Dynamical Breaking to Special or Regular Subgroups in $SO(N)$ Nambu--Jona-Lasinio Model

It is recently shown that in 4D $SU(N)$ Nambu--Jona-Lasinio (NJL) type models, the $SU(N)$ symmetry breaking into its special subgroups is not special but much more common than that into the regular subgroups, where the fermions belong to complex representations of $SU(N)$. We perform the same analysis for $SO(N)$ NJL model for various $N$ with fermions belonging to an irreducible spinor representation of $SO(N)$. We find that the symmetry breaking into special or regular subgroups has some correlation with the type of fermion representations; i.e., complex, real, pseudo-real representations.

Recently, in Ref. [20], present authors have examined dynamical symmetry breaking pattern in 4D SU (N ) Nambu-Jona-Lasinio (NJL) type models in which the fermion belongs to an irreducible representation of SU (N ) n or n(n − 1)/2. The potential analysis has shown that for almost all cases at the potential minimum, the SU (N ) group symmetry is broken to its special subgroups such as SO(N ) or USp(N ) when symmetry breaking occurs. (Note that special subgroups may be referred to as non-regular or irregular subgroups depending on literature. For more information about Lie subgroups, see, e.g., Refs. [28][29][30][31][32].) Also, in Ref. [19], one of the author (T.K.) and J. Sato have performed the same analysis in a 4D E 6 NJL type model in which the fermion belongs to an irreducible representation 27 of E 6 . The potential analysis showed that for all cases at the potential minimum the E 6 group symmetry is broken to its special subgroups such as SU (3), USp(8), G 2 , F 4 when symmetry breaking occurs. These results clearly show that the symmetry breaking into special subgroups is not special at least in 4D NJL type models.
One might think that the above results may highly depend on a particular feature of the NJL type models, that is, fermion pair condensation in a specific type of effective theories, so the symmetry breaking into special subgroups may still be special, e.g., in Higgs mechanism [4][5][6]. In Ref. [33], however, L.-F. Li investigated the symmetry breaking patterns in gauge theories with Higgs scalars in several representations. The analysis showed that SU (n) symmetry is broken to its regular subgroups for the cases of Higgs field in an SU (n) fundamental and adjoint  Table 1: SO(N ) spinor representations: R, PR, C stand for real, pseudo-real, and complex representations. For even N , there are two irreducible spinors, denoted here as r and r ′ (orr), with opposite 'chirality'.
representation, while SU (n) symmetry is broken to its regular or special subgroups for the cases of Higgs field in an SU (n) rank-2 symmetric or anti-symmetric tensor representation. Therefore, the symmetry breaking into special subgroups is not so special even in the Higgs models.
From the usage in GUT model buildings in 4D framework, we are primarily motivated in studying 4D NJL type models in which the fermion matter belongs to complex representations of SU (N )(N ≥ 5), SO(4n+2) (n ≥ 2), and E 6 . We have already known some examples for SU (N ) and E 6 cases in Ref. [19,20], so if we investigate SO(4n + 2)(n ≥ 2) cases, our primary purpose will be completed at least for 4D cases. As is well-known in e.g., Refs. [31,32,35], the irreducible SO(N ) spinor representations are classified into three types depending on N (mod 8); i.e., complex representations for N = 8ℓ ± 2 (ℓ ≥ 1)); real representations for N = 8ℓ, 8ℓ ± 1(ℓ ≥ 1); pseudo-real representations for N = 8ℓ±3, 8ℓ+4(ℓ ≥ 1), which are summarized in Table 1. Some higher dimensional models, e.g., 5 and 6 dimensional SO (11) gauge-Higgs GUTs [50,56] contain the fermions in the spinor representation of SO(11) 32, which is a pseudo-real representation. So, we are also interested in symmetry breaking pattern caused by the pair condensation of fermions not only in complex representations but also in real and pseudo-real representations. Through the investigation of 4D SO(N ) NJL type models in which the fermion matter belongs to the spinor representation, we find that the symmetry breaking into special or regular subgroups has certain correlation with the type of representations of the fermion. (Note that even Ref. [33] has not investigated the Higgs potential models whose scalar field belongs to the SO(N ) spinor representation except for N = 5 case because the dimension of the SO(N ) spinor representation, 2 [(N −1)/2] , rapidly becomes too large for large N .) In this paper, we analyze the SO(N ) symmetry breaking patterns in 4D SO(N ) NJL type models with fermions of SO(N ) irreducible spinor representation which is complex for N = 4n+2 and self-conjugate otherwise. The main purpose of this study is to show that for N = 4n+2 (n ∈ Z ≥1 ), SO(N ) symmetry is broken to its special subgroups such as SO( N 2 ) × SO( N 2 )(S), G 2 (S) for N = 14 and F 4 (S) for N = 26. For N = 4n + 2, on the other hand, the prime number cases N ∈ P ≥3 and the other composite number cases N ∈ Z ≥3 \(P) are distinguished. For prime numbers N ∈ P ≥3 such as 3,5,7,11,13, · · · , SO(N ) symmetry is always broken to its regular subgroups such as SO( For composite numbers N ∈ Z ≥3 \(P) (other than N = 4n + 2) such as 4,8,9,12,16, · · · , SO(N ) symmetry is broken to its regular or special subgroups such as SO( ( Note that (R) and (S) here stand for regular and special subgroups, respectively.) This paper is organized as follows. In Sec. 2, we quickly review the SO(N ) NJL models and examine SO(N ) spinor properties and maximal little groups of SO(N ) for the irreducible representations of the fermion bilinear composite scalar fields. By using the knowledges in Sec. 2, we discuss addition-type, product-type, and embedding-type subgroups in Sec. 3, 4, and 5, respectively. Section 6 is devoted to summary and discussion.

