The CKM matrix from anti-SU(7) unification of GUT families

We estimate the CKM matrix elements in the recently proposed minimal model, anti-SU(7) GUT for the family unification, $[\,3\,]+2\,[\,2\,]+8\,[\,\bar{1}\,]$+\,(singlets). It is shown that the real angles of the right-handed unitary matrix diagonalizing the mass matrix can be determined to fit the Particle Data Group data. However, the phase in the right-handed unitary matrix is not constrained very much. We also includes an argument about allocating the Jarlskog phase in the CKM matrix. Phenomenologically, there are three classes of possible parametrizations, $\delq=\alpha,\beta,$ or $\gamma$ of the unitarity triangle. For the choice of $\delq=\alpha$, the phase is close to a maximal one.


I. INTRODUCTION
At present, the unitarity triangle is determined with a very high precision such that any flavor unification models can be tested against it. Therefore, we attempt to see whether the recently proposed unification of grand unification families (UGUTF) based on anti-SU(7) [1] is ruled out or not, from the determination [2] of the Cabibbo-Kobayashi-Maskawa(CKM) matrix elements [3][4][5][6]. A simple CKM analysis can be performed in the Kim-Seo(KS) parametrization [7] where only complex phase gives the invariant Jarlskog phase itself [8]. This phase is called the CKM phase δ CKM .
The anti-SU(7) solution of the family problem is to put all fermion representations in where the indices inside square brackets imply anti-symmetric combinations, and the bold-faced numbers are the dimensions of representations. The color indices are 1, 2, 3 and weak indices are 4, 5. With U(1)'s, it is possible to assign the electromagnetic charge Q em = 0 for separating the color and weak charges at the location [45], which is the key point for realizing the doublet-triplet splitting in the GUT BEH multiplets [1]. The merits of the UGUTF of Ref. (1) are, (i) it allows the missing partner mechanism naturally based on a suitable µ parameter [23], (ii) it is obtained from string compactification, and (iii) it leads to plausible Yukawa couplings. The first merit has been already discussed in Ref. [1]. The second merit is the following. The R-parity in SUSY and the Peccei-Quinn symmetry are greatly used for proton longevity and toward a solution of the strong CP problem and cold dark matter [24]. Because of the gravity spoil of such symmetries in general [25,26], discrete gauge symmetries were considered in the bottom approach [27,28]. It can be a discrete subgroup of some gauge group. In the top-down approach, such as in models from string compactification, the resulting approximate discrete and global symmetries are automatically allowed since string theory describes gravity without such problems [29,30].
In this paper, we focus on the third merit by adopting the spectra obtained in Ref. [1], and explicitly calculate the CKM matrix. Here, we do not use the full description of Yukawa couplings dictated from string theory [31], but use just the supergravity couplings including non-renormalizable terms 1 suppressed by the string scale, M s . Thus, every nonrenormalizable term introduces an undetermined coefficient of O(1). Here, we reduce the number of couplings, using the Z 12−I discrete symmetry implied from its origin of Z 12−I orbifold compactification [1].

