Majorana neutrino masses in the scenario of gauge–Higgs unification

for the Majorana masses, which is the counterpart of the well-known SU ( 2 ) L × U ( 1 ) Y invariant higher-mass-dimensional ( d = 5) operator. As the ﬁrst possibility we discuss the seesaw mechanism by assigning leptonic ﬁelds to the adjoint representation of the gauge group, so that a d = 5 gauge-invariant operator can be formed. It turns out that the mechanism leading to the small Majorana masses is the admixture of the Type I and Type III seesaw mechanisms. As the second possibility, we consider the case where the relevant operator has d = 7, by introducing a matter scalar belonging to the fundamental representation of the gauge group. Reﬂecting the fact that the mass dimension of the operator is higher than usually expected, the Majorana masses are generated by a “double seesaw mechanism.”


Introduction
The standard model (SM) possesses a few serious theoretical problems. A well-known important problem is that of gauge hierarchy. The attempts to solve this problem have led to representative scenarios of physics beyond the standard model (BSM). The most well-studied scenario is supersymmetry, whose concrete realization is minimal supersymmetric standard model (MSSM). In this paper we focus on the scenario of gauge-Higgs unification (GHU), where the Higgs boson is originally a gauge boson. To be precise, the Higgs field is identified with the (Kaluza-Klein (KK) zero mode of) an extra-dimensional component of a higher-dimensional gauge field [1][2][3][4]. A nice feature of this scenario is that, by virtue of the higher-dimensional local gauge symmetry, the quantum correction to the Higgs mass is UV-finite, thus opening a new avenue for the solution of the hierarchy problem [5].
Another basic theoretical problem in the standard model is that there is no principle to restrict Higgs interactions, such as Yukawa couplings. Namely, there is no guiding principle to determine the quark and lepton masses theoretically. From this viewpoint, again the GHU scenario is hopeful: the GHU scenario may provide a natural mechanism to restrict Higgs interactions, relying on the gauge principle. Let us note that in GHU, Yukawa couplings are originally gauge couplings since the Higgs field is originally a gauge field.
Neutrino masses are expected to play special roles in the investigation of the viability of the various BSM scenarios. First, it should be noticed that if neutrinos are assumed to be Majorana fermions, the neutrino mass matrix (in the basis of weak eigenstates) is directly determined by the observed neutrino mass eigenvalues, generation mixing angles, and (physical) CP phases, some of them having PTEP 2018, 073B01 K. Hasegawa and C. S. Lim already been fixed (with some errors) or restricted experimentally. Thus it is possible to compare the prediction of each BSM scenario with such determined mass matrices. This is in contrast to the case of quark mass matrices; here, because of the freedom of unitary transformations in the sector of right-handed quarks, even though we know all of the observables mentioned above, the mass matrices cannot be uniquely fixed.
It should also be noticed that the mass matrices of the lepton sector show very characteristic features: neutrino mass eigenvalues are remarkably small compared to those of quarks and charged leptons. Also impressive is that (two of the) generation mixing angles in the Maki-Nakagawa-Sakata matrix are considerably greater than the corresponding angles in the Kobayashi-Maskawa matrix. These interesting features may also have their origin in the Majorana nature of neutrinos. Let us recall that only neutrinos, being electrically neutral, can be Majorana fermions without contradicting charge conservation.
Based on these observations as the first step, in this paper we study systematically how small Majorana neutrino masses can be realized in the GHU scenario. In the literature, the most popular mechanism for realizing small neutrino masses is the seesaw mechanism [6][7][8]. In this paper, we will propose possible models to realize this idea concretely in the framework of GHU.
What is special for the GHU scenario in the discussion of Majorana neutrino masses? We may easily note that to realize the aforementioned mechanisms for inducing small Majorana neutrino masses in GHU is a little challenging. First, since the Higgs field is originally a gauge field belonging to the adjoint representation of the gauge group, it is non-trivial to form a gauge-invariant operator with mass dimension d = 5 (from a four-dimensional (4D) point of view) corresponding to the well- where L is a left-handed lepton doublet and φ is the Higgs doublet. Also, the Yukawa coupling coming from the covariant derivative of higherdimensional gauge theory usually preserves fermion number, and to break the lepton number is a non-trivial task, though if we extend our discussion to the grand GHU [9][10][11], the gauge interactions there may lead to the violation of baryon and/or lepton number.