NJL-type model and SO(N ) spinor properties
We consider the NJL-type model where the fermion ψ I (I = 1, · · · , d) belongs to the irreducible SO(N ) spinor representation R of dimension d := 2 [ N− 1 2 ] , denoted by using Dynkin label as Since the fermion ψ I , for each I, actually stands for 2-component Weyl spinor of Lorentz group, the fermion bilinear scalar Φ IJ ∼ ψ I ψ K gives symmetric product (R × R) S decomposed into the following [ N 8 ] irreducible representations R p and also [ N +1 8 ] irreducible representations R ′ p for odd N : ) are the representations denoted by the following Dynkin labels and correspond to the denoted anti-symmetric (AS) tensors: (0 n−4q , 1, 0 4q−1 ) for 4q ≤ n (0 n ) = 1 for 4q = n + 1 non-existing for 4q ≥ n + 2 ∼ ψ I (CΓ a 1 ···a n−4q+1 ) IJ ψ J rank-(n − 4q + 1) AS tensor, where C is SO(N ) charge conjugation matrix and Γ a 1 ···ar are rank-r AS tensor gamma matrices defined by Γ a 1 a 2 ···ar := Γ [a 1 Γ a 2 · · · Γ ar] = Γ a 1 Γ a 2 · · · Γ ar if a 1 , a 2 , · · · a r are all different 0 otherwise (2.4) in terms of the SO(N ) gamma matrices Γ a (a = 1, 2, · · · , N ) given explicitly later below. (For SO(N ) tensor products, see e.g., Ref. [32].) In the following, R k denotes R 0 and R p for even N ; R 0 , R p and R ′ q for odd N , respectively. Now the Lagrangian of the NJL model we discuss is given by where ψ I ψ J R k is the projection of the fermion bilinear scalar ψ I ψ J into the irreducible channel R k , and G R k is the 4-Fermi coupling constants in that channel. As in the previous papers [19,20], we introduce a set of irreducible auxiliary scalar fields Φ R k IJ standing for the composite operators −(G R k /2)(ψ I ψ J ) G R k . Then, we can equivalently rewrite this Lagrangian into the form and obtain the effective potential of this system in the leading order in 1/N as [20] where M 2 R k := 1/G R k , and Φ R k 's are constrained symmetric d × d matrices subject to nontrivial condition belonging to the irreducible representation R k , so satisfying the orthogonality tr(Φ † R k Φ R k ′ ) = 0 for k = k ′ , while Φ := k Φ R k without the irreducible suffix R k is the general (unconstrained) symmetric d × d matrix which appears in the Yukawa interaction terms in Eq. (2.6) and hence in the 1-loop part potential V 1-loop . Here m 2 I are d eigenvalues of the Hermitian matrix Φ † Φ, which stand for d mass-square eigenvalues of the fermion ψ I , and the function f (m 2 ) is given by where Λ is UV cutoff on the Euclideanized momentum. We also use F (x; M 2 ) defined as To discuss spontaneous symmetry breaking, we have to check little groups, where a little group H φ of a vector φ in a representation R of G is defined by (2.10) This little group H φ of G depends not only on the representation R of φ but also the vector (value) φ itself. The vector φ must be an H φ -singlet, so that a subgroup H can be a little group of G for some representation R only when R contains at least one H-singlet. Following so-called Michel's conjecture [58], we assume that the potential minimum of the scalar field in an irreducible representation R of G is located at one of the little group H of R. We classify little groups H of G = SO(N ) into three types: "addition-type," "product-type," and "embedding-type" subgroups . The examples of these three types of subgroups are listed in  Tables 2 and 3, Table 4, and Table 5, respectively, from which the meaning of the name for these three types will be clear. An addition-type subgroup is usually a regular subgroup except for the cases N = 4ℓ + 2 for which n := N/2 is odd and the subgroup H = SO(n + 4p) × SO(n − 4p) has lower rank than G = SO(N = 2n). Any embedding-type subgroup is a special subgroup.