II. SOME COMMENTS RELATED TO THE JARLSKOG DETERMINANT
It is known that δ CKM ≈ 90 o in the KS parametrization [33]. The Particle Data Group(PDG) compilation gives δ CKM = (85.4 +3.9 −3.8 ) o , i.e. our δ CKM is their α PDG [2]. We consider this as a maximal phase with the prescribed real angles. The Jarlskog determinant J is the area of the Jarlskog parallelogram which has two angles whose sum is π. The area of the parallelogram has the form: (combination of real angles) · sin δ CKM . So, the Jarlskog phase can be taken as δ CKM or π − δ CKM . Let us define 'Jarlskog triangle' by cutting the parallelogram along a diagonal line, and the Jarlskog invariant phase is the angle opposite to the cutted line. One crucial question is whether the Jarlskog phase is parametrization-independent or not. This is because the Jarlskog determinant, i.e. the area, can be made the same in different parametrization schemes by appropriately changing (combination of real angles) and sin δ CKM . We argue that there are only three classes of the CKM parametrizations from length sides of O(λ 3 ) unitarity triangle. From the unitarity triangle of B s meson decay with O(λ 3 ) lengths, there are three angles α, β and γ, and we can define three classes of parallelograms with the same area. Since there are six different unitarity triangles, three vertical cases in choosing two columns and three horizontal cases in choosing two rows, the total number of possibilities is 18. Out of these 18 angles, 4 angles (having δ CKM 0) with side lengths of O(λ) and O(λ 2 ) from horizontal and vertical cases are phenomenologically excluded. Furthermore, the invariant phase appears in all six triangles. Since the unitarity triangle of B s meson decay is known rather accurately, only three angles are suitable for δ CKM . The KS parametrization uses α of the unitarity triangle of B s meson decay as δ CKM while the Chau-Keung-Maiani (CK) parametrization [6] uses γ as δ CKM . Since α 90 o , it is minimal (also, see below) to adopt the KS parametrization since there can be one Jarlskog phase π 2 . In the other classes, there are two Jarlskog phases, γ and π − γ, or β and π − β.
If the phase is parametrization-dependent, it is not so important to try to determine very accurately α, β, γ in the unitarity triangle of B s meson decay in Particle Data Book [2]. Here, we argue that the CKM phase δ CKM is scheme independent up to three classes. Assume that the weak CP violation is introduced spontaneously [34] by a complex vacuum expectation value of the standard model (SM) singlet X [35]. Suppose the phase of X is 2πn/N DW where n and N DW do not have a common divisor. Thus, the vacuum has N DW different domains which are separated by domain walls [36]. Depending on the value of δ CKM , we can similarly define the domain wall number of the CKM matrix, N CKM . Let the phase δ of the SM singlet X vary continuously from 0 to 2π. Along this variation, one passes through the different domains of number N DW . Now, suppose we perform weak CP violation experiments looking at the Jarlskog phase. Observe that δ CKM must be proportional to the phase of X since there will be no CP violation if X is real. In the same domain, measurements on the weak CP phase must be identical. So, we obtain N CKM ≤ N DW . In addition, in the two adjacent domains, measurements on weak CP phase must be different, leading to N CKM ≥ N DW . Thus, we obtain N CKM = N DW . In this Gedanken experiment with spontaneous CP violation, |δ CKM | is the magnitude of the phase of the VEV X . So, it must be scheme independent, up to three classes, since in any CKM parametrization the VEV of fundamental field X is not introduced. In this argument, it is better to use the parametrization scheme where the phase of X sits at origin, regardless of the product of the combinations of real angles. Namely, the phase δ CKM has an invariant meaning, up to three classes. It is a topological argument and may be applicable for the case of complex Yukawa couplings also because one can mimick the complex Yukawa couplings by VEVs of SM singlets. Thus, the first result from the Jarlskog determinant is that the phase δ CKM has a parametrization-independent meaning, up to three classes. The second result is that the product of the lengths of two sides enclosing the Jarlskog phase δ CKM is invariant.
