DOI: 10.22661/AAPPSBL.2018.28.5.47
The Mechanism of Neutrino Mass
HIROSHI OKADA
ASIA PACIFIC CENTER FOR THEORETICAL PHYSICS
ABSTRACT
We review several types of mechanisms for neutrino mass, which is much smaller than the masses found in the quark sector and in the chargedlepton sector. We will also show original results and their related phenomenologies.
INTRODUCTION
Several recent experimental results suggest that neutrinos definitely have nonvanishing masses, and oscillate among them. From a theoretical perspective, neutrino mass has historically been used to explore mechanisms, due to their tiny mass and neutrality under an electrical charge. The standard model (SM) cannot describe the origin of neutrino masses in a renormalizable theory, although the SM predictions are almost consistent with current experimental data, such as Large Hadron Collider (LHC) at the energy frontier.
One elegant scenario to explain the tiny neutrino masses is the seesaw mechanism, and the mechanism requires a high energy scale beyond the SM. The simplest achievement is to introduce heavier neutral fermions with righthanded chirality; Majorana fermions. Then the neutrino mass is suppressed by the ratio between the vacuum expectation values (VEVs) at electroweak scale ~100 GeV and the mass scale of heavier neutral fermions ~10^{15 }GeV, supposing order 1 Yukawa couplings. Thus, the order 0.1 eV of neutrino masses is realized, where neutrinos are also Majorana fields in light of the nature of heavier neutral fermions. In this case, one can successfully explain baryon asymmetry of the Universe (BAU) via "leptogenesis" as a big bonus [1]. In principle, the nature of the leptogenesis is to require violation of the lepton number in the Lagrangian that is typically caused by heavier Majorana fermions. In this sense, neutrinos are in favor of being Majorana particles. Also, a signal of neutrinoless double beta decay could be detected in an experiment [2], if the neutrino would be a Majorana particle. Thus I assume neutrinos to be Majorana fermions, and I will review some concrete scenarios to describe the neutrino mass below.
11. Canonical seesaw model [3]:
This is the simplest idea and has a long history. All we do is to introduce at least two Majorana heavy fermions in order to explain the current neutrino oscillation data. Below we fix its number to be three. Then the relevant Lagrangian is given by
where flavor indices are abbreviated for simplicity, H̃ ≡ iσ_{2 }H^{*}, σ_{2} being the second component of the Pauli matrices, and M_{N} (as well as the chargedlepton Yukawa coupling y_{l}) can be taken to be a diagonal basis without loss of generality. L_{L}=[ν_{L}, _{L}]^{T} is the gauge doublet lepton under SU(2)_{L} symmetry in the SM, while e_{R} and N_{R} are respectively the chargedlepton and Majorana (heavy) fermion with a gauge singlet. After the spontaneous electroweak (EW) symmetry breaking of the SM Higgs H, one finds the two by two symmetric block mass matrix of neutral fermions in the basis of [ν_{L}, N_{R}^{C}]^{T} as

where m_{D} consists of y_{D} and the vacuum expectation value (VEV) of H. Under the approximation m_{D} ≪ M_{N}, M can be blockdiagonalized, and its lighter part is given by m_{ν} ≡ m^{T}_{D} M_{N}^{1}m_{D}. Thus, the tiny neutrino mass matrix is given, when the hierarchy between M_{N} and m_{D} is large enough. Then, m_{ν} is diagonalized by the three by three unitary matrix U_{MNS}, as D_{ν}=U_{MNS} m_{ν} U^{T}_{MNS}, and D_{ν} and U_{MNS} have been almost measured by current experiments such as RENO [4], where the chargedlepton mass is expected to be diagonal.
NOTE:
When one replaces the SU(2)_{L} gauge singlet neutral fermion N_{R} into gauge triplet fermions Σ_{R} in the framework of a typeI seesaw model, the form of the neutrino mass matrix does not change. This is known as the "typeIII seesaw model" [5]. Although the lepton sector is similar to the typeI seesaw, different types of interactions for Σ_{R} are induced from the kinetic term and the Higgs potential due to the triplet SU(2)_{L} gauge symmetry, and one could distinguish the typeIII model from the other types of models.
12. TypeII seesaw [6]:
As another example to induce neutrino mass, we present a typeII seesaw model. The typeII seesaw model requires an isospin triplet boson ∆ with 1 under the U(1) hypercharge. Then one can write the following Lagrangian
After the EW symmetry breaking of ∆, the neutrino mass is given by m_{ν} ≡ y_{∆} v_{∆}, where v_{∆} is VEV of ∆. The scale of v_{∆} is restricted by the rho parameter; v_{∆}<O(1) GeV. Even though v_{∆} ≪ v~246 GeV, y_{∆}~O(10^{9}) is needed to obtain the neutrino mass. In this sense, ∆ physics could be more interesting.
