Radiative neutrino mass via fermion kinetic mixing

We propose that the radiative generation of the neutrino mass can be achieved by incorporating the kinetic mixing of fermion fields which arises radiatively at one-loop level. As a demonstrative example of the application of the mechanism, we present the particular case of the Standard Model extension by $U(1)_D$ symmetry. As a result, we show how neutrino masses can be generated via a kinetic mixing portal instead of a mass matrix with residual symmetries responsible for the stability of multicomponent dark matter.

On the other hand, neutrino might be Dirac in nature, in which case it would require additional symmetries and different high energy realizations for the smallness of neutrino masses. Three years ago a systematic study of small Dirac neutrino mass realizations [25] at tree and one loop level have been performed. It has been shown that there are only and only four possible cases at tree level and two possible realizations at one loop order. Neutrino mass generation has also been realized in the context of left-right symmetric models [26,27].
The purpose of this work is to deviate from this canonical approaches to the neutrino mass problem which is achieved by relying on the new idea of fermion kinetic mixing we introduce and to show how it can be realized in the context of the neutrino mass generation with dark matter(DM), i.e. scotogenic scenario.
The paper is organized as follows: in section II we introduce and explain the idea of kinetic mixing, following with section III which describes the model and presents radiative neutrino mass generation via kinetic mixing of fermions, section IV gives the generation of the fermion and boson masses, DM candidates are discussed in section V, section VI concludes.

II. FERMION KINETIC MIXING
In order to present the idea of kinetic mixing for further generation of neutrino masses, we introduce the minimal set of fields in the conext of U (1)D gauge symmetry exntension of the SM. The fermion, call it AL with mA = 0, and the massive fermion CL, with mC = 0 are introduced. The Feynmann diagram representing the mixing mechanism is shown in Fig.1. As can be seen from the diagram, in order to complete the loop, we add the ΨL fermion field and a pair of scalars, namely s7 and s11. φ serves the purpose of U (1)D gauge symmetry breaking in a specific manner, such that there is a residual symmetry left which stabilizes the DM. This residual symmtry is also needed in order to prevent the collapsing of the loop down to tree level, aka to prevent the s7 and s11 obtaining the vacuum expectation values(VEV's). In our case it is achieved by choosing the specific QD charge assignments for the fields in the loop. Quantum number assignments for these fields are presented in the table I. The result of the diagram in produces the effective kinetic mixing between two fermion fields which leads to the lagrangian kinetic term where a represents loop structure and will be given in eq. (14). In order to bring kinetic part to the canonical form we consider all relevant kinetic terms L = ıĀL ∂AL + ıCL ∂CL In order to bring kinetic terms into canonical form, the first step is to rotate by π/4 so that the kinetic matrix can be diagonalized. When the kinetic terms are diagonal but still not properly normalized, we rescale or renormalize the corresponding fermion fields, so that the kinetic part becomes canonicaly structured.
After we have worked out the kinetic part, we need to diagonalize back the mass matrix of AL and CL fermions in the new basis. This rotation will differ from π/4 due to presence of rescaling. The final form of the relevant lagrangian is given by (4) where FaL and FcL fermions correspond to now properly normalized mass eigenstates. The relation between AL, CL and the canonicaly normalized mass eigenstates is given by and their mass eigenvalues are given by Here U (θ, ∆), unitary 2×2 transformation, and other relevant parameters given as shown below And finally a is given by where where Yukawa's are given with flavour indicies suppressed. Here mΨ is the Dirac mass of the Ψ Dirac fermion, s and c stand for sinus and cosinus respectively, and correspond to the mixing of s7 with s11 scalars, which is proportional to µ3 term in the lagrangian(see eq. (25)). The scalar mixing angles and mass eigenvalues, ms i[R/I] , are given in section IV, equation (43).