The third aspect is the following. Firstly, as an illustration, consider the Kobayashi-Maskawa(KM) parametrization [4]. Its determinant is not 1, but −e iδ . A proper redefinition, making the determinant real and rotating all six Jarlskog triangles without changing the shapes, is to multiply e i(π−δ )/3 to every element. It is equivalent to multiplying e iδ0 (δ 0 = (π − δ )/3) toū L ,c L , andt L fields such that the newly defined primed fields areū L = e iδ0ū L ,c L = e iδ0c L , and t L = e iδ0t L . Then, obviously the shapes of all six triangles are not changed. But this introduces a factor e iπ/3 in every elements. To keep the shapes of at least three vertical Jarlskog triangles, including the familiar one in the PDG book, multiply diag. (1, 1, −e −iδ ) on the right-hand side of the KM matrix, leading to from which we obtain Note that α π − δ , i.e. the KM parametrization uses α of PDG as δ CKM . 2 It is very useful if the CKM matrix itself contains the invariant phase δ CKM in a visible manner. J is always arising with O(λ 3 ) multiplied. Since the (13) and (31) [33]. 3 It is convenient to make the first row real, i.e. the (13) element real. Then, the denominator V ud V * ub in Eq. (3) is real and the Jarlskog triangle has one side on x-axis. The angle at the origin is δ CKM . On the other hand, the CK parametrization [6] has real values for both the first row and first column, and its determinant = 1. Thus, J must be contributed from the phase in the (22) element. The V (cd) · V * (cb) component (for the (22) element, or the (cc) element in the mass eigenstate bases) appears for β and for γ in Eq. (3). For the one in the numerator, i.e. in β, V * tb in the denominator is also complex, and V (cd) · V * (cb) alone cannot determine δ CKM . On the other hand, the one in the numerator, i.e. in γ, the numerator is real, and V (cd) · V * (cb) alone determines δ CKM . Thus, δ CKM is γ in the CK parametrization. We can generalize this statement. Let us use the parametrizations such that the large components of the diagonal elements are real. Then, if the first row or first column is real, δ CKM = α. If both the first row and first column are real, then δ CKM = γ. To have β as δ CKM , we need that the (22) element contains a large impaginary part. In this analysis, it was useful to remember the formula of Ref. [33]: The invariant Jarlskog phase appears in all Jarlskog triangles, not necessarily at the origin. Let us take, as an illustration purpose, α 90 o = 2π 4 , β 22.5 o = 2π 16 , and γ 67.5 o = 2π 16 × 3 which are within the experimental bounds. If these phases appear from some Z N symmetry, we can choose three kinds of N depending on which angle is used for δ CKM . If α, β, and γ are used for δ CKM , N DW of X must be 4, 16, and 16, respectively. In this paper we use the KS parametrization [7,33], which is a kind of minimal one, Note that J is given as, For a numerical study, we can choose a vertical Jarlskog triangle of the first and second columns, where two O(λ) side lengths are |c 1 c 3 s 1 |, |c 2 s 1 (c 1 c 2 c 3 + s 2 s 3 e −iα )|, and an O(λ 5 ) side length is e iα s 1 s 2 |(c 1 c 3 s 2 − c 2 s 3 e −iα )| with the phase explicitly written for the O(λ 5 ) side to rotate it freely. The corrected area depending on θ 2 , θ 3 and α is J/c 1 sin 2 θ C = 1 2 sin(2θ 2 ) tan(θ 3 ) sin α. For given sin 2θ 2 and tan θ 3 , we can rotate α to 90 o to obtain the largest δ CKM since in our choice of α ∼ 90 o is allowed. We cannot give this argument for δ CKM = γ, where γ is far from 90 o .
It is pointed out that if δ CKM = ±δ PMNS is empirically proved then the idea of spontaneous CP violationà la Froggatt and Nielsen with a UGUTF makes sense [38]. In this case, the value δ PMNS will choose one class of the CKM parametrizations we discussed here.

