Is Symmetry Breaking into Special Subgroup Special?

The purpose of this paper is to show that the symmetry breaking into special subgroups is not special at all, contrary to the usual wisdom. To demonstrate this explicitly, we examine dynamical symmetry breaking pattern in 4D $SU(N)$ Nambu--Jona-Lasinio type models in which the fermion matter belongs to an irreducible representation of $SU(N)$. The potential analysis shows 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.

For SU (n) and its breaking via the Higgs mechanism [26], it is well-known that SU (n) symmetry is broken to SU (n − 1) and SU (m) × SU (n − m) × U (1) by the non-vanishing vacuum expectation value (VEV) of a scalar field in an SU (n) fundamental representation n and an SU (n) adjoint representation n 2 − 1, respectively. On the other hand, SU (n) symmetry is broken to SU (n − 1) or SO(n) and SU (n − 2) or USp(2 )( := [n/2]) by the non-vanishing VEV of a scalar field in an SU (n) 2nd-rank symmetric tensor representation n(n + 1)/2 and an SU (n) 2nd-rank anti-symmetric tensor representation n(n − 1)/2, respectively.
The above subgroups SU (n − 1), SU (m) × SU (n − m) × U (1), SU (n − 1), and SU (n − 2) are regular subgroups of SU (n), while the SO(n) and USp(2 ) are special subgroups (or irregular subgroups) of SU (n) [27,28]. Note that a subgroup H of a group G is called a regular subgroup if all the Cartan subgroups of H are also the Cartan subgroups of G; otherwise, the subgroup H is called a special subgroup. For example, SU (2) × U (1) of SU (3) is a regular subgroup, while SO(3) SU (2) of SU (3) is a special subgroup. If we use the familiar Gell-Mann matrices λ a (a = 1 − 8) for the SU (3) generators, the regular subgroup SU (2) × U (1) has the generators λ 1 , λ 2 , λ 3 , λ 8 when the SU (2) is the usual isospin subgroup, while the generators of the special subgroup SO(3) are the three anti-symmetric (hence, purely imaginary) matrices λ 2 , λ 5 , λ 7 . Note that all regular subgroups are obtained by deleting circles from (extended) Dynkin diagrams, while all special subgroups are not done so. (For review, see e.g., Refs. [29][30][31].) For grand unified theories (GUTs) in 4 dimensional (4D) theories [30][31][32][33][34][35][36][37][38] and higher dimensional theories [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53], a lot of GUT models use the Lie groups and their regular subgroups in a series: where G SM := SU (3) C × SU (2) L × U (1) Y , and we omitted several U (1) subgroups. A few GUT models [54][55][56] are known to use not only the regular subgroups but also special subgroups such as where we omitted several U (1) subgroups for regular subgroups. The SU (16) group has a maximal special subgroup SO(10), 16 spinor of which is identified with the defining 16 representation of SU (16). The SU (16) symmetry can be broken to SO(10) via the VEV of the SU (16) 5440 representation corresponding to a Young tableau . Note that a subgroup H of G is called maximal if there is no larger subgroup containing it except G itself. For example, U (1) × U (1) of SU (3) is not a maximal subgroup because one of U (1) is contained in the regular subgroup SU (2) ⊂ SU (3). Some typical examples of the maximal special subgroups of SU (n) are listed in Table 1. Table 1: H is a maximal special subgroup of G = SU (N ). This table is a part of Tables 4 and  5 of Ref. [31]. Note that this table is not a complete list of maximal special subgroups.
When we discuss spontaneous symmetry breaking, it is important to know not only subgroups but also little groups. A little group H φ of a vector φ in a representation R of G is defined by H φ := g gφ = φ, g ∈ G . (1.3) 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. For example, the maximal little groups of SU (3) 3, 6, and 8 representations are SU (2)(R), SU (2)(R) and SO(3)(S), and SU (2) × U (1)(R), where (R) and (S) stand for regular and special subgroups, respectively. Practically, the so-called Michel's conjecture [57] are very useful. The Michel's conjecture tells us that a potential that consists of a scalar field in an irreducible representation R of a group G has its potential minimum that preserves one of its maximal little groups H of R. This conjecture drastically reduces the number of states especially for higher rank group cases.
Many people vaguely believe that symmetry groups are broken to only regular subgroups, not to special subgroups. The main purpose of this paper is to show that symmetry breaking into special subgroups are not special by using 4D Nambu-Jona-Lasinio (NJL) type model in the framework of dynamical symmetry breaking scenario [58].