Seesaw mechanism in the GHU scenario
We discuss how the seesaw mechanism is realized in the GHU scenario, by taking the minimal unified electro-weak GHU model, i.e. the 5D SU(3) model [12,13]. The extra dimension is assumed to be an orbifold S 1 /Z 2 in order to break SU(3) into the gauge group of the SM and also to realize a chiral theory. Let us note that in GHU the gauge group of the SM should inevitably be enlarged, and the simplest choice is SU(3). The Higgs field behaves as an octet, the adjoint representation of SU (3). Then, in this model, assigning leptonic fields in an SU(3) triplet will be unrealistic. First, the charge assignment in this model is such that the fields in the triplet all have fractional charges, being identified with those of quark fields. (The situation will change if the gauge group has an additional U(1) factor [14,15].) Second, the d = 5 operator (A y L) 2 (L: lepton triplet, A y : the fifth component of the 5D gauge field) contained in (D y L) 2 (D y denotes the gauge covariant derivative), which should be the counterpart of (φ † L) 2 , clearly cannot be gauge invariant.
For these reasons, we assign lepton fields to an SU(3) octet , whose component fields have integer charges. Also, by taking this choice of representation, we can immediately find a d = 5 operator Tr{[A y , ] 2 } stemming from a gauge-invariant operator Tr{(D y ) 2 } (with spinor indices being omitted for brevity), responsible for the Majorana mass of ν L . One may wonder why we do not introduce an SU(3) singlet field to be identified with ν R . Let us note that such a ν R cannot form a Dirac mass with ν L through the vacuum expectation value (VEV) of A y , since ν R and ν L belong 2/9 PTEP 2018, 073B01 K. Hasegawa and C. S. Lim to different representations of gauge group and therefore cannot communicate with each another through Yukawa coupling. The octet possesses both the SU(2) L doublet containing ν L and the SU(2) L singlet containing ν R in a single representation.
In order to complete the seesaw mechanism, in addition to the Dirac mass term mentioned above, the Majorana mass term for ν R is needed. At first glance the Majorana mass term seems to be provided by a gauge-invariant operator Tr 2 . Unfortunately, the story is not so straightforward. First, although the adjoint representation is a real representation and therefore it seems to be natural to assign Majorana particles to this representation, it is known that we cannot regard each component of as an ordinary 4D Majorana field, satisfying the Majorana condition ψ c = ψ with ψ c = C(ψ) t (C = iγ 0 γ 2 ) being 4D charge conjugation, even though the number of components of the spinor is the same as in the case of 4D space-time. It may be worth noting the fact that there does not exist a Majorana spinor in 5D space-time. In fact, even if we try to form a Majorana mass term for a generic four-component spinor ψ, ψ c ψ, the mass term is known to be non-invariant under the 5D Lorentz transformation, which connects 4D space-time coordinates with the extra space coordinate.
