Extending two Higgs doublet models for two-loop neutrino mass generation and one-loop neutrinoless double beta decay

We extend some two Higgs doublet models, where the Yukawa couplings for the charged fermion mass generation only involve one Higgs doublet, by two singlet scalars respectively carrying a singly electric charge and a doubly electric charge. The doublet and singlet scalars together can mediate a two-loop diagram to generate a tiny Majorana mass matrix of the standard model neutrinos. Remarkably, the structure of the neutrino mass matrix is fully determined by the symmetric Yukawa couplings of the doubly charged scalar to the right-handed leptons. Meanwhile, a one-loop induced neutrinoless double beta decay can arrive at a testable level even if the electron neutrino has an extremely small Majorana mass. We also study other experimental constraints and implications including some rare processes and Higgs phenomenology.


I. INTRODUCTION
The massive and mixing neutrinos have been confirmed by the precise measurements on the atmospheric, solar, accelerator and reactor neutrino oscillations [1]. This fact implies the need for new physics beyond the SU (3) c × SU (2) L × U (1) Y standard model (SM). On the other hand, the cosmological observations have indicated that the neutrino masses should be in a sub-eV range [1]. In order to understand the small neutrino masses, people have proposed various ideas, among which the tree-level seesaw [2][3][4][5] mechanism is very popular [2][3][4][5][6][7][8][9][10][11][12]. However, the seesaw will not be easy to verify unless it is not at a naturally high scale. Alternatively, the neutrinos can acquire their tiny masses at loop order . These models for the radiative neutrino mass generation contain additionally charged scalars so that they may be tested at colliders.
In principle the neutrinos can have a Majorana nature [46] since they do not carry any electric charges. One hence can expect a neutrinoless double beta decay (0νββ) [47] process mediated by the Majorana electron neutrinos. This 0νββ process is determined by one unknown parameter m ee , i.e. the 1 − 1 element in the Majorana neutrino mass matrix, so that it can be seen in the running and planning experiments unless the m ee parameter is big enough [48,49]. However, there are other possibilities for a 0νββ process [5,10,[50][51][52][53][54][55][56][57][58][59][60]. For example, some left-right symmetric models for a linear seesaw of tree-level neutrino mass generation can offer a nonconventional tree-level 0νββ process with an testable lifetime, which simply depends on the scale of the left-right symmetry breaking rather than the details of the Majorana neutrino mass matrix [61]. One may also consider other models which accommodate an observable 0νββ process at tree level and then give a negligible contribution to the neutrino masses at loop order [62]. These 0νββ processes, which are related to quite a few arbitrary parameters, thus can be free of the constraint from the neutrino mass matrix [63][64][65].
It should be interesting if an enhanced 0νββ process originates from a tiny m ee . Some people have realized this scenario [37][38][39][40][41][42][43][44][45]. In a realistic model [37], after the SM Higgs doublet develops a vacuum expectation value (VEV) for spontaneously breaking the electroweak symmetry, a Higgs triplet without any Yukawa couplings can acquire an induced VEV up to a few GeV, meanwhile, its doubly charged component can mix with a doubly charged scalar singlet. Thanks to the gauge interactions, a two-loop induced Majorana neutrino mass matrix then can have a structure fully determined by the symmetric Yukawa couplings of the doubly charged scalar singlet to the right-handed leptons. As for the 0νββ process, it can appear at tree level through the same Yukawa interactions and the related gauge interactions.
In this paper we shall extend some two Higgs doublet models [66], where the Yukawa couplings for the charged fermion mass generation only involve one Higgs scalar, to generate the required neutrino masses and the enhanced 0νββ process. Specifically we shall introduce two scalar singlets among which one carries a singly electric charge while the other one carries a doubly electric charge. The singly charged scalar without any Yukawa couplings has a cubic term with the two Higgs scalars. The doubly charged scalar has the Yukawa couplings with the right-handed leptons, besides its trilinear coupling with the singly charged scalar. After the electroweak symmetry breaking, we can obtain a dominant Majorana neutrino mass matrix at two-loop and a negligible one at three-loop level. The symmetric Yukawa couplings of the doubly charged scalar to the right-handed leptons can fully determine the structure of the neutrino mass matrix. The 0νββ processes can be induced at tree, one-loop and two-loop level. The amplitudes of these 0νββ processes are all proportional to the electron neutrino mass. The one-loop 0νββ process can arrive at an observable level even if the electron neutrino mass is extremely small. We will also study other experimental constraints and implications including some rare processes and Higgs phenomenology.