This paper is organized as follows. In Sec. 2 we first review a 4D NJL type model to show the method of potential analysis. In Secs. 3 and 4, we apply the method for two cases in which the fermion belongs to the defining representation and rank-2 anti-symmetric representations of SU (n), respectively. For the latter NJL model with rank-2 anti-symmetric fermion, we will show, in particular, that SU (16) symmetry breaks into two degenerate vacua of special subgroups SO (16) and SO(10) for a certain region of coupling constants. However, this degeneracy actually turns out to cause the mixing of the two vacua and leads to the total breaking of the SU (16) symmetry, generally. Some detailed identification of the scalar VEVs is necessary to discuss this mixing phenomenon of the degenerate vacua, so the task will be given in the Appendix. Section 5 is devoted to a summary and discussions, where we also note the similarity of the present results to the previous one in Ref. [58] for the E 6 NJL model with fundamental 27 fermion.

Nambu-Jona-Lasinio type model
We consider a 4D Nambu-Jona-Lasinio (NJL) type model [1,2] in which the fermion matter ψ = (ψ I ) (I = 1, 2, · · · , dimR = d) belongs to an irreducible representation R of dimension d of the group G. The each fermion field ψ I is the two-component left-handed spinor ψ Iα with an undotted spinor index α running over 1 to 2. Then the Lorentz scalar fermion bilinears ψ I ψ J := ψ α I ψ Jα := ε αβ ψ Iβ ψ Jα andψ IψJ :=ψ IαψJα = (ψ I ψ J ) * are symmetric under exchange I ↔ J owing to the Fermi statistics of ψ I . Assume that the symmetric tensor product (R × R) S is decomposed into n R irreducible representations R p : Then the NJL Lagrangian has n R independent 4-fermion interaction terms: where ψ I ψ J Rp denotes the projection of the fermion bilinear into the irreducible component R p . Introducing auxiliary complex scalar fields Φ Rp (p = 1, · · · , n R ) standing for each of the irreducible components −(G Rp /2) ψ I ψ J Rp [59,60], we rewrite this Lagrangian into where/ ∂ :=σ µ ∂ µ , M 2 Rp := 1/G Rp , Φ IJ without irreducible index R p was introduced in the second line to denote the sum which now stands for the general symmetric d × d complex matrix with no more constraint. Now, noting that the kinetic and Yukawa terms of the fermion can be rewritten intō up to the total derivative terms, one can calculate the effective potential in the leading order in 1/N 1 as where det 4⊗d denotes the determinant of 4d × 4d matrix. Now inserting the 1-loop potential part reads where tr d denotes the trace of d × d matrix and the last relation follows from ΦΦ † = (Φ † Φ) T since Φ is a symmetric matrix. Since the Φ-independent constant V 1-loop (0) can be discarded for our purpose finding the potential minimum, we henceforth redefine the 1-loop part This formula is valid even when m 2 is a general Hermitian matrix if f (m 2 ) is understood to be a matrix-valued function of the matrix. So the final form of the 1-loop part is where m 2 I are d eigenvalues of the Hermitian matrix Φ † Φ, which stand for d mass-square eigenvalues of the fermion ψ I . This 1-loop function f (m 2 ) is monotonically increasing upward-convex function. In Fig. 1, we plot the rescaled dimensionless functionf (x) := 16π 2 Λ 4 f (m 2 ) of x = m 2 /Λ 2 ≥ 0 as well as the first and second derivatives: They lead tof (x) > 0,f (x) < 0 in the whole region x > 0. For the single component ΦΦ † = v 2 case the leading potential is given by From the behavior off (x) in Fig. 1, we see that the critical coupling constant G crt = M −2 crt for d = dimR = 1 case is given by as determined by the decreasing condition of the function It is convenient to rewrite the tree part potential into the following form by picking up one particular representation, say R 1 , from R p 's: This is because Φ = p Φ Rp is the general (unconstrained) symmetric d × d matrix which solely appears in the 1-loop part potential V 1-loop , while Φ Rp 's are constrained matrices subject to nontrivial condition belonging to the irreducible representation R p , so satisfying the orthogonality tr(Φ † Rp Φ R p ) = 0 for p = p . Whether a symmetry breaking pattern G → H is possible or not is found as follows. Expand each G-irreducible representation R p into H-irreducible components r (2.17) If there is an H-singlet contained in this decomposition for one p or more, then the possibility for the breaking G → H exists. So assuming the non-zero VEV for all the H-singlets and identifying how those singlet VEV('s) is contained in the scalars Φ Rp , we can calculate the potential and find the potential values at the minimum points of the potential. We do this calculation for all possibilities of the subgroup H. Then we can find the true minimum, comparing those minimum values for all possible choices of H. To find the symmetry breaking that realizes the lowest minimum of the potential, we should note that the present potential V (Φ) in Eq.(2.6) consists of negative definite monotonically decreasing 1-loop potential  [30,57] which claims that the group symmetry can breaks down only to one of the maximal little groups of the considered scalar field Φ Rp . Our system does not fall into such a restricted system, so that the lowest potential needs not be realized by one of the maximal little groups. But we can anyway consider the breaking possibilities starting with maximal little group cases, and consider their successive breakings into smaller subgroups if necessary in view of the above criterion (2.18).