13. Inverse seesaw model [7] and linear seesaw model [8]:
We will now describe two other types of seesaw models, the Inverse seesaw model and the Linear seesaw model, that are realized by three by three block mass matrices. The motivation of these scenarios comes from a policy that in theory both types of chiralities should be present; in particular, lefthanded neutral fermion S_{L}, as well as righthanded one N_{R}. In this sense, these theories are often discussed in larger gauge group theories such as
SU(2)_{L }× SU(2)_{R} [9], SO(10) [10], and so on, which will be realized in a larger energy scale. Here, apart from some specific details of the theories, let us start from the following form of the neutral mass matrix, on the basis of
[ν_{L}, N_{R}^{C}, S_{L}]^{T} :

Depending on the mass hierarchies among masses (m_{D}, m_{DS}, m_{NS}, μ_{S}), each of the resulting neutrino mass matrices is given by Inverse seesaw : m_{ν} ≈ m^{T}_{D }m_{NS}^{1} μ_{S}(m^{T}_{NS})^{1}m_{D}, for m_{DS} = 0; μ_{S} ≪ m_{D} ≤ m_{NS}, Linear seesaw :
m_{ν} ≈  m^{T}_{DS}(m^{T}_{NS})^{1}m_{D}  m^{T}_{D }m_{NS}^{1 }m_{DS}; μ_{S} = 0, m_{DS} ≪ m_{D} ≤ m_{NS}.
Nonunitarity [11]: At this point, we should mention the possibility of nonunitarity matrix U_{MNS}. This is typically parametrized by the form U'_{MNS} ≡(1FF^{†}/2) U_{MNS}, where F is a Hermitian matrix that is determined by each model, U_{MNS} is the three by three unitarity matrix, while U'_{MNS} represents the deviation from the unitarity. Then F is respectively given by Canonical seesaw : F = M^{1}m^{T}_{D}, Inverse(Linear) seesaw : F = (m^{T}_{NS})^{1}m^{T}_{D}. The global constraints are found via several experimental results such as the SM W boson mass M_{W}, the effective Weinberg angle θ_{W}, several ratios of Z boson fermionic decays, the invisible decay of Z, EW universality, the mixing matrix via quark sector that is corresponding to U_{MNS} in the lepton sector, and lepton flavor violations (LFVs). The result is evaluated by FF^{†}≤O(10^{5}~10^{3}).
Even though these scenarios can describe the neutrino sector very well, an extremely high energy scale is typically expected and it cannot directly be tested by any current experiments whose reachable energy scale is at most 100 TeV. In order to overcome this limitation, various models have widely been proposed by theoretical physicists from around the world. In the section below, I will introduce some interesting mechanisms at the low energy scale.
Fig. 1
14. Radiative seesaw model:
This mechanism provides neutrino mass not at the tree level but at the loop level. Thanks to the loop effect, the tiny neutrino mass can naturally be described. Furthermore, new particles inside the neutrino loop can be dark matter candidates (DMs), when they are neutral under the electric charges. Note here that DM is also one of the attractive mysteries beyond the SM, since it has not been directly detected by any current experiments yet, in spite of sustained efforts by experimentalists. Below, we will show a simple example of a radiative seesaw model [12].
First, we introduce a new Higgs boson (η), where all the quantum numbers are the same as the SM Higgs H. In addition, we impose a discrete Z_{2} symmetry for new particles; η and N_{R}. Namely, + of Z_{2} is assigned for SM particles, while  is assigned for the new one. Thus the y_{D} that appears in Eq.(1) is forbidden by Z_{2} symmetry. Instead, a new Lagrangian is given by
where η does not have nonzero VEV. Then, the resulting neutrino mass is generated at oneloop level in Fig.1. As can be seen in this figure, a neutral component of η or the lightest mass of N_{R} can be a DM candidate, because it does not decay into SM particles. Furthermore, one finds that this stability is assured by Z_{2} symmetry.
Here, one might be able to give an interesting interpretation of the origin of tiny neutrinos, that is, that neutrinos interact with DM that is likely to be a weakly interacting massive particle (WIMP). This interpretation provides not only a reasonable explanation of tiny neutrino mass but also provides a verifiable new physics such as DM in the low energy scale.