II. THE MODELS
In the fermion sector, the quarks and leptons are as same as the SM ones, Here and thereafter the brackets following the fields describe the transformations under the SU (2) c × SU (2) L × U (1) Y gauge groups. The scalar sector contains two charged singlets, besides two Higgs doublets, We impose a softly broken discrete symmetry S so that the Yukawa couplings for generating the charged fermion masses will only involve one Higgs doublet. For example, we can take S = Z 2 under which one type or three types of the right-handed fermions and one Higgs scalar carry an odd parity while the other fermions and scalars carry an even parity. Alternatively, the S symmetry can be a global one. For instance, we can take S = U (1) X under which only one Higgs scalar is non-trivial. We further assume a softly broken lepton number, under which the doubly charged scalar carries a lepton number of two units while the Higgs scalars and the singly charged scalar do not carry any lepton numbers.
We then summarize the allowed Yukawa interactions as below, • Case-1 : The right-handed up-type quarks, down-type quarks and charged leptons couple to a same Higgs doublet, • Case-2 : The right-handed up-type quarks and down-type quarks couple to a same Higgs doublet, • Case-3 : The right-handed up-type quarks and charged leptons couple to a same Higgs doublet, • Case-4 : The right-handed down-type quarks and charged leptons couple to a same Higgs doublet, We also write down the full scalar potential with the following quadratic, cubic and quartic terms, Note only the µ 2 12 -term, the ω-term and the σ 12 -term can softly break the additional S symmetry. So, if the S symmetry is a global one, the λ ′′ 12 -term should be absent.

III. ELECTROWEAK SYMMETRY BREAKING AND PHYSICAL SCALARS
The two Higgs scalars φ 1,2 can be always rotated by For a proper choice of the rotation angle β, only one of the newly defined Higgs scalars χ and ϕ will develop a nonzero VEV to spontaneously break the electroweak symmetry. Without loss of generality, we can denote and then take This means the Higgs scalar ϕ will be responsible for the electroweak symmetry breaking.
In the base with χ = 0, we can rewrite the scalar potential to be Meanwhile, the Yukawa interactions can be expanded by • Case-1 : • Case-2 : • Case-3 : • Case-4 : After the Higgs scalar ϕ develops its VEV for the electroweak symmetry breaking, we can take The scalar potential (12) then should give the mass terms as follows, Now the singly charged scalars δ ± and χ ± mix with each other. Their mass eigenstates should bê where the rotation angle θ is determined by The charged fermions can get their masses through the Yukawa couplings involving the Higgs scalar ϕ, i.e. [1] Note the perturbation requirement • Case-1 : • Case-2 : • Case-3 : • Case-4 : Moreover, the cubic couplings ω and σ in the potential (12) should also match the perturbation requirement. Roughly speaking, we have

IV. NEUTRINO MASSES
As shown in Fig. 1, the left-handed neutrinos can obtain a Majorana mass term at two-loop level, It is easy to see that the two-loop neutrino mass matrix should have the structure as follows, Although the three-loop diagram given in Fig. 2 also generates the neutrino masses, its contribution should be much smaller than the two-loop contribution (29). So, the neutrino mass matrix can be well described by We calculate Here I 2 is given by the two-loop integral, where the function F 2 (x 2 1 , x 2 2 ) is defined by withq 1,2 = q 1,2 /m ξ ±± being the reduced momentum. In Fig. 3, we show the numerical results of the two-loop integral I 2 (x 2 1 , x 2 2 ) as a function of the difference x 2 − x 1 for a given x 1 . The lines from top to bottom correspond to x 1 = 0.1, x 1 = 1, x 1 = 10 and x 1 = 100, respectively. For a proper parameter choice, the two-loop integral I 2 can be of the order of O(1). Note that I 2 (x 2 1 , x 2 2 ) ≡ 0 for x 1 = x 2 . We should also keep in mind that a bigger x 2 − x 1 leads to a bigger I 2 but a smaller sin 2 2θ, see Eq. (21). The product I 2 sin 2 2θ thus cannot be very large. We hence would fail in enhancing the neutrino masses by choosing a bigger cubic coupling ω, which is not allowed to be much bigger than the charged scalar masses. In other words, the charged scalars cannot be far above the electroweak scale. Now the neutrino mass matrix has a structure fully determined by the symmetric Yukawa couplings f αβ = f βα (α, β = e, µ, τ ), i.e.
In turn, we can parametrize the Yukawa couplings f by the neutrino mass matrix,