III. MODEL
In order to present a complete model for the kinetic mixing mechanism presented in section II, we We give all fermions as left handed fields. The last column in the table I shows the number of copies besides the flavor count. AL, CL fermions and ηL, ηD scalars are added for the generation of Dirac radiative neutrino mass mechanism (see Fig. 3), NL is needed as a Dirac mass partner for νL , aka to produce effective Dirac mass term m loop νLNL. Next, ΨL fermion and s7,11 scalars are needed for the kinetic mixing mechanism as was explained in section II. φ is introduced for the spontaneous symmetry breaking of the U (1)D dark gauge symmetry. It carries even QD charge in order to generate a residual Z2 symmetry(as will be shown at the end of this section) upon breaking of U (1)D. ΨR is added to produce a Dirac mass for ΨL. Lastly, NR, multiple copies of F2L and CL fermions are added to render the model chiral anomaly free. Even though, in general all five copies of CL can kineticly mix with one copy of AL, without loss of generality and for the sake of simplicity we take the bases of CL fermions in which only one particlar CL mixes with AL.
Considering triangular anomalies, SU (3)C × SU (2)L ×U (1)Y anomalies are canceled in the same way as in canonical SM case. Since there are no fermions that transform non-trivialy under SM and dark sector simultaniously, any cross anomalies be-tween SM and U (1)D are trivialy absent. The only anomalies to consider for cancelation are U (1)DGrav and [U (1)D] 3 . For this purpose, multiple copies of CL and F2L, N c R , and Ψ c R fields are added. Considering U (1)D sector anomalies, they cancel in the following way The interaction terms of the lagrangian for the GSM ⊗ U (1)D gauge symmetry case are given by where (−1) qx = −1 for H and φ, and 1 for the other scalars. Here a, b, etc. indicate the flavour index and run over 1,2,3, whereas α, β indicate the copy index, which exists only for CL and F2L, and run over 1-5 and 1-4 for CL and F2L respectively.
Here important points to notice are that L lepton doublet couples only to AL new fermion field, µ3 trilinear scalar term mixes s7 with s11 when φ = 0 which is needed for the kinetic mixing, λ Hηφ quartic scalar term which mixes η 0 L with ηD needed to generate Dirac radiative neutrino mass, and that all terms which could give AL fermion a mass, even at 1-loop order, are forbidden by symmetry and field content of the model.
Diagram representing radiative neutrino mass generation via kinetic mixing for GSM × U (1)D gauge symmetry is shown in Fig. 3. Crosses between A L and C L fields correspond to kinetic mixing given by diagram in Fig. 1.
The Dirac mass mixing of ν and NL, leading to Dirac neutrino mass (see eq. (31)) is given by where s, c correspond to the sinus and cosinus, respectively, of the mixing between η 0 L and ηD (eq. (40)) generated by the quartic scalar coupling λ Hηφ . The rest of the functions and parameters are defined as and mF c is given in eq. (8).
The plot showing how m loop depends on its parameters is depicted in Fig. 4. As can be seen the extra suppression from allows for wider range of masses, mass splittings and yukawa couplings. Here ∆m12 = m1 − m2 indicates the mass spliting between ηL,D scalar eigenvalues, whereas scalar and pseudo-scalar splitings are labeled by ∆mRI = mR − mI .
After the electroweak and dark symmetry breaking, due to symmetry and field content of the model, we obtain 2 residual dark Z2 symmetries whcih are not ad hoc. The first Z2 symmetry is analogous to the one from canonical Scotogenic model, but here is it obtained from U (1)D spontaneous symmetry breaking. The other Z2 symmetry is new and present here due to fractional charge assignments of the particles involved in the kinetic mixing which will be explained in the section II. This gives us opportunity for multicomponent DM case(as will be discussed in section V). Fields odd, (−1) Q D =odd , under the first Z2 symmetry include ηL, ηD, AL, CLα. The ones that are odd under the second Z2 symmetry are, the ones with fractional QD charges, s7, s11, ΨL,R. This is summeriezed in table II.