3 G = SU (N ), R = ; defining representation ψ i case First consider the simplest case in which the fermion belongs to the defining representation R = of G = SU (N ); ψ I = ψ i . Then, d := dim = N and the irreducible decomposition of the symmetric product of R × R is now trivial, since R p is unique: So, in this case, the irreducible scalar Φ IJ is identical with the general unconstrained symmetric complex N × N matrix Φ IJ , so that the leading order potential is given by where v 2 I is the eigenvalues of the d × d Hermitian matrix Φ † Φ. The point here is that the d eigenvalues v 2 I are all independent and are independently determined by the minimum condition of the common function F (x ; M 2 ). Since the minimum point x 0 is uniquely fixed by f (x 0 ) = M 2 , we can conclude that v 2 I = x 0 for ∀I → N fermions ψ I all have a degenerate mass-square x 0 .
This common mass-square is, of course, non-vanishing only when G = 1/M 2 is larger than the critical coupling G c = 16π 2 /Λ 2 . That is, as far as the dynamical spontaneous breaking occurs, the subgroup H to which the G = SU (N ) is broken down must be such that The first condition alone already excludes the dynamical breaking into regular subgroup H! This is because, if H is a regular subgroup of SU (N ), the defining representation necessarily splits into plural H-irreducible representations. And, the special subgroups H of G = SU (N ) satisfying this condition i) are only SO(N ) and USp(N ) (for only even N cases for the latter), aside from very special subgroups like SO(10) for the case of G = SU (16). In any cases, it is only SO(N ) that can also satisfy the second condition ii), since the symmetric tensor Φ IJ realizes the common mass Φ IJ ∝ δ IJ for ψ I but δ IJ is an invariant tensor only of SO(N ).
We thus conclude: For G = SU (N ) NJL theory with fermion ψ I in defining representation R = , SU (N ) is spontaneously broken to the special subgroup H = SO(N ).
in which the N -plet fermion ψ I of SU (N ) becomes N -plet of SO(N ) and the N (N + 1)/2 dimensional scalars Φ IJ ∈ splits into an SO(N ) singlet trace part tr Φ = I Φ II and traceless symmetric part Φ IJ − (1/N )δ IJ tr Φ of dimension N (N + 1)/2 − 1 = (N − 1)(N + 2)/2; the latter scalars are the Nambu-Goldstone bosons for this breaking Before closing this section, we note an interesting general conclusion valid for a special coupling case, which can be drawn from this simple example; that is, for the general NJL model with fermions of general irreducible representation R, we always have dynamical breaking into a special subgroup, if the coupling constants G Rp = 1/M 2 Rp for G-irreducible channels R p are all degenerate (i.e., R p -independent). Indeed, in such a case, potential V depends only on the unconstrained scalar Φ because of the identity (2. 16), so that all the fermions get a common mass just in the same way as in the simplest model in this section.
Next consider the case where the fermion belongs to the rank-2 anti-symmetric representation R = , so that the index I now stands for the anti-symmetric pair [ij] (i, j = 1, · · · , N ; N ≥ 2); ψ I = ψ ij = −ψ ji . Then the fermion bilinear scalar Φ IJ ∼ ψ I ψ J gives symmetric product (R × R) S decomposed into the following two irreducible representations R p : Namely, we have two irreducible auxiliary scalar fields in this case: There are the following six maximal little groups H of G = SU (N ), under which these two SU (N ) irreducible scalars have H-singlet components listed in Table 2.  USp (4) is really the maximal little group of , but not of ; for the latter , the maximal little group is the case 2) (Regular) SU (4) which contains the USp(4) as a subgroup.
As explained before, we start the analysis of the potential with these breakings into maximal little groups and consider the possibility of successive breakings into further smaller subgroups when necessary.
First, we consider symmetry breaking of the cases 1), 3), 5), and 6) since their breakings are caused by the condensation alone, so, independent of the coupling constant G = M −2 .
As far as the coupling constant G = M −2 is larger than its critical coupling, we can compare the potential energies for those breaking cases with one another irrespectively of the coupling strength G . From Tables 3 and 4, we see that the original fermion N (N −1) , and also in the very special case 6) of N = 16, H = SO(10). The potential for those cases is clearly given by, for any N , Since V 2 can be chosen to be the minimum of the function F (V 2 ; M 2 ), then this potential clearly realizes the lowest possible value for the breakings into this channel scalar Φ . We can thus forget about the other possibilities of 1) and 5), henceforth. For the other coupling strength cases, M 2 ≤ M 2 , we need to consider the condensations into the channel Φ also and evaluate the potential in more detail by identifying the explicit form of the scalar VEVs. So let us now turn to this task.

Scalar VEV and potential for each case
Here we identify the explicit form of the scalar VEVs for the cases 2), 3), 4), and 6) one by one to evaluate the potential in detail.