In this paper, we follow the notation and convention of Ref. [32]. More explicitly, we normalize the Dynkin index of the SO(n) defining representation n as T (n) = 1, the Dynkin index of the SU (n) defining representation n as T (n) = 1/2, and the Dynkin index of USp(2n) 2n representation as T (2n) = 1/2. Note that T  Table 2: "Addition-type" little groups H of SO(N = 2n) for irreducible representation R p scalars possessing H-singlet are listed, where p = 0, 1, 2, · · · , N 8 . (Regular/Special) in front of each little group name shows that it is a regular or special subgroup of SO(N = 2n) depending on whether n is even or odd, respectively. Note that the rank of the subgroup is lowered than that of SO(n) when n is odd. Table 3: "Addition-type" little groups H of SO(N = 2n + 1) for irreducible representation R p and R ′ q scalars possessing H-singlet are listed, where p = 0, 1, 2, · · · , N 8 and q = 1, 2, · · · , N +1   Table 6 (Note there the isomorphism USp(2) ≃ SO(3) ≃ SU (2), SO(5) ≃ USp(4) and U (1) ≃ SO(2)). (Special) and (Regular) in front of each little group name show that they are all special or regular subgroups of SO(N ) with N indicated.
In the next section, we discuss the effective potential VEV of addition-type, product-type, and embedding-type subgroups one by one.
Before starting the discussion, let us here recapitulate the basic properties of spinor representation briefly.   Table 6. (Special) in front of each little group name indicates that they are all special subgroups of SO(N ).
The SO(N = 2n) or SO(N = 2n + 1) spinor representation can be described by 2n + 1 Hermitian 2 n × 2 n gamma matrices Γ a satisfying When necessary in this paper, we will use the following expression for the SO(N ) gamma matrices Γ a and the corresponding charge conjugation matrix C: where σ j (j = 1, 2, 3) are the Pauli matrices, (2.14) and Γ 2n+1 plays the role of 'chirality' operator for SO(2n) case. The charge conjugation matrix C (2.13) in this representation satisfies As we will see below, the auxiliary scalar field Φ R k in any SO(N ) irreducible representation R k has an H-singlet of an addition-type little group listed in Tables 2 and 3. Also, in any breaking into an addition-type subgroup, squared masses of the SO(N ) spinor fermion will be found all degenerate. As has been discussed in Refs. [19,20], if the fermion masses are all degenerate, the corresponding vacuum realizes the global minimum of the effective potential in the leading order in NJL model, and so giving a candidate for the true vacuum.