Interestingly, we realize that if we add γ 5 to the mass term to form ψ c γ 5 ψ, the modified mass term turns out to be invariant under the full 5D Lorentz transformations. So, the linear combination ψ + γ 5 ψ c seems to be a self-conjugate spinor, correctly transforming under the 5D Lorentz transformation. Unfortunately this is not the case, since γ 5 (γ 5 ψ c ) c = −ψ. However, this in turn means that once we form an eight-component spinor ψ SM , it is self-conjugate in the following sense: ψ SM represents a "symplectic Majonara" spinor [16,17]. Just as the 5D SUSY gauge theory can be naturally obtained from the SUSY (pure) Yang-Mills theory in 6D space-time by naive dimensional reduction, it may be useful to construct a Lagrangian for ψ SM , as if the space-time is 6D, and then perform a naive dimensional reduction to 5D. Adopting the following basis for 6D gamma matrices, (with the 6D chiral operator being given by 7 = I 4 ⊗ σ 3 ), the symplectic Majorana condition reads In 5D space-time with the S 1 /Z 2 orbifold as the extra dimension, the Z 2 transformation is a sort of chiral transformation from the 4D point of view, and hence ψ and ψ c should have opposite Z 2 parities. Thus, the 4D Majorana spinor is not compatible with the orbifolding. (This is a reflection of the fact that in 4D space-time there is no Majorana-Weyl spinor.) For the symplectic Majorana spinor, the Z 2 transformation can be modified into that in the orbifold T 2 /Z 2 , the extra dimension of 6D space-time: where y is the extra-dimensional coordinate and the 3 × 3 matrix P defines the Z 2 -parities of each component of the fundamental representation of SU (3), i.e. triplet, as Since −i 5 6 = γ 5 ⊗ σ 3 , ψ and ψ c are now allowed to have opposite 4D chiralities, as they should. The transformation −i 5 6 is a rotation of an angle π in the 2D extra dimension. So, it should be equivalent to ordinary Z 2 transformation after the dimensional reduction to the 5D space-time. In fact, it is easy to check that the bilinear form ψ SM M ψ SM for M = μ (μ = 0-3), 5, the transformation in Eq. (5) is equivalent to the transformation without 6 . If the 4D spinor ψ of Eq. (1) only has the left-handed Weyl spinor ψ L , for instance as the result of the orbifolding mentioned above, which just reduces to the four-component 4D Majorana spinor ψ M : Then the mass term for ψ SM just reduces to the 4D Majorana mass term for ψ M : where ψ SM = ψ † SM 0 , while ψ M = ψ † M γ 0 . Now we are ready to discuss our model more concretely. The SU(3) octet contains symplectic Majorana spinors ψ a SM (a = 1-8) as its component fields: where λ a (a = 1-8) are Gell-Mann matrices. The free Lagrangian for with Majorana mass M , after naive dimensional reduction into 5D space-time, is given as Let us note that the condition in Eq. (4) is compatible with 5D Lorentz transformation, but not compatible with the Lorentz transformation connecting a sixth (extra space) coordinate with 5D coordinates, reflecting the fact that in 6D space-time there does not exist a Majorana spinor. So, Eq. (11) is invariant only under 5D Lorentz transformation, which is sufficient for our purpose. The gauge interaction of is described by As the result of the orbifolding, the sector of the KK zero mode is given as follows (we show only the part with +1 eigenvalue of 7 , (+) ): where N γ , N Z are associated with the generators, which are identical to those for the neutral gauge bosons γ , Z, and hence both are mixtures of the SU(2) singlet (associated with λ 8 ), corresponding to ν R , and triplet (associated with λ 3 ) leptons. Also relevant is the KK zero mode of the extra-dimensional component of the gauge field, A y : where (φ + , φ 0 ) is nothing but the Higgs doublet in the SM, whose VEV, , spontaneously breaks the gauge symmetry of the SM through the Hosotani mechanism [2][3][4]. Now let us move to the discussion of how the seesaw mechanism is realized in this model. For that purpose, we restrict our discussion to electrically neutral leptons. Our task is to realize the small Majorana mass for ν L , belonging to the SU(2) doublet L = (ν, e − ) L , through the seesaw mechanism. For the mechanism to work, the "exotic" left-handed doubletL = (ν,ẽ + ) L is redundant, as it does not exist in the standard model. Also, if it remains in the low-energy effective theory, it will form a gauge-invariant Dirac mass term with the doublet (ν, e − ) of our interest, The coefficient M , supposed to be the mass scale of the ν R Majorana mass, is assumed to be much larger than the weak scale M W for the seesaw mechanism to work, and hence ν L will decouple from our low-energy world. A possible way out of this problem is to introduce a brane-localized SU(2) doublet L b = (ν b , e + b ) R to form a brane-localized Dirac mass term at one of the fixed points of the orbifold (where the gauge symmetry is reduced to that of the SM by the orbifolding), The "brane-localized mass" M b is assumed to be much larger than the Majorana mass M , M b M . In an extreme limit, M b → ∞,ν is completely decoupled from the theory forming a Dirac mass with ν b , thus leaving ν alone as a massless state. By the way, all the fields appearing in Eqs. (13)- (16) should be understood to be 4D fields with proper mass dimension and kinetic terms after the dimensional reduction to 4D space-time.