2) For the regular breaking case 2) into H = SU (4) × SU (N − 4) (N ≥ 4), the H-singlet scalar is contained only in Φ and the VEV takes the form: where ijkl56···N is a rank-N totally anti-symmetric tensor of SU (N ) so that it is non-vanishing only when the first four indices i, j, k, l all take the values 1 to 4 belonging to the SU (4) subgroup. This VEV (4.5) gives the following form of fermion mass matrix for the 6 independent components So, in this case of regular breaking into SU (4) × SU (N − 4), only these six fermions get mass square v 2 , so the potential is given by For N ≥ 6, the remaining subgroup SU (N − 4) can be further broken by the nonvanishing VEV of the scalar field components Φ ij,kl and Φ ij,kl with 5 ≤ i, j, k, l ≤ N , keeping the first SU (4) intact. This breaking again lowers the potential energy since more fermions becomes massive. This successive breaking also can be discussed by simply applying our present argument for SU (N ) to the case N → N − 4.
3) We already know the potential (4.3) for the third case 3) breaking into H = SO(N ). For completeness, however, we explicitly write the form of the H-singlet scalar component in Φ , which is easily guessed to take the form where the multi-index Kronecker's delta is defined by 4) The breaking into USp(N = 2n) for even N = 2n is most non-trivial, since both the G-irreducible components Φ and Φ of the scalar Φ have an H-singlet component. We should note that USp(2n) groups have, aside from the usual SU (N ) invariant tensors δ i j and i 1 i 2 ···i N , an additional invariant tensor Ω ij , U T ΩU = Ω for ∀U ∈ USp(2n), called symplectic metric whose explicit 2n × 2n matrix form can be taken to be Then the H-singlet component in Φ is clearly given by using the totally anti-symmetric tensor i 1 i 2 ···i N and the symplectic metric Ω ij n − 2 times: Note that this VEV for N = 4, possessing no symplectic metric Ω, is SU (4)-invariant rather than USp(4)-invariant.
The H-singlet component in Φ is given by using Ω twice and by acting the Young symmetrizer Y to satisfy the required index symmetry: with (ij) denoting transposition operator between the indices i and j. So we have With these H-singlet VEVs, we can calculate the fermion mass terms by a straightforward calculation. But, before doing so for general N = 2n case, it is helpful to calculate these VEV matrices explicitly for the simplest G = SU (6) (i.e., n = 3) case. Then, among the independent fermions ψ I = ψ i<j , we find it convenient to distinguish the 'diagonal' components ψ 2 −1,2 ( = 1, 2, · · · , n), which appear in the symplectic trace (1/2)Ω ij ψ ij = ψ 12 + ψ 34 + · · · + ψ 2n−1,2n , from the other 2n(n − 1) 'off-diagonal' fermions ψ 2 −1,j or ψ 2 ,j with j ≥ 2 + 1. We put them in the following order explicitly for n = 3 case: With this independent fermion basis, the H-singlet VEV matrices are explicitly written as Note that these matrices are orthogonal to each other, tr Φ Φ = 0, as they should be.
Taking these explicit matrix forms into account, we can now write down the result for the general n case: where the first lines of Eqs. (4.17) and (4.18) are for the terms containing only the n 'diagonal' fermions ψ 2 −1,2 ( = 1, 2, · · · , n), and the second lines are for the bilinear terms of the other 2n(n − 1) 'off-diagonal' fermions. Note that the second lines consist of n(n − 1) bilinear terms so that all the off-diagonal fermions appear only once there.
We can now find the eigenvalues of these matrices Φ and Φ . Calculating separately the 'diagonal' component sector and 'off-diagonal' component sector, we find the eigenvalues for SU (2n) case Recall that the fermion mass-square eigenvalues are given by the eigenvalues of Φ † Φ with the total scalar field Φ = Φ + Φ . We, therefore, have the fermion mass-square eigenvalues as Note that this splitting pattern of fermion mass-squared eigenvalues correctly reflects the de- where the USp(2n)-singlet component is given by the symplectic trace ∝ Ω ij ψ ij . Then, noting we thus find the potential for this breaking SU (2n) → USp(2n): where the identity (2.16) has been used in going to the second and third expressions. 6) Finally, for the case 6) of SU (16) → SO (10), the potential is the same as that in Eq. (4.3) with N = 16 for the case 3) of SU (16) → SO (16). But the form of the H-singlet scalar component in Φ is of course different from the latter case one (4.8), and is given by (G > G ), separately. It is also necessary to discuss even and odd N (≥ 3) cases, separately, since the maximal little group USp(N = 2n) for the case 4) is also a maximal subgroup of SU (N = 2n) for even N , but not so for odd N = 2n + 1. In evaluating the potential henceforth, we assume that the theory shows the spontaneous symmetry breaking; that is, the larger coupling constant, at least, is larger than the critical coupling constant, Min(G , G ) > G cr .