15. Neutrino model induced by higher representation of a SU(2) lefthanded gauge group:
Another intriguing way to generate tiny neutrino masses is to introduce Higgs bosons with higher representation of a SU(2) lefthanded gauge group, where neutrino masses are given by VEV of this Higgs boson. If the boson has a triplet or more than triplet representation, the VEV is restricted by the rho parameter even at the tree level that comes from deviation between the mass of neutral gauge boson Z and singlycharged gauge boson in SM. Namely, the typical mass scale of VEV is 1 GeV, which is lighter than the SM scale by two orders of magnitude, as discussed in the TypeII model.
In addition, we have several interesting features of higher representation fields.
The first one is that these fields can be welltested by current colliders such as LHC because of cascade decays of multielectric charged fields with quasidegenerated masses among the components, where the cascade decays come from the kinetic term. Notice here that their degeneracies come from the constraints of oblique parameters.
The second one is that the cutoff scale of SU(2) lefthanded gauge coupling g_{2} determines the valid energy scale of a theory, since the scale might blow up at lower energy such as TeV.
Therefore, one might insist that the theory can be valid up to a scale that reaches the current experiments, and testability is quite feasible.
Below, we show an interesting neutrino model with higher representations of the SU(2) lefthanded gauge group [13]. First of all, we prepare new fields in Table 1.

L^{a}_{L} 
e^{a}_{R} 
ψ^{a} 
Σ^{a}_{R} 
H_{2} 
H_{4} 
H_{5} 
SU(2)_{L} 
2 
1 
4 
7 
2 
4 
5 
U(1)_{Y} 
1/2 
1 
3/2 
0 
1/2 
3/2 
2 
Table I.
Then renormalizable Yukawa Lagrangian under these symmetries is given by
where the SU(2)_{L} index is omitted, assuming it is contracted to be gauge invariant, and y_{l}, M, and M_{Σ} are assumed to be diagonal matrices with real parameters. Each of the mass matrix (m_{l}, m_{D}) consists of (y_{l}, v) and (y_{D}, v_{5}) after the EW symmetry breaking, where (v, v_{4}, v_{5}) is respectively VEVs of (H_{2}, H_{4}, H_{5}). The rho parameter then gives the following relation: ρ ≈ (v^{2}_{2} + 7v^{2}_{4} + 10v^{2}_{5} ) / (v^{2}_{2} + v^{2}_{4} + 5v^{2}_{5} ), where the experimental value is given by ρ = (1.0000  1.0007) at at 2σ confidence level [14]. To satisfy the constraint, we have the following solutions; (v, v_{4}, v_{5}) (245.9, 1.67, 1.72) GeV.
Neutral fermions: The neutral fermion mass matrix in terms of the neutrino sector (ν_{L}, ψ^{0C}, ψ^{0})^{T} is given by
where μ_{L}_{(}_{R}_{)} is given by v^{2}_{4} f_{L}_{(}_{R}_{) }M_{Σ}^{1 }f^{T}_{L}_{(}_{R}_{)} on a manner analogical to the canonical seesaw mechanism, as shown in Fig. 2.
Fig. 2
Then the active neutrino mass matrix can approximately be found as m_{ν} ≈ m^{*}_{D }M^{1 }μ^{*}_{R}(M^{T})^{1}m^{†}_{D}, where μ_{L}_{/}_{R} < m_{D} ≪ M is naturally expected due to the constraint of the ρ parameter and the seesawlike mechanism of μ_{R}_{/}_{L}. We thus obtain correlation in terms of the size of neutrino mass and other mass parameters such as m_{ν} ≈ (v_{4}/M_{Σ})(v_{5}/M)^{2} f^{2}_{R }y^{2}_{D} v_{4}. Note that M_{Σ} and M cannot be much larger than the TeV scale, since v_{4} and v_{5} are in the GeV scale, which requires a perturbative limit for Yukawa coupling constants. Now, one finds that the hierarchies μ_{L}_{(}_{R}_{) }≪ m_{D }< M, M_{Σ} are naturally realized. Thus, the neutrino mass scale can be generated with natural parameter spaces.
Beta function of SU(2)_{L} gauge coupling g_{2} : Due to larger representational fields, a large contribution to the renormalization group equation(RGE) will be provided. In our case, the cutoff scale is of the order 1 TeV~ 100 TeV, depending on the threshold masses 0.5 TeV~5 TeV. It implies this scenario is valid for the 1~100 TeV scale, and that the theory will be drastically changed for energy that is greater than 1~100 TeV scale.