Fermion masses
Masses of SM fermions are generated identical to SM case, so we focus on mass genenations of  ∼ (+, +) even sector. Next, Z 1 2 ∼ (−) odd sector, similar to the one present in the canonical Scotogenic paper [22], (AL, CLi) fields. Lastly Z 2 2 ∼ (−) odd fermions, special for this model, due to the presence of kinetic mixing mechanism, aka Ψ fermion. Starting with Z2 even fermions we have, considering the La-grangian given in eq. (25), we get the following mass matrix for these new fermions and neutrinos where m loop is generated radiatively at 1 loop level, as shown in Fig. 3 and is given by eq. (26). Here we choose the basis for F2L in which the linear combination of four F2L that couples to NR apears in the mass matrix and the other 3 orthonormal combinations do not couple to NR. Determinant of the Z2 even fermion mass matrix is given by As can be seen, before U (1)D symmetry breaking, N is a vector-like fermion with a mass MN and neutrinos together with F2L are massless. Where as after U (1)D symmetry breaking we obtain one heavy Dirac fermion, mostly N , and most importantly neutrino combines with F2L to become a Dirac fermion. The eigenvalues, to the leading order in the m loop MN , YNF v φ limit, are approximately given by In the case if m loop does not provide enough suppression for the neutrino masses, the ratio of mN /v φ can provide extra suppression for the neutrino masses. For example if m loop ∼ 10 −4 GeV then mN /v φ ∼ 10 6 would give mν ∼ O(0.1eV). The other 3 F2L states, orthonormal to the F2L state coupled to NR fermion, obtain their masses radiatively through the diagram shown in Fig. 5. Next, considering the Z 1 2 ∼ (−) odd sector, AL remains massless till after neutrinos generate their masses, such that AL mass is generated through neutrino mass (m loop ) as shown in Fig. 6. mA is then given by where F (xi, xj) and x are given in eqs. (27,28). Fig. 7 shows the dependence of mA on its parameters. This is important point becuase if AL obtained its mass in some other way, neutrinos would generate their masses through AL's mass and the kinetic mixing would contribute in the sub-leading order and be unnessesary. One more important point to mention is that, this predicts one dark fermion to be naturaly lighter than the neutrino, since its mass is one loop suppressed with respect to the Dirac neutrino mass as can be seen from Fig. 6.
The five copies of CL dark fermions obtain their masses through φ at tree level by incorporating Lastly, considering the Z 2 2 ∼ (−) odd sector, we have only Ψ fermion which is vector-like and has an invariant mass of mΨ.

Boson masses
The only new vector gauge boson, corresponding to U (1)D gauge symmetry of dark sector, gets its mass through a cannonical higgs mechanism during spontaneous symmetry breaking of U (1)D gauge symmetry in the dark sector. Mass of dark U (1)D gauge boson is given by the corrsponding would-be Nambu-Goldstone boson is Im[φ]. Due to absence of scalars, with non-zero VEV, that simultaneously transform under GSM and dark U (1)D gauge symmetry, there is no tree level mixing between A µ D and SM neutral gauge bosons. Mixing will apear at one loop order through η ±,0 L running in the loop but it is loop suppressed and we will ignore the mixing here. The rest of gauge bosons obtain their masses just like in SM.
Coming to scalar sector. The charged higgs scalar from SM, H ± , corresponds to would-be Nambu-Goldstone boson and gets eatten up by W ± . The other electrically charged scalar, η ± L , part of ηL doublet needed for the neutrino mass generation, does not mix with H ± due to presence of Z 1 2 under which ηL ∼ − and H ∼ +. The charged scalar mass is given by Besides that, the real components of H 0 and φ mix with each other and after using scalar potential minimization conditions, ∂V /∂[H/φ] = 0 to eliminate m 2 H and m 2 φ , their corresponding 2×2 mass square matrix is given by with their corresponding eigenvalues and mixing angle given as Mass matrix corresponing to H 0 I and φI is given by zero, meaning that they become the longitutional degrees of freedom of Z and AD gauge bosons, as expected.
Real and imaginary components of η 0 L and ηD mix with each other, respectively. Their mass squared matrix is given by The corresponing eigenvalues and mixing angles are given by The mass spliting, which is needed for the non-zero neutrino mass matrix, of real and imaginary parts is accomplished by λ Hηφ HηLηDφ * operator, which is analogous to the η † H 2 operator from [22].
As the last piece, the s7 and s11 mix with each other, real and imaginary parts, respectively. Their mass matrix is given by The eigenvalues and mixing angles corresponding to s scalars are given by Here, the important point is the splitting of masses of real and imaginary parts which is needed for the kinetic mixing and generated in this case by operator µ3φs11s7.
As can be seen above, there are 3 separate scalar sectors that do not mix with one another. This is due to 2×Z2 symmetries present here. The first one is analogous to the canonical Scotogenic model and plays the same role here, whereas the second Z2 symmetry in this case is unique and is present here due to the kinetic mixing mechanism and the fractional charges of the particles involved. It can be thought of as a dark stabilizing symmetry for the dark sector within dark secotor of Scotogenic model. In this way, corresponding neutral scalars can be categorized as (H 0 , φ) ∈ {+, +}, (η 0 L , ηD) ∈ {−, +}, and (s7, s11) ∈ {−, −} under the 2 Z2 symmetries.

