraints in the numerical analysis later. To avoid the global minimum with electromagnetic charge-breaking V(r = 0) > 0, the following condition should be at least satisfied: If all these quartic couplings are of the order as λ i ≈ O(π) 3 , the following condition can be given by where m 2 Φ and λ Φ are neglected. Note that the vacuum stability conditions take the following forms in the Zee-Babu model, 3 λ 0 is excluded, because λ 0 is negative and the maximum value is 0.

D. Neutrino mass matrix
The Majorana neutrino mass matrix m ν is derived at the three-loop level from the diagrams depicted in Fig. 1, which is described by an effective operator, ] and the loop function F 1 is computed as The neutrino mass eigenstates and their mixings can be straightforwardly given by applying them to the Zee-Babu analogy [66], since the structure of the fermion line is the same as the the Zee-Babu model [3], that is, a rank two model of the neutrino mass matrix due to the antisymmetricity of y L . Let us define the neutrino mass matrix as in our case, one can derive some simple formulae 5 . When we consider the normal ordering, 4 We assume m ℓ i/j ≈ 0 in our numerical analysis, since these masses are much smaller than the other masses inside the loops. 5 More details are given in Ref. [66] for both cases.  We fix and take the following parameter ranges: GeV (for our model in NH, Zee-Babu model in NH and IH),    Figure 4 shows the quartic couplings of λ h 1 ,h 2 ,k in terms of the allowed mass range of the DM candidate m χ 0R for both ordering cases, in which one finds that the allowed region for the IH decreases drastically than the one for the NH. Since the mass of the DM should be the smallest among the particles with negative Z 2 parity, its value is bounded from above via the allowed mass range of m h ± 2 typically as m h ± 2 O(10 2 ) TeV. In this sense, our DM can naturally explain the observed relic density [78] and the direct detection searches [79] which typically lies on the O(100) GeV mass scale. The details of the DM properties can be found in Ref. [42], just replacing m 5 → v ′ / √ 2 6 . Figure 5 shows the allowed region of m k ±± and m h ± in the original Zee-Babu model in order to compare with our allowed regions, where one finds that the allowed parameter configurations are much wider than ours in both of the orderings. Here, we take the same parameter set in Eq. (II.30).

G. Collider-related issues
In this subsection, we discuss possible collider-related issues. In our model, apart from the Zee-Babu model, the masses of the charged scalars are apt to be very heavy as typically above 10 TeV. They are highly decoupled and it is very difficult to detect a signature of our model even in the 14 TeV Large Hadron Collider (LHC) at CERN. However, still some 6 Since the lower bound of the DM mass is assumed to be larger than the tau lepton mass ≈ 1 GeV to simplify the neutrino mass formula in Eq. (II.17), the typical order could be larger than O(10) GeV. In our model, we find three charged scalar particles, k ±± , h ± 1 , h ± 2 coupling with the photon and the Higgs boson and their contributions modify the signal strength of the Higgs diphoton With taking into account the points, we can write down the following form like in the Zee-Babu model [66,[84][85][86], where the function δR(m x , λ Φx ) (x standing for a type of the charged particles) is defined with the loop functions A 0 , A 1/2 , A 1 as , (II.32) A 0 (x) = −x + x 2 f (1/x), (II.33) Here, τ i means 4m 2 i /m 2 h and the concrete form of f (x) is arcsin 2 √ x (when x ≤ 1) 7 . Here in our model, all the charged scalars are decoupled and contributions from these particles are negligible as µ γγ ≈ cos 2 α. It is very easy to estimate the lower bounds on | cos α| with 2σ confidence level from the ATLAS and the CMS experiments are 0.794 and 0.825, respectively, while no upper bound arises on | cos α|. This issue highly suggests that the value of cos α should be close to one. One possibility for realizing cos α ≃ 1 is that the mass of the additional CP-even scalar is very large (see Eq. (II.5)).
Next, we consider the prospects in future like-sign electron linear collider. For generating a doubly-charged scalar as an s-channel resonance, a like-sign linear collider is a fascinating option 8 . As discussed in [62], after accumulating a total luminosity of 50 fb −1 , more than a few tens of signal events of e − e − → k −− → ℓ − ℓ − can be expected for a doubly-charged scalar with m k ±± 10 TeV (even) with the center of mass energy √ s = 500 GeV in a like-sign electron linear collider, where possibly, we cannot reconstruct the mass of the doubly-charged scalar since this particle seems to be off-shell 9 . Even though detailed expectations depend on the interrelation among the matrix elements of y R , we can expect that our model and the Zee-Babu model are widely tested up to the parameter region with a large value in m k ±± (and also in m h ± 1,2 ). Now, we try to evaluate the prospects for the discovery of our model in a like-sign electron linear collider through the process e − e − → k −− → e − e − . We choose the template value of m k ±± as 20 TeV (being rather close to the minimum) and assume (y R ) ee = 1, these values are not favored for evading the bounds from lepton-flavor violation processes, but it would be realizable with fine tuning in the other elements of y R . For other values, we assign 10 −6 to circumvent the bounds. We consider the cases of √ s = 1, 3, 5, 10 TeV and ignore the two singly-charged scalars in calculating the decay width of k ±± . For estimating the production cross section, we implement our model with the help of FeynRules 2.1 [87,88] to generate the model file in the UFO format [89] for simulations in MadGraph5 aMC@NLO [90,91]. The values of σ e − e − →k −− →h − 2 h − 2 are 0.0959 fb, 0.898 fb, 2.71 fb and 16.9 fb, respectively. Then, 7 In our case, every particle including the W boson and the top quark inside the loop fulfills the condition, 4m 2 i > m 2 h , being equivalent to τ i ≥ 1. 8 Lots of works have already been done about the physics in a like-sign linear collider, e.g., see [92][93][94][95][96][97][98][99]. 9 Note that we can differentiate the the Zee-Babu model from SU (2) L triplet models (including a doublycharged scalar) by measuring the processes e − e − → ℓ − α ℓ − β with various final states and analyzing the patterns [62].
we can conclude that a multi-TeV like-sign electron linear collider could hold reasonable potential for exploring our model, at least in the specified choice. After accumulating a large amount of data, the case with (y R ) ee 1 could be reachable.

III. CONCLUSIONS
We have constructed a three-loop induced neutrino model with a global U(1) symmetry, in which we naturally accommodate a DM candidate. Taking into account for all the constraints, namely, the vacuum stability of these charged bosons, all the lepton fla- 150 TeV for the IH case, respectively. We have found that the NH case is prone to have wider allowed region.
In our model, the masses of the charged scalars tend to be very large as around O(10) TeV because of the required large trilinear couplings among these particles leading to the menace to the positivity of scalar quartic couplings at the one-loop level. After considering the stability of the DM, its mass is naturally bounded from above. In this sense, our DM candidate can naturally explain the observed relic density and the direct detection searches which typically lies on the O(100) GeV mass scale.
We have discussed possible collider-related issues, in which, the masses of the charged scalars tend to be very heavy as above 10 TeV that is apart from the Zee-Babu model.
Hence, they are highly decoupled from the SM particles and it is very difficult to detect a signature of our model even in the 14 TeV LHC. However, we expect that a future likesign electron linear collider could detect such heavy particles such as k ±± (and also h ± 1,2 , probably).