In the following sections, we will often use the "traceless condition" for the mass matrix of our SO(N ) spinor fermion ψ. The fermion mass term is given by the VEV of the auxiliary scalar field Φ = k Φ R k as is seen from the Yukawa term (2. 16) in the Lagrangian (2.6). This VEV is developed in an H-singlet component of one of irreducible scalar fields Φ R k , which, as already shown in Eq. (2.3), corresponds to a certain rank-r AS tensor Γ-matrix Γ a 1 a 2 ···ar , so that the fermion mass matrix takes the form with charge conjugation matrix C and H-singlet linear combination M r of rank-r tensor gamma matrices. Note that C should be present here since the scalar Φ is symmetric. Note, however, that both Φ † R k = C M r and M r realize the same quadratic mass matrix which actually determines the value of the effective potential: Moreover, the mass matrix which becomes proportional to a unit matrix on each H-irreducible sector of the fermion ψ is not CM r but M r . This is because the SO(N ) invariance of the Yukawa term Eq. (2.17) implies that their transformation law under the g ∈ SO(N ) transformation ψ → ψ ′ = gψ are given by where use has been made of C −1 g T −1 C = g which follows from the fact that the SO(N ) spinor rotation g is written as g = exp(θ ab Γ ab ) with real angle θ ab and rank-2 gamma matrix Γ ab , and the property of the charge conjugation matrix taking the form for which Schur's lemma can be applied and tells us that, when the components of the fermion ψ I are decomposed and ordered into H-irreducible blocks, then the matrix M r becomes block diagonal and is proportional to an identity matrix in each H-irreducible block (so being actually diagonal matrix on this basis). Therefore we can call M r 'fermion mass matrix', since it gives a constant mass eigenvalue on each H-irreducible sector of ψ and its square M r M † r gives square mass eigenvalues m 2 I appearing in the effective potential. Now we can state our "traceless condition" for the fermion mass matrix M r : Since the rank-r AS tensor gamma matrices are traceless 1 tr [Γ a 1 a 2 ···ar ] = 0, (2.22) 1 These equations can be understood as special (s = 0) case of the general orthonormal relation among antisymmetric tensor gamma matrices: for r = s (2.21) the fermion mass matrix M r given as their linear combination (2.17) is also traceless and hence the sum of mass eigenvalues should vanish. This conclusion is valid for SO(N =2n+1) cases, but does not apply as it stands for SO(N =2n) cases in which the irreducible spinor fermion ψ is 'Weyl fermion' possessing 'chirality' Γ 2n+1 = +1 or −1. When we use the representation of Γ 2n+1 given in Eq. (2.12), Weyl spinors with chirality Γ 2n+1 = ±1 have only upper-half or lower-half non-vanishing components. We refer to Γ-matrices possessing non-vanishing matrix elements only on diagonal (off-diagonal) blocks as "γ 5 -diagonal (γ 5 -off-diagonal)". Then, assuming positive chirality for our spinor ψ, the fermion mass matrix in Eq. (2.17) is now replaced by the following chiral projected one: In order for this matrix M r not to vanish, Γ a 1 a 2 ···ar must be γ 5 -diagonal and hence r must be even 2 . Moreover, in order for the mass term ψ T (C M r )ψ not to vanish, the matrix CM r should also be γ 5 -diagonal, and hence C as well as M r must be γ 5 -diagonal. From Eq. (2.13), we find that C is γ 5 -diagonal only for even n; that is, the cases where SO(N = 2n) spinor representations are real for n ≡ 0 (mod.4), or pseudo-real for n ≡ 2 (mod.4). For those cases, the mass matrix M r in (2.23) is still traceless since both tr [Γ a 1 a 2 ···ar ] = 0 and tr [Γ a 1 a 2 ···ar Γ 2n+1 ] = 0 hold, and can give non-trivial constraint on the fermion mass that the sum of mass eigenvalues should vanish. Note, therefore, that the traceless condition on the mass eigenvalues is valid except only for N = 4n + 2 cases where the SO(N ) spinor representation is complex.

Addition-type subgroups
We can write down the form of the auxiliary scalar VEV which breaks SO(N ) into an additiontype subgroup H: where we have factored out the charge conjugation matrix C from the VEV Φ † R k in conformity with Eq. (2.17), so V SO(N )→H corresponds to the fermion mass matrix M r to which the "traceless condition" can be applied. SO(N ) spinor representations has N modulo 8 structure. H represents one of the little group of SO(N ). From Tables 2 and 3 and by using the expression of the gamma matrices in Eq. (2.12), we find the following VEVs in general;
It should be understood that the chiral projection matrix is multiplied to these expressions for V SO(N )→H when N is even N = 2n. Any VEV of these is proportional to Γ 1···r of a certain rank r, so the quadratic mass matrix of the SO(N ) spinor fermion is found to be with understanding that P = 1 for SO(N ) with odd N = 2n + 1 case. That is, all the d = 2 [ N−1 2 ] components of ψ obtain a degenerate mass square m 2 realizing the minimum of F (x, M 2 R k ) for all cases. Therefore the global minimum of the potential is realized by the VEV of Φ R k for which the coupling constant G R k is the strongest. So the symmetry breaking pattern is determined depending on which coupling constant G R p is the strongest: We find the following breaking pattern. For even N = 2n, For odd N = 2n + 1,