V. DARK MATTER
The GSM ⊗ U (1)D model discussed in section III can accomodate multicomponent DM scenario. The lightest of the particles that transform as Z 1,2 2 ∼ (−, +) is one component, the stability is provided by the Z 1 2 symmetry which is exact. Assuming mΨ > ms 7,11 , the second component is the lightest eigenstate of the s7, s11 sector, which transforms as Z 1,2 2 ∼ (+, −). In this case Ψ ∼ (−, −) under Z 1,2 2 would decay into lighter s7,11 eigenstates∼ (+, −) and AL ∼ (−, +) through YA,C yukawa couplings. Assuming that the lightest stable particle (LSP) of (+, −) sector is s, the dominant contribution to its relic abundance would be come from effective ss → HH diagram, where H is the SM Higgs scalar field. The quartic λ coupling between s and H would need to be suppressed to avoid elastic scattering between s and nuclei, which is mediated by sHH trilinear coupling. However, the trilinear coupling ssφ and φHH which are proportional to v φ are not suppressed and would contribute to ss → HH, assuming the mixing between φ and H is small enough to avoid the same elastic scattering off nuclei.
The LSP of the (−, +) sector is naturaly AL which mass is suppressed by 1-loop order with respect to m loop . If m loop is to be taken as 10 −4 GeV, mA can be of the order O(100−1000keV). The process contributing to its relic abundance would be ALAL → νν where ν is the Dirac neutrino. There would be no tree level bound from elastic scattering off nuclei. The detailed analysis for the relic density is beyond the scope of this work.

VI. CONCLUSIONS
We have presented the mechanism of neutrino mass generation via kinetic mixing in the context of the anomaly free SU (3)c ⊗SU (2)L ⊗U (1)Y ⊗U (1)D gauge symmetry. In this case Dirac netrino mass is generated after electroweak and U (1)D symmetry breaking via the kinetic mixing of 2 fermions in the dark sector. As a consequence the neutrino mass is naturaly suppressed by the radiative nature of the generation mechanism(similar to Scotogenic scenario). This model includes 2 dark sectors, first one is the same as in Scotogenic scenario and the second is unique to this mechanism, present here due to the kinetic mixing mechanism, wchich allows for the multiparticle DM scenario.
Despite presenting the particular example with GSM ⊗ U (1)D gauge symmetry, the kinetic mixing idea is more general and can be realized in cases with other gauge symmtries as well. In principle, the kinetic mixing of fermions does not need to be carried out in the dark sector and we could kineticaly mix neutrino with other fermion but this would require to include sterile neutrinos into the model, further more in this case in order to increase the neutral fermion mass matrix rank(give netrino a mass) one would need to follow the scenario like: neutrino mixes with another neutral fermion leading to mass generation of dark sector particle and then using the same diagram neutrino would get a mass from this same dark sector particle. These and other prospects and phenomenology is among further possible research directions of this work.