Even N ≥ 4 We have already known that for N = 16 the potentials for the cases 3) and 6) are the same.
Here, we need to consider only the potentials for the cases 2), 3) and 4).
where V 2 0 is the minimum point x = V 2 0 of the function F (x; M ) as introduced above. Note that F (V 2 0 ; M 2 ) < 0 because of the symmetry breaking assumption. The above inequality holds for ∀v. Therefore, we find for N ≥ 4 (4.26) Next, we compare the potential for 3) SO(N ) and 4) USp(N ) cases. From Eq. (4.23) So, since (M 2 − M 2 )n(n − 1)V 2 > 0 in this case, we have for even N = 2n, Thus, the SO(N ) vacuum realizes the lowest potential value and we can conclude that the symmetry breaking in this case is also a breaking to special subgroup: where the equality holds only for N = 4. Next, we compare the potential for 3) SO(N ) and 4) USp(N ) cases. This case of degenerate couplings was already discussed generally at the end of the previous section. We know that all the fermions get a common mass after symmetry breaking so that the breaking must be down to a special subgroup. In this case, we have two possibilities for the special subgroup, SO(N ) and USp(N = 2n), which correspond to cases 3) and 4) breaking, respectively. At first sight, the latter SU (N ) → USp(N = 2n) breaking case seems not realizing a common mass for all the fermions ψ ij but gives two mass square values, since the N (N − 1)/2-plet fermion ψ ij splits into a singlet 1 and the rest (2n + 1)(n − 1) under H = USp(N = 2n) as already seen in Eq. (4.21). In the absence of the term (M 2 − M 2 ) 3n(n − 1)V 2 , however, V USp(2n) potential (4.23) takes the form Since v and V are two independent variables corresponding to the VEVs Φ and Φ , respectively, the two mass-square parameters ((2n + 1)V + (n − 1)v) 2 and (V − v) 2 can be varied independently so as to choose the minimum V 2 0 of the function F (x; M 2 ). Indeed, two points and realize the minimum, and then 1 + (2n + 1)(n − 1) = N (N − 1)/2 fermions all have a degenerate mass-square V 2 0 also in these USp(N = 2n) vacua. (We notice that the latter USp(N = 2n) vacuum (4.32) for N = 4 reduces to the SU (4) × SU (N − 4) = SU (4) vacuum realized by Φ = v alone, i.e., with V = 0).
Recalling the expression (4.3) for the SO(N ) potential, we see that both USp(2n) and SO(N ) vacua realize the degenerate lowest potential minimum in this case: and we again conclude the breaking into special subgroups also in this case: and, for N = 4 case, in particular, although the last SU(4) vacuum breaks no symmetry but is merely a bilinear fermion condensation.

(c) M 2 < M 2 case
Since the coupling G becomes stronger in this region, we can intuitively guess that the USp(N = 2n) vacuum realizes the lower potential value than the SO(N ) one. It can indeed be shown explicitly as follows. If we put the above two points (4.31) and (4.32) into the expression (4.23) for the potential V USp(2n) (v, V ), then, we have Since the first terms on the RHSs are negative in this case, USp(2n) potential V USp(2n) (v, V ) at these points already take values lower than the minimum of the SO(N ) potential. The true minimum of V USp(2n) (v, V ) must be lower than these, implying V USp(2n) min < V SO(N ) min . We should first consider a special case N = 2n = 4 (i.e., n = 2), in which H = SU (4) × SU (N − 4) is just H = SU (4) implying no breaking of G = SU (4). However, the SU (4) vacuum is realized by the condensation into the channel Φ = v and the potential is given by 6F (v 2 ; M 2 ). All the 6 components of fermion get a common mass square v 2 0 realizing the minimum of the function F (x; M 2 ), so it is clear that this SU (4) vacuum realizes the lowest potential in this coupling region M 2 < M 2 . (As noted above, the second USp(4) vacuum (4.32) for n = 2 is identical with this SU (4) vacuum since V = 0.) We thus conclude for N = 4 that V SU (4) min < V USp(4) min < V SO(4) min . (4.38) Now, we have to consider the general cases N = 2n ≥ 6 (i.e., n ≥ 3). We here want to show that the opposite to the N = 4 case holds for this general case N ≥ 6; that is, V SU  The derivative of ∆(M 2 ) with respect to M 2 is evaluated as where v 1 := (2n + 1)V + (n − 1)v and v 2 :=v −V are (square root of) the arguments of the two f functions in Eq. (4.23) at the minimum point. Inserting the inverse relation Eq. (4.44) can be rewritten into In order for this simultaneous Eqs. (4.46) and (4.47) to have non-vanishing solution, must vanish, so that we obtain where we have defined a parameter .