III. YUKAWA COUPLINGS AND MASSES
A. U(1) charges in anti-SU (7) To check the Yukawa couplings, it is useful to have U(1) charges in the anti-SU(7) model. For completenes, therefore, we list them. For the fundamental representation 7, the U(1) charges belonging to SU(5) and SU(7) are defined as The extra U(1) charge beyond SU (7) is For the matter 7, therefore, we represent it as 7 −5/7 . The electroweak hypercharge Y of the SM and the U(1) charge X of the flipped-SU(5) are defined as When 21 branches to SU(5) representations 10, 2 · 5, and 1, the SM U(1) charges are required to be the familiar ones, determining subscripts a, b, c in the following , where we used Eqs. (5) and (7) where we used Eqs. (5) and (7) and used quantum numbers of 35 = Ψ [ABC] . Because of the compact group nature, the naive U(1) charge calculation given in the bracket just by the tensor representation components is not exact. We use the |X| ≤ 1 for the fundamental representation Eq. (7). With two SU(5) indices, the |X| charge are redundantly added, and we subtract ±1. With one more indices in addition to the two indices, again we subtract ±1 once more. The rule to use in Eqs. (8) and (9) is to subtract (N − 1) from X for N SU (5) indices. Because d = − 1 5 and e = 1 5 , one vectorlike pair of 10 and 10 are removed at the GUT scale and we obtain two 10 1/5 's from two 21 of Eq. (8) and one 10 1/5 from 35 of Eq. (9). In particular, note that 10 −1/5 of Eq. (9) contains N which can develop a VEV. Thus, there result three SM families. Therefore, for the chiral representations we treat the anti-SU(5) representations as usual. For the BEH scalars, we need U(1) charges of the anti-SU(5) as 5 −2/5 which houses H d and 5 +2/5 which houses H u . Now we can calculate the Yukawa coupling matrices for the quark sector. Here, we attempt to calculate V CKM , and comment on U PMNS in the end. For charged leptons including e + , µ + , τ + , which appear as SU(7) singlets, we must obtain all SU(7) singlet spectra. These singlets are not available at present. Thus, we try to calculate V CKM and U PMNS without the knowledge on the singlets. The CKM matrix is obtained if we know the Q em = 2 3 and −1 3 quark mass matrices, u L M (2/3) u R = 10 1/5 5 −3/5 Φ BEH,2/5 · (· · · ) d L M (−1/3) d R = 10 1/5 10 1/5 Φ BEH,−2/5 · (· · · ) (10) where we used the anti-SU(5) notation. For the PMNS matrix, we do not need information on the Yukawa couplings of the charged leptons. On the other hand, we need to know for the Dirac mass of N i (the (45) element of 10) and ν j and N i − N k masses. The Dirac mass coupling is shown in Fig. (a) and the Majorana mass term of N is shown in Fig. (b). The seesaw is a double seesaw as depicted in Fig  1 (c), which is obtained from Fig. 1(a) and Fig. 1(b). But, we do not need the charged lepton mass matrices. As commented in Ref. [1], the b-quark mass is expected to be much smaller than the t-quark mass, O( T 21 3,BEH T

B. A democratic submatrix of M weak
The multiplicity 2 of the fields from T 3 leads to a democratic form for the submatrix of the mass matrix. Thus, we consider 1 2 which can be diagonalized to give the eigenvalues 0 and 1. The democratic form can be extended to have a permutation symmetric form S 2 which has only singlet representations. Introducing two small numbers x and y (for the two independent singlets) for breaking the S 2 symmetry, it can be diagonalized to by A 3 × 3 mass matrix is changed, using a U 3×3 matrix, where U 3×3 contains the U 2×2 submatrix. Here, u's denote small parameters, breaking S 2 spontaneously by the GUT scale VEVs of some SM singlet fields: In view of the worry on the gravity spoil of discrete symmetries [26][27][28][29], two singlet fields are better to be two components of a doublet representation Φ of a hypothetical gauge group SU (2) in the bottom-up scenario. 4 The VEV Φ breaks the S 2 symmetry spontaneously [29]. Then, the trace of Φ quantum number is zero. Thus, trace of Eq. (13) is 1, leading to = 0. Thus, for the gravity-safe correction, which is our case arising from string compactification, let us diagonalize the democratic form to Therefore, from the information on the origin of families in the untwisted and twisted sectors (U 1 , T 3 , T + 5 ) [1], we can write the up-and down-type mass matrices as where the parameters in Eqs. (17,18) can be complex in general. Note that M (u) is not a Hermitian matrix and M (d) is a symmetric matrix. In the bases where Eqs. (17,18) are written, we proceed to calculate the CKM and PMNS matrices. Parameter 1 is given in the democratic form of the 2 × 2 matrix. But Eq. (18) is written in the bases where the democratic