L^{a}_{L} 
e^{a}_{R} 
ψ^{a} 
Σ^{a}_{R} 
H_{2} 
H_{5} 
SU(2)_{L} 
2 
1 
4 
7 
2 
5 
U(1)_{Y} 
1/2 
1 
1/2 
0 
1/2 
0 
Table 2.
16. A hybrid model:
A radiative seesaw model in 14 and a higher representation model in 15 can be combined, retaining both models' features.
Moreover, a wellcombining model sometimes supplies an additional feature such that we do not need any additional symmetries that would not achieve the typical radiative seesaw model with lower representation of the SU(2) lefthanded gauge group. Below, we show an intriguing scenario [15].
First of all, let us introduce vectorlike quartet fermions ψ with 1/2 U(1) hypercharge, and righthanded quintet fermions Σ_{R} and boson H_{5} with zero U(1) hypercharge, as shown in Table 2. Note here that H_{4} has not been mentioned that it is very important to obtain a radiative neutrino mass model and the big difference between this model and 15. If H_{4} would be there, one could write the term of [H_{4}^{*}H_{2}H_{2}H_{2}] in model 15 that will lead to nonzero VEV of H_{4}. Therefore, H_{4} cannot be an inert boson unless additional symmetry would be imposed. On the other hand, H_{5} does not have such kind of terms.
Under these field contents and assignments, the renormalizable Yukawa Lagrangian is given by
where (y_{l}, M_{D}, M_{Σ}) are assumed to be diagonal matrices with real parameters without loss of generality. Then active neutrino mass is generated at the oneloop level, where ψ_{α} and H_{5} propagate inside a loop diagram, and then were just replaced by N_{R} and η in Fig.1. As a result, the active neutrino mass matrix is given by m_{ν} ~ f M f^{T} F(m_{H}_{5}, M) / (32π^{2}), where M is nineheavier neutral fermion mass eigenstate in basis of (ψ_{0}, ψ_{0}^{C}, Σ_{0})^{T}, m_{H}^{5} is the mass eigenstate of neutral component of H_{5}, F is a loopfunction~1, and 32π^{2} in the denominator comes from the loop factor.
Beta function of SU(2)_{L} gauge coupling g_{2} : This model also has a large contribution to the RGE. In our case, the cutoff scale is of the order 100 TeV~ 1 PeV, depending on the threshold masses 0.5 TeV~5 TeV. Compared with 15, the cutoff scale increases. This is a direct consequence of fewer fields with higher order representations.
Other aspects and summary of this model:
Thus, DM stability is automatically assured by a remnant symmetry after the spontaneous EW symmetry breaking, where the neutral component of Σ_{R} is the one. Here, we have found a fermionic DM with 10 TeV mass originating from a neutral component of quintet fermions [16]. Additionally, a neutrino mass matrix is induced at the oneloop level as a dominant contribution, because the neutral component of a quintet Higgs boson does not have VEV in a renormalizable theory, which is also assured by the remnant symmetry of the SU(2) lefthanded gauge group. Because of several fields with higher representations, the valid cutoff scale is fixed to be 100 TeV in the analysis of renormalization group equation of g_{2}. In addition, we have discussed other issues; LFVs, the muon anomalous magnetic moment (muon g2), which should frequently be discussed, where LFVs are nothing but constraint from current experiments, while muon g2 has large deviation from the SM prediction. Even though muon g2 cannot completely be explained by our model, it could be an attractive scenario because a lot of features are involved in a rather simple framework.
We hope that we have clearly explained several ways of generating tiny neutrino mass and their related phenomena, even though these models that we have previously discussed in the article are just small parts of the neutrino scenarios. However, since there are still a number of unresolved issues experimentally and/or theoretically, we are excited to continue to explore and contribute in developing this field.
Acknowledgements: I would like to express my sincere gratitude toward my Korean colleagues (especially the extremely talented and warmhearted supporting staff members at APCTP), and for the incredibly hospitable research environment at APCTP.
And I would be very happy if I could have large contribution to continued success and prosperity of APCTP and Korean physics society.
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Hiroshi Okada is a junior research group leader (JRG) at the Asia Pacific Center for Theoretical Physics (APCTP). After receiving a D.Sci from the University of Kanawaza, he worked at the British Univ. in Egypt, the Korea Institute for Advanced Study (KIAS), and the National Center for Theoretical Sciences (NCTS) in Taiwan before joining APCTP in 2018. His research field is particle physics phenomenology. 