, respectively, since f (x) is a monotonically decreasing function and M 2 = f (v 2 0 ). However, the case ii) is inconsistent with Eq. (4.51), which says v 2 1 > v 2 2 since (n − 1)α −1 > 1 for α < 1, n ≥ 2. Thus we have only the case i), which is consistent with Eq. (4.51) if n − 1 > α > 1. Now to prove the positivity of Eq. (4.42), we need an inequality. Recall that f (x) is a monotonically decreasing downward-convex function, so it satisfies the following inequality for ∀ λ ∈ [0, 1], (4.52) Noting the ordering v 2 2 < v 2 0 < v 2 1 , we take  . However, one can convince oneself that the latter solution with α < 0 has the size-ordering v 2 1 < v 2 < v 2 2 < v 2 0 f (v 2 ) := M 2 and realizes higher potential value than that realized by the former solution with α > 0 discussed here. In any case, it is enough to prove VUSp|min < V SU (4)×SU (N −4) |min for one solution for the present purpose. Now, we can evaluate d∆(M 2 )/dM 2 in Eq. (4.42); the first term is given by where we have used Eqs. (4.51) and (4.55). An elementary analysis for the function G(α −1 ) over the region n − 1 > α > 1, i.e., (n − 1) −1 < α −1 < 1, shows that G(α −1 ) is maximum at the starting point α −1 = (n − 1) −1 and is a monotonically decreasing function in this region. So the minimum of the function G(α −1 ) is located at α −1 = 1 for n ≥ 3; G(1) = 27n 2 /(n 2 − 3n + 5). Therefore, we have We thus again conclude the breaking into special subgroups also in this case N ≥ 6:  Here, however, we should comment on the possibility of further breaking of the SU (N − 4) part of SU (4) × SU (N − 4), which exists for n ≥ 3 and can actually make the potential lower as remarked before. However, in this coupling region, we now know that the breaking SU This is because the number of the massive fermions on the USp(2n) vacuum is much larger than that on the SU (4) × USp(N − 4) vacuum; the difference is which is larger than 8 already at the lowest value n = 3 here. So the above conclusion of the SU (2n) → USp(2n) breaking is still valid even if the possibility of the breaking into non-maximal little groups is taken into account.
Obviously, for N = 3, SU (3) is broken to the maximal special subgroup SO(3) as far as G is larger than its critical coupling. We will discuss the potentials for N ≥ 5 in detail.
with equality for the case (b). But, since Eq. (4.3) tells us the inequality we have anyway Thus, the SO(N = 2n + 1) (N ≥ 5) vacuum realizes the lowest potential value and we can conclude that the symmetry breaking in these cases M 2 ≥M 2 is also a breaking to special subgroup: In this coupling region, the condensation into Φ is preferred to into Φ . Here we first compare the potentials for 2) SU (4) × SU (N − 4) and 4) USp(N = N − 1) for (N ≥ 5). The same discussion as in even N , given from Eq. (4.39) to Eq. (4.58), holds if n ≥ 3, so that we have, for N = 2n + 1 ≥ 7, (4.70) for N ≥ 7, (4.71) The SU (2n + 1) phase diagrams are shown in the coupling constant (G , G ) plane for n = 2 and n ≥ 3 cases, in Figure 3.

Summary and discussions
We have performed the potential analysis of the SU (N ) NJL type models for two cases with a fermion in an SU (N ) defining representation R = and an SU (N ) rank-2 anti-symmetric representation R = , respectively.
The former case with R = fermion shows that at the potential minimum the SU (N ) group symmetry is always broken to its special subgroup SO(N ) as far as the symmetry breaking occurs. The latter case with R = also shows that the SU (N ) symmetry for N ≥ 4 is, if broken, always broken to its special subgroup SO(N ) or USp(2[N/2]) aside from some exceptional cases; for N = 4 the SU (4) symmetry is broken to its special subgroup SO(4) or is not broken although the condensation into SU (4)-singlet occurs; for N = 16 the SU (16) is broken to its special subgroup SO (16) or SO (10) or USp (16); for N = 5 the SU (5) is broken to its special subgroup SO(5) or to a regular subgroup SU (4); for N = 3 the SU (3) is broken to its special subgroup SO(3). That is, aside from the only breaking SU (5) → SU (4) for N = 5, all the SU (N ) symmetry breakings for N ≥ 3 is down to its special subgroups in the case R = .
This result clearly shows that symmetry breaking into special subgroups is not special at all at least for the dynamical symmetry breaking in the 4D NJL type model. One might, however, suspect that this may be a special situation specific to the classical group G = SU (N ) model. But, actually, this tendency of symmetry breaking to special subgroups was found previously for the exceptional group G = E 6 model in Ref. [58]. They analyzed the potential in the 4D E 6 NJL model with fundamental representation R = 27 fermion, which have two coupling constants  This result is very similar to the breaking pattern in our SU (16) case with fermion shown in Figure 2.