Orthogonal-type
and SO(n) B vector index I (1 ≤ I ≤ n); more explicitly, the SO(mn) Gamma matrices Γ a are denoted in both way as (4.1) Then the SO(m) A and SO(n) B generators J A ij and J B IJ are given by the following linear combi- The quadratic Casimir operator of SO(m) A is calculated as where the first terms ∝ 1 come from I = J terms and the second term 2M 4 from I = J, which is a (linear combination of) rank-4 anti-symmetric tensor Γ {iI}{jI}{iJ}{jJ} since i = j, I = J. Note that since SO(m) A generators J A ij are SO(n) B -invariant by construction and so the Casimir Note also that the quadratic Casimir of the other factor group SO(n) B reproduce the same rank-4 H-singlet M 4 with minus sign as This should be so because the rank-4 H-singlet is unique in the SO(mn) rank-4 tensor Γ abcd , or more technically, because First, we show that the SO(9) → SO(3) A × SO(3) B breaking VEV of the SO(9) 126 (rank-4 AS tensor) scalar field Φ R 0 generates a common mass for the SO(9) spinor fermion 16. The SO(9) spinor 16 is decomposed into two representations under (See e.g., Ref. [32].) With SO(3) generator denoted in 3-vector notation J = (J 23 , J 31 , J 12 ) as usual, the quadratic Casimir operator ( J A ) 2 of SO(3) A subgroup possesses the following eigenvalues on the SO(9) spinor 16: where the SO(3) Casimir eigenvalues C 2 (4) = 15/8, C 2 (2) = 3/8 here are halves of the SU (2) ones j(j + 1) for spin-j representation listed in Ref. [32] because of our convention T (3) = 1 for SO (3). The quadratic Casimir operators (4.3) and (4.4) in this m = n = 3 case are given by with the rank-4 tensor part M 4 which more explicitly reads by using SO(9) vector index a = 2i+I in place of {iI} notation (a = 1, 2, · · · , 9; i, I = 1, 2, 3), Since the M 4 is an SO(3) A × SO(3) B -singlet in the rank-4 tensor Γ abcd as noted above, the fermion mass matrix is proportional to M 4 and acts on the SO(9) spinor 16 as This leads to the fermion mass square matrix proportional to (M † 4 M 4 ) ∝ 1 16 . Thus, we find that all the fermion states in (4, 2) and (2, (9) get the same mass from the condensation of SO(3) A × SO(3) B -singlet combination (4.8) of rank-4 Γ abcd . Note, however, that we could have obtained this result (4.9) more quickly from the "traceless condition" of the fermion mass matrix M 4 since it can be applied for SO(9) with odd 9 and the H-irreducible sectors (4, 2) and (2, 4) have the same dimensions 8.
One may wonder whether for other cases such as SO (15) where use has been made of, for SO (3)   This again confirms the validity of our traceless condition of the fermion mass matrix. In this case, however, the original spinor is decomposed into three H-irreducible components so that the traceless condition alone cannot determine the three unknown mass eigenvalues even aside from the overall scale. In general, when the breaking SO(mn) → SO(m) × SO(n)(m, n ≥ 3) occurs, the SO(mn) spinor representation is decomposed into at least two different H-irreducible representations.

Symplectic-type
Here we discuss the SO(4mn) → USp(2m) × USp(2n) breaking case. A close parallelism holds between the previous breaking SO(mn) → SO(m) × SO(n) in Sec. 4.1 and the present one From the tables in Ref. [32] and additional searches, we find that under the subgroup H = USp(2m) × USp(2n) ⊂ SO(4mn) except the m = 1, n = 2 and m = n = 2 cases, the SO(4mn) spinor representation is decomposed into two essentially different H-irreducible representations. So the quadratic mass matrix M r M † r of the fermion cannot be proportional to identity. Thus, they cannot realize the global minimum of the effective potential.
Consider  The pair condensation of the 128 spinor fermion cannot realize the global minimum of the potential since the fermion mass eigenvalues are not degenerate in this case. This can be shown as follows.
The traceless condition holds in this case SO(16) possessing real spinor. For the case of 128 spinor which decomposes into more than two H-irreducible components, the traceless condition alone is not enough to calculate the fermion mass eigenvalues. Since the USp(4)×USp(4)-singlet exists in the channel Φ R 1 (rank-4 tensor), the eigenvalue of the mass matrix M 4 in the rank-4 tensor can be calculated by computing the USp(4) quadratic Casimir C 2 which must be a linear combination of G-singlet matrix 1 ∈ Φ R 2 and the H-singlet M 4 ∈ Φ R 1 . Hence we have M 4 ∝ C 2 − x1: where the USp(4) Casimir eigenvalues C 2 (14) = 5, C 2 (10) = 3, C 2 (5) = 2, C 2 (1) = 0 have been used and, in going to the last expression, the G-singlet coefficient x was found to be x = 5/2 by using the traceless condition: Note that we were able to skip the laborious explicit identification of the U Sp(4) generators in terms of the original SO(16) generators Γ ab . For the case of the spinor r ′ = 128 ′ , on the other hand, the traceless condition is enough to determine the (ratio of) the mass eigenvalues since r ′ = 128 ′ has only two H-irreducible components under H = USp(4) × USp(4) ⊂ SO (16). Again the H-singlet is in the Φ R 1 (rank-4 tensor) so the mass matrix M 4 must be traceless: .