V. NEUTRINOLESS DOUBLE BETA DECAY
It is well known that the electron neutrino with a Majorana mass can mediate a 0νββ process. This most popular 0νββ picture can be described by the effective operator as below, where q = 100 − 200 MeV is the transfer momentum. In the present model, the above 0νββ process actually is a two-loop effect since the neutrino masses are induced at two-loop level. As shown in Figs. 4a, our model can also generate the other 0νββ processes at tree level. The effective operators should be where the coefficients c 1,2,3 are defined by Furthermore, we can realize a 0νββ process at one-loop level. The relevant diagram is shown in Fig. 4b. The effective operator should be Here I 1 is given by the one-loop integral, ) as a function of the difference x 2 − x 1 for a given x 1 . The lines from top to bottom correspond to x 1 = 0.1, x 1 = 1, x 1 = 10 and x 1 = 100, respectively.
with the function F 1 (a, b) being a double integral, Clearly, Fig. 5, we show the numerical results ofĨ 1 ( ) as a function of the difference x 2 − x 1 for a given x 1 . The lines from top to bottom correspond to x 1 = 0.1, x 1 = 1, x 1 = 10 and x 1 = 100, respectively. The one-loop integral I 1 = I 1R − I 1I can be of the order of O(1) for a proper parameter choice.
From Eqs. (38), (39) and (41), we can conclude for a reasonable parameter choice. The 0νββ thus should be dominated by the one-loop contribution. The lifetime is determined by [67] 1 T 0ν where G 0ν is the phase space factor, M F,N and M GT,N are the nuclear matrix elements, while f V ≈ 1 and f A ≈ 1.
Similarly, we can give the effective Lagrangian for the muonium-antimuonium conversion µ − e + → µ + e − as below, For the rare three-body decay µ → 3e, it can be described by the effective Lagrangian, and then its decay width can be computed by By taking into account the SM result of the muon total decay width, we can read the branching ratio, Clearly, we can get the similar formula for the other rare three-body decays τ − → 3µ, µ + µ − e − , e + µ − µ − , µ − e + e − , µ + e − e − , 3e by replacing the related parameters m αβ in Eq. (52). We also consider the lepton flavor changing decay ϕ ϕ ϕ ϕ µ → eγ. The decay width should be and then the branching ratio, For simplicity, we do not show the similar formula for the other lepton flavor changing decays τ → µγ, eγ. We then calculate the muon anomalous magnetic momnet (g − 2) µ , i.e.

VII. SUMMARY
In this paper we have demonstrated the Majorana neutrino mass generation in some extended two Higgs doublet models, where the Yukawa couplings for the charged fermion mass generation only involve one Higgs scalar. In our scenario, a singly charged scalar without any Yukawa couplings has a cubic term with the two Higgs scalars. Another doubly charged scalar has the Yukawa couplings with the right-handed leptons, besides its trilinear coupling with the singly charged scalar. After the electroweak symmetry breaking, we can obtain the desired neutrino masses at two-loop level. The symmetric Yukawa couplings of the doubly charged scalar to the right-handed leptons can fully determine the structure of the neutrino mass matrix. Meanwhile, a one-loop 0νββ process can arrive at an observable level even if the electron neutrino mass is extremely tiny. We have also checked the other experimental constraints and implications including some rare processes and Higgs phenomenology.