First of all, all the groups F 4 , USp(8), G 2 and SU (3) appearing here in Figure 4 are special subgroups, and the breaking into the regular subgroup SO(10) = E 5 does not occur at all despite that SO(10) is one of the maximal little groups of scalar Φ 27 or Φ 351 . Moreover, the 27 fermion falls into a single irreducible representation 27 under the special subgroups USp(8), G 2 and SU (3) while it splits into two 26 + 1 under F 4 . This is very much parallel to the situation in our SU (16) case that the fermion 120 falls into a single representation 120 also under the SO(16) and SO(10) subgroups, while it splits into 119+1 under USp (16). In particular, the fact that the irreducible representation fermion of G also falls into a single irreducible representation under distinct plural subgroups H implies in this NJL model the special existence of degenerate broken vacua; USp(8), G 2 and SU (3) vacua for the G = E 6 case, and SO (16) and SO(10) vacua for G = SU (16) case. For the E 6 case, however, numerical study showed the surprising fact that the general vacuum does not show any of the symmetries USp(8), or G 2 or SU (3). The authors of Ref. [58] conjectured the existence of the continuous path in the scalar Φ 351 space connecting those three vacua of USp(8), G 2 and SU (3) through which the potential is flat and the E 6 symmetry is totally broken in between those three points. Although this was a conjecture for the E 6 case, we can show explicitly that it is really the case for our SU (16) → SO (16) which also realizes the degenerate mass-square eigenvalue for 16 · 15/2 = 120 fermions ψ ij determined by the minimum of M 2 x − f (x), so realizing the same lowest vacuum energy value as the above SO(16) vacuum. The 16 × 16 matrix σ abc C will be explained shortly below.
To understand the reason why these two vacua, SO (16) and SO (10), can realize the same degenerate 120 fermion mass-square is interesting and important, since these two vacua turn out to be continuously connected with each other via one-parameter family of vacua with nonvanishing VEV in 5440 Φ which all realize the same degenerate 120 fermion mass-square but nevertheless violate completely the SU (16) symmetry.
Similar phenomenon was previously observed in Ref. [58] which considered the G = E 6 NJL model with 27 fermion: there, the system has three degenerate broken vacua into USp(8), G 2 and SU (3), respectively, which all realize the degenerate 27 fermion mass-square and hence the lowest vacuum energy for the coupling region G 351 > G 27 . The authors of Ref. [58] performed the numerical search for the potential minimum and actually found the degenerate mass-square for the 27 fermion there. But, they also computed the E 6 gauge boson mass eigenvalues on those vacua to identify the residual unbroken symmetries, and, surprisingly found that the gauge bosons are all massive and non-degenerate, implying no symmetries remain there. They interpreted it that there exist a path in Φ 351 space connecting those three vacua of USp(8), G 2 and SU (3) through which the potential is flat and the E 6 symmetry is totally broken in between those three points. This was merely their interpretation of the numerical results but was not shown analytically. Here, in this G = SU (16) case, we can show this explicitly as we now do so.
The SU (16) indices i, j, · · · taking values 1, 2, · · · , N (=16) are identified with the spinor indices of the special subgroup SO (10). So, it is now necessary to recall some properties of the SO(10) Clifford algebra, which was explicitly constructed in the Appendix of Ref. [58]: Its ten generators, i.e., ten 32 × 32 gamma matrices Γ a and charge conjugation matrix 10 C are given in the following form in terms of the 16 × 16 'Weyl' submatrices σ a and C: (a = 1, 2, · · · , 10), 10 The matrix C is chosen real as The anti-symmetric spinor pair index [ij] can equivalently be expressed by the rank-3 antisymmetric SO(10) vector indices [abc] (a, b, c, · · · = 1, 2, · · · , 10) by the transformation tensor (σ abc C) ij and (Cσ abc ) ij , where This is because the 10 C 3 = 120 matrices (σ abc C) ij (or their complex conjugates (Cσ abc ) ij ) span a complete set of anti-symmetric 16 × 16 matrices for which exist 16 · 15/2 = 120 independent ones, and satisfy the completeness relation: Thus our scalar field Φ ij,kl can be equivalently expressed by (A.10) They both possess the same norms: Φ abc,def 2 = Φ ij,kl 2 , where Φ abc,def 2 := 1 2 Now, using the relation (A.9), we can express the SO(16) VEV (A.1) and SO(10) VEV (A.2) in terms of Φ abc,def of SO(10) rank-3 antisymmetric tensor basis: We can now see that these VEVs are simple diagonal matrices ∝ δ def abc in this SO(10) tensor basis, whose 120 diagonal elements are all v/2 for SO(10) vacuum while 60 v/2 and 60 −v/2 for SO (16) vacuum. The sign factor ε(abc) = ±1 for the latter in Eq. (A.12) came from Eq. (A.5) for rewriting Cσ abc into σ abc C for the SO(16) vacuum. For both vacua, the fermion mass square matrix Φ † Φ becomes exactly the same one (v/2) 2 δ def abc = (v/2) 2 1 120 for both vacua. Now we can find the one-parameter family of more general vacua connecting these two vacua: that is, the vacua |0 t parameterized by t ∈ [0, 1] which realize the scalar field VEV Φ abc,def t := t 0|Φ abc,def |0 t as Φ abc,def t = [ε(abc)] t v 2 δ def abc . (A.14) If we introduce a diagonal unitary 120 × 120 matrix U t (U t ) abc, def = [ε(abc)] t/2 δ def abc , (A. 15) this VEV can be written as Let us now show that 1. Although being a unitary matrix, U t does not belong to the SU (16) transformation so that the G = SU (16) symmetry is totally broken on the vacua |0 t for t ∈ (0, 1).