To calculate the SU (n) decomposition of the SO(2n) spinor fermion, we here follow the well-known trick to rewrite the SO(2n) gamma matrices Γ a into the creation and annihilation operators of SU (n) fermion, a † k , a k (k = 1, 2, · · · , n) [59]: .

(4.24)
Then the SO(2n) (reducible) 2 n -dimensional spinor is represented in the form giving a decomposition into irreducible representations of SU (n). The SO(2n) chirality matrix (γ 5 -analogue) is represented by with the total fermion number operatorN := k a † k a k . So SO(2n) irreducible chiral spinors |ψ ± with Γ χ = ±1 are given by half sums of states in the RHS of Eq. (4.25) possessing even and odd number of fermion excitations, respectively.
Since the gamma matrices Γ 2k−1 are symmetric and Γ 2k are anti-symmetric in the spinor basis of Eq. (4.25), 3 the charge conjugation matrix C satisfying can be chosen to be the operator 3 In this spinor basis, the creation and annihilation operators a † and a are represented by 2 × 2 Pauli matrices as σ + = (σ1 + iσ2)/2 and σ − = (σ1 − iσ2)/2, respectively, in each k sector. So, for the present case with n species of fermions, the gamma matrices are represented as This representation of the Γa matrices are different from that in Eq. (2.12).
where |ψ + and |ψ − correspond to (1, 2) and (2, 1) in Eq. (4.30), respectively. The bispinor representation R 0 is now a rank-2 tensor Γ a 1 a 2 and so can be described by the operators {a † i a † j , a † i a j , a i a j }. The SU (2)-singlet in R 0 is clearly given by a linear combination {a † 1 a † 2 , a 1 a 2 }: c 1 a † 1 a † 2 +c 2 a 2 a 1 with arbitrary coefficients c j (j = 1, 2). The SU (2)-singlet mass matrix M 2 should also satisfy M † 2 M 2 = M 2 M † 2 since it is diagonalizable as we have shown before, which further leads to a condition |c 1 | 2 = |c 2 | 2 . We choose the phase convention of a 1 and a 2 such that c 1 = c 2 =: m is satisfied with real m and take M 2 as a Hermitian operator: Acting this on the positive chiral spinor |ψ + , We can diagonalize the mass matrix M 2 as The pair condensation of the (1, 2) spinor fermion thus yields a degenerate mass square eigenvalue for the spinor (1, 2) and hence realizes the global minimum of the potential. The SO(6) chiral spinors are given in the SU (n) fermion representation as (4.37) The mass matrix M 3 (10) (M 3 (10)) in the rank-3 tensor bispinor 10(R 0 ) (10(R 0 )) corresponds to an SU (3) singlet operator: For odd n, the mass matrix M n and the charge conjugation matrix C are both Γ χ -off diagonal, while CM n is Γ χ -diagonal. So we directly have to calculate the fermion bilinear mass term ψ T ± CM n ψ ± . Noting ψ T ± is represented by the state and using the expression (4.29) of the charge conjugation operator C for n = 3, the mass terms ψ T ± CM 3 ψ ± can be calculated as follows: That is, only the singletψ 0 in 4 (ψ 0 in 4) gets the mass m while the rest triplet 3 (ψ i ) ∈ 4 (3 (ψ i ) ∈ 4) remains massless. Thus, the pair condensation of the 4 (4) spinor fermion cannot realize the global minimum of the potential. This argument actually holds for all cases of odd n = 2ℓ + 1, for which the chiral spinor is complex (4.41) and the symmetric product of SO(2n) = SO(4ℓ + 2) spinor contains the SU (n) singlet only in the R 0 (rank n tensor). The mass operator M n is thus given by M n = ma † 1 a † 2 · · · a † n or ma n a n−1 · · · a 2 a 1 . Then the fermion mass term ψ T ± CM n ψ ± is calculated in quite the same way as above; for instance, we have for positive chiral spinor ψ + : In either case of ± chiral spinors, only the singlet componentψ 0 or ψ 0 obtains the mass while all the other SU (n) non-singlet components remain massless, so that the pair condensation of the SO(2n = 4ℓ + 2) chiral spinor fermion ψ ± cannot realize the global minimum of the potential for ℓ ≥ 1. Also for the cases of SO(2n) with even n = 2ℓ, it is easy to see that the SU (n)-invariant mass operator M n (4.42) in the channel R 0 can give a mass only to the SU (n) singlet components and leaves the other non-singlet components (existing for ℓ ≥ 2) massless. For this even n = 2ℓ cases, however, the symmetric tensor product of SO(2n) chiral spinors also contains other channels R p (p ≥ 1) that contains SU (n)-singlets. Since the only invariant tensors of SU (n) is δ j i other than ε i 1 i 2 ···in and ε i 1 i 2 ···in , the only possible SU (n)-invariant mass operators of lower rank than a 1 a 2 · · · a n = ε i 1 i 2 ···in a i 1 a i 2 · · · a in /n! must be constructed with the total number operator N = δ i j a † i a j (rank 2). So the SU (n)-invariant mass operator contained in the channel R p (rank n − 4p tensor) must be a polynomial m(N q + c 1N q−1 + · · · + c q ) of degree q = (n − 4p)/2.