After the spontaneous gauge symmetry breaking due to φ 0 = v √ 2 , ν L belonging to the SU(2) doublet L forms a Dirac mass term of order gv ∼ M W with a right-handed neutral lepton in Eq. (13), behaving as either a singlet or a triplet of SU(2) L (2 × 2 = 1 + 3). It turns out that the partner of ν L to form the Dirac mass is N Z (not N γ ). This is basically because the VEV of A y is electrically neutral From these lessons, we learn that though we have five neutral leptonic states to start with, ν L ,ν L , ν bR , N γ R , N ZR , in the limit M b → ∞, the Majorana mass of ν L is effectively determined by the diagonalization of the mass matrix M 2×2 in the basis of subsystem (ν L , (N ZR ) c ): The The physical reason for getting the small Majorana mass for ν L is the decoupling of N Z due to its large Majorana mass. An important remark here is that the state N Z is a mixture of SU(2) singlet and triplet states. Namely, the seesaw mechanism operating in this model is the admixture of two types of seesaw mechanism, i.e. Type I [6][7][8] and Type III [18].

Majorana neutrino masses due to higher-mass-dimensional operator
As was discussed in the introduction, in GHU it seems to be impossible to form a gauge-invariant operator corresponding to the dimension d = 5 and SU(2) L ×U(1) Y invariant operator, (φ † L) 2 , when the lepton doublet L is assigned as a member of the fundamental representation of the gauge group. Actually, this is based on our implicit assumption that only the extra-dimensional component of the gauge field A y is the field developing the VEV, which breaks the gauge symmetry of the standard model. Once we relax this constraint and allow the introduction of a scalar field, which also develops a VEV, the situation will change. If the VEV of the introduced scalar is a singlet concerning the gauge group of the SM, it has nothing to do with the weak scale M W and hence the gauge hierarchy problem. By introducing such a scalar field, together with the lepton multiplet and A y , it will become possible to form a gauge-invariant operator with mass dimension d higher than 5, typically 7.
As an example to make this idea concrete, we discuss a 5D SU(4) unified electro-weak GHU model. In this model, the Higgs doublet is contained in the triplet representation of the subgroup SU(3), in contrast to the case of the SU(3) model in the previous section, where the Higgs doublet is a member of the SU(3) octet. As a result, in this model the predicted weak mixing angle is a desirable one, sin 2 θ W = 1/4 [19]. The fields responsible for the Majorana neutrino mass generation are denoted as follows: We have introduced a scalar field . Leptons have now been assigned to the fundamental representation of SU(4), whose chirality is tentative and will be fixed by the orbifolding discussed below. We also introduce a gauge singlet fermion χ 0 . The Lagrangian, relevant for the neutrino mass, is where (y), with y being the coordinate of the extra space assumed to be an orbifold S 1 /Z 2 , is the "sign function": (y) = 1 and −1 for positive and negative y, respectively, and the "Z 2 -odd" bulk mass term (y)M bψ ψ causes exponential suppression of theYukawa coupling f g(π RM b )e −π RM b (R: the radius of S 1 ), which is desirable in order to get a small Dirac mass, relevant for the lighter (first or second) generation.
The remaining KK zero modes as the result of the orbifolding are We have put the overall phase as −1 for in its Z 2 transformation, so that onlyŝ 0 has the even parity (+, +).
The higher-mass-dimensional (d = 7) gauge-invariant operator relevant for the Majorana mass of ν L is known to be It is easy to confirm that this operator does contribute to the Majorana mass after A y and are replaced by their VEV: Here some comments are in order on the issue of how the scalar field can develop its nonzero VEV, V . One possibility is just to add a gauge-invariant potential term for , −μ 2 † + λ( † ) 2 (μ 2 , λ > 0). We notice, however, that this may cause a problem, since the mass-squared term for is not protected by any symmetry and the radiatively induced VEV, V , may be quite large (unless we perform some fine-tuning). This in turn may cause too-large spontaneous symmetry breaking of SU(4) (though the breaking of SU(4) itself is realized by orbifolding, as was discussed above), leading to a too-large positive mass-squared for the Higgs doublet, belonging to the "broken generator" of SU(4), through the term (D M ) † (D M ). Thus, realizing the spontaneous breaking of the SM gauge symmetry and getting a small Higgs mass becomes non-trivial.
Hence, a desirable alternative choice would be to embed the field as part of A y by adopting a larger gauge group, so that the potential of becomes under control thanks to the higher-dimensional gauge symmetry. Two VEVs, V and v, both being generated by the Hosotani mechanism [2][3][4], may be naturally comparable in their orders of magnitude.
After the spontaneous breaking, the mass terms relevant for ν L can be read off from the Lagrangian in Eq. (19), and are summarized here in a form using a mass matrix:  We assume a hierarchical structure, fv αV M . fv αV is achieved by the exponentially suppressed small Yukawa coupling f , and the mass scale M can be much larger than v and V , since it is a singlet with respect to the SM gauge symmetry. Then, the diagonalization of the 3 × 3 mass matrix is straightforward. Namely, by an orthogonal rotation in the 2 × 2 submatrix for the lower two components with the small angle of O(αV /M ), we get the approximate form ⎛ We now immediately get three approximated mass eigenvalues (their absolute values), M , (αV ) 2 M , and which is identified with the Majorana mass of ν L . In this derivation we have made the additional assumption (fv)M (αV ) 2 . Here are two steps of a seesaw-like mechanisms, which are seen schematically in Fig. 1. Figure 1 clearly shows that the operator giving rise to the Majorana mass is the one given in Eq. (21). A similar mechanism to get the small Majorana neutrino mass has been discussed in Ref. [20].

Summary and discussion
In this paper we have considered possible mechanisms to generate small Majorana neutrino masses for (active) left-handed neutrinos in the scenario of gauge-Higgs unification, one of the attractive scenarios of physics beyond the standard model. A specific feature of this scenario in the construction of the Majorana mass term is that it is non-trivial to find an operator, responsible for the Majorana neutrino masses, which is a counterpart of the SU(2) L × U(1) Y invariant higher-mass-dimensional d = 5 (from 4D point of view) operator (φ † L) 2 (L: left-handed lepton doublet, φ: Higgs doublet). For instance, in the minimal unified electro-weak SU(3) GHU model [12,13], we cannot get a gauge-invariant operator by just replacing the lepton doublet L by a triplet field, the fundamental representation of SU(3), since the Higgs field in this scenario, corresponding to φ, is A y (the extradimensional component of the gauge field), which of course belongs to the adjoint representation of the gauge group.
As the first possible mechanism to generate small Majorana neutrino masses we discussed the seesaw mechanism [6][7][8]. Leptonic matter fields are assigned to the adjoint representation of SU(3), i.e. the SU(3) octet, so that the component fields have integer charges and a d = 5 operator Tr{[A y , ] 2 } stemming from a gauge-invariant operator Tr{(D y ) 2 } can be formed. Though the Majorana spinor 8/9 PTEP 2018, 073B01 K. Hasegawa and C. S. Lim seems to fit naturally to the octet, i.e. the real representation of the gauge group, it has been known that in 5D space-time there does not exist a Majorana spinor. Thus we formulated the Lagrangian for leptons by use of the symplectic Majorana spinor [16,17], which has eight components and naturally fits 6D space-time. Interestingly, in our model the partner of ν L to form a Dirac mass is the admixture of SU(2) L singlet (corresponding to ν R ) and triplet fermions. Thus the mechanism operating in this model based on the GHU scenario turns out to be the admixture of Type I [6][7][8] and Type III [18] seesaw mechanisms.
As the second possibility we considered the case where Majorana neutrino masses are generated in a form of higher-mass-dimensional (d > 5) gauge-invariant operator. We argued that once the implicit constraint that the VEV to break the gauge symmetry should be given only by the VEV of A y is relaxed, introducing a matter scalar belonging to the fundamental representation of the gauge group (together with an additional singlet fermionic field), we can form a higher-mass-dimensional (d = 7) gauge-invariant operator, responsible for the Majorana neutrino masses. Reflecting the fact that the relevant operator has a mass dimension than usually expected, the Majorana neutrino masses are generated by, say, the double seesaw mechanism.