2. The vacua |0 t have non-vanishing VEV only in the channel Φ : The first point immediately follows from the fact that the vacuum |0 t is SO(10) vacuum at t = 0 and SO(16) vacuum at t = 1. That is, the isometry group changes as t changes, while the isometry group cannot change if U t is an SU (16) transformation.
The second point is proved as follows. Since the general 120 × 120 symmetric matrix Φ is decomposed into two irreducible components, Φ and Φ , it is sufficient to show that Φ component is vanishing on the vacua |0 t , which is given by where A ijkl and B ijkl are the sum over the 60 sets of (a, b, c) with ε(abc) = ±1, respectively. We already know that Φ t belongs to Φ at the end points t = 0 and t = 1, so we have Thus, Φ t vanishes for any t, proving the second point.
This property Φ t = 0 guarantees that all the vacua |0 t realize the lowest energy states degenerate with the SO(10) and SO(16) vacua at the endpoints t = 0 and t = 1; this is because the potential is commonly calculated by M 2 tr(Φ † Φ) − tr f (Φ † Φ) since Φ = Φ for these vacua.

A.1 Mass square matrix of SU (16) gauge boson
In order to see which symmetry actually remains on a vacuum with given VEV, one way is to see the mass spectrum of the gauge boson for (gauged) G symmetry. It is also necessary to calculate the gauge boson masses in order to see how the degeneracy of the vacuum energy due to the fermion loop is lifted by the gauge boson loop contribution. It is actually difficult to analytically calculate the gauge boson mass square matrix for the general vacua |0 t given above, since all the G = SU (16) symmetry is expected lost there. So we calculate it only at the two end points, SO (16) and SO(10) vacua and guess the spectrum by interpolation.
The scalar kinetic term (D µ Φ) † (D µ Φ) gives the gauge boson mass term (1/2)M 2 AB A A µ A B µ by substituting the VEV for the scalar field Φ. Since the derivative term ∂ µ Φ does not contribute for the constant VEV, this implies that we can find the mass square matrix M 2 AB by simply calculating the square of the gauge transformation δ(θ): The G = SU (N = 16) transformation for this case is given by Here g is the gauge coupling and the second line is the particular choice of the SU (16) generators respecting the SO(10) subgroup: SU (16) adjoint 255 = 45 + 210 of SO (10). We adopt the convention tr(T A T B ) = (1/2)δ AB for Hermitian generators T † A = T A , then, The norm square is computed as with N = 16. Note that Θ+Θ T = 2gθ A S T SA is given by the sum only over the symmetric matrices T SA , which stand for the broken generators for G = SU (16) → SO (16) and recall that the generators of unbroken SO(16) consist of all the antisymmetric N ×N matrices whose dimension is N (N − 1)/2 = 120. Using also tr Θ = tr Θ T = 0 for SU (N ) case, and tr(T SA T SB ) = (1/2)δ AB , we find A,B∈SU (16)/SO (16) θ A S θ B S δ AB .   while the other SO(10) 45 gauge bosons for the unbroken generators T ab remain massless. Finally, two comments are in order: First, for the interpolating vacua |0 t , the gauge transformation (A.33) is replaced by Since the total sum of gauge boson mass squares is the same between the two vacua, 135 × 14 = 210×9 = 1890, the upward convexity of the function f (x) leads to the inequality, 210f ( 9 2 g 2 v 2 ) > 135f ( 14 2 g 2 v 2 ). This implies that the gauge boson 1-loop contribution lifts the degeneracy between the two vacua SO (16) and SO (10), and SO(16) vacuum will be realized as the lowest energy vacuum.  : n 2 (n + 1)(n − 1) 12 = n(n + 1)(n + 2)(n − 3) 12 ⊕ (n − 1)(n + 2) 2 ⊕ 1 : n(n − 1)(n − 2)(n − 3) 24 = n(n − 1)(n − 2)(n − 3) 24 .