(See Appendix A for the explicit form of this SU (n)-singlet operators.) In any case, those mass operators have eigenvalues depending on the SU (n) fermion number so the SU (n) multiplets have non-degenerate masses so that the spinor pair condensation in those channels R p (p ≥ 1) cannot realize the global minimum of the potential. Although this general discussion is enough, let us demonstrate this general feature in an explicit example of n = 10 case in which two bispinor channels R 1 and R 2 containing SU (n)singlets exist other than the usual R 0 of rank n.
For n = 10, i.e., SO(20) → SU (10) × U (1), the SO(20) irreducible spinors are represented as follows in terms of SU (10) fermion operators giving the branching rule: (4.45) The symmetric tensor product of these SO(20) spinors gives (4.46) 92378 ′ (R 0 ) (rank-10 tensor) contains the usual SU (10) singlet which gives a mass operator m 0 a † 1 a † 2 · · · a † 10 + a 10 a 9 · · · a 1 (4.47) for the spinor ψ + (512 ′ ), while 92378(R 0 ) does not. The other channels R 1 (rank-6 tensor) and R 2 (rank-2 tensor) both contain SU (10) singlets of the form (A.5) and (A.1), respectively, which give the following mass operators for the both spinors ψ + (512 ′ ) and ψ − (512): Thus the mass matrix M + of the positive chiral fermion ψ + is given by The mass matrix M − of the negative chiral fermion ψ − is given by (4.50) These masses proportional to m 0 , m 1 and m 2 are generated by the fermion pair condensation in the bispinor channels R 0 , R 1 and R 2 , respectively. Consider the case of pair condensation into a single channel. For the ψ + case, It is interesting that all the non-singlet components 45, 45, 210, 210 have a degenerate mass for the R 1 condensation. The situation is also similar for the ψ − case. In any case, however, the fermion mass spectrum does not realize the total degeneracy, so the pair condensation in any channels R 0 , R 1 and R 2 cannot realize the global minimum of the potential.
In the following, we summarize the branching rules of these sub-type subgroups. Several examples can be found in Ref. [32].

Orthogonal adjoint-type (rank-2 anti-symmetric tensor-type)
We consider SO N = n(n−1) SO(15) spinor 128 is real and so a tensor product (128 ⊗ 128) S has a singlet of SO (15). SO (6) 64 is also a real representation, so a tensor product [(64 ⊕ 64) ⊗ (64 ⊕ 64)] S of SO(6) has three independent singlets of SO(6) ⊂ SO (15). One of them is an SO(15) singlet. From Table 6, the other SO(6) singlets are in R 0 = 6435 and R 1 = 455 representations, which correspond to rank-7 and rank-3 anti-symmetric tensor representations, respectively. The fermion mass matrix from the rank-3 tensor representations M 3 is traceless as noted before, so we have which yields the quadratic mass matrix M † 3 M 3 proportional to identity matrix. This can also be confirmed in another way as follows. Since M 3 is given by a linear combination of rank-3 gamma matrices Γ a 1 a 2 a 3 , the quadratic mass matrix (M † 3 M 3 ) takes the form M † 3 M 3 = (rank-6 part) + (rank-4 part) + (rank-2 part) + 1 × (# of terms) (5.5) but the rank 6, 4, 2 parts must vanish since there are no such H-singlets now, thus leaving only the part proportional to the identity matrix. Also, the mass matrix from the rank-7 tensor representation M 7 is traceless, so the quadratic mass (M † 7 M 7 ) is proportional to the identity matrix. Therefore, the VEVs of the auxiliary field Φ R 0 =6435 and Φ R 1 =455 lead to the identity quadratic mass matrix (M † j M j ) ∝ 1 (j = 3, 7). Third, consider the n = 7, N = 21 case. There is an SO(21) → SO(7) breaking via the pair condensation of the SO(21) spinor fermion. This is because the branching rule of SO(21) ⊃ SO(7) for the spinor representation is given by 1024 = 2(512).
First, consider SO(7) ⊃ G 2 case. There is an SO(7) → G 2 breaking via the pair condensation of the SO(7) spinor fermion. This is because the branching rule of SO(7) real spinor representation 8 under G 2 ⊂ SO(7) is given by G 2 singlet exists in 35 = R 0 (rank-3 tensor) in the fermion pair (8 ⊗ 8) S of SO (7) spinor. We can apply the traceless condition to the mass matrix from the rank-3 tensor representation M 3 also here and obtain SO (14) spinor 64 (or 64) is complex, so a tensor product (64 ⊗ 64) S has no SO(14) singlet, while G 2 64 is a real representation, so a tensor product (64 ⊗ 64) S of G 2 has one G 2 singlet. From Table 6, this G 2 singlet exists in R 1 = 364 (rank-3 tensor) representation. The VEVs of the scalar field Φ R 1 must give the quadratic mass matrix (M † 3 M 3 ) ∝ 1 proportional to identity since 64 is also irreducible under G 2 . (Recall that the traceless condition does not apply to the chiral projected mass matrix M r for the SO(N = 8ℓ ± 2) cases with complex spinor.) Third, for SO(26) ⊃ F 4 case, the situation is almost the same as for the previous SO (14) Table 6, the F 4 singlet exists in R 1 = 3124550 (rank-9 tensor) representation. The VEV of the scalar field Φ R 1 must give the quadratic mass matrix (M † 9 M 9 ) ∝ 1 proportional to identity since the SO(26) spinor 4096 is also irreducible under F 4 .

Summary and discussions
We summarize the symmetry breaking patterns in the SO(N ) NJL models whose fermions belong to an irreducible spinor representation of SO(N ). The summary Table 7 shows which symmetry vacuum realizes (or vacua realize) the lowest potential energy when which 4-Fermi coupling constant G R k beyond critical is the strongest. For all of them, the spinor fermions get a totally degenerate mass so realizing the possible lowest potential energy. All the vacua are written there if they realize such a lowest potential energy in the 1/N leading order. The perturbation will determine the true vacuum among those degenerate vacua in the leading order. From Table 7, we find that when SO(N ) spinor representations are complex, the SO(N ) symmetry is always broken into special subgroups. When SO(N ) spinor representations are pseudo-real, the SO(N ) symmetry is broken into regular subgroups in most cases and into special subgroups in some cases. When SO(N ) spinor representations are real, the SO(N ) symmetry is unbroken or broken into regular subgroups in most cases and into special subgroups in some cases. That is, our analysis have shown that the symmetry breaking into special or regular subgroups is strongly correlated with the type of representations; i.e., complex, or real, or pseudo-real representations.
Note that R, PR and C in the R/PR/C column indicate that the SO(N ) spinor representation is real, pseudo-real and complex, respectively. For ( * ) attached subgroup R 0 for N = 4k cases, which has two real (or pseudo-real) irreducible spinor r and r ′ , only one R 0 either in r × r or in r ′ × r ′ contains the H-singlet. A, P and E in the A/P/E column indicate addition-type, product-type, and embedding-type subgroup, respectively. (R), (S) and (No) indicate that the subgroup H is "regular", "special" and "No breaking" (i.e.,G = H), respectively. This list omitted the possible maximal little groups which are not maximal subgroups, since those vacua in any case cannot realize the global minimum of the potential.  k denotes both channels R k and R ′ k , the latter channel R ′ k with prime exists only for odd N . In the odd N row, therefore, R k : and R ′ k : are put in front of the H names to denote which channel of the two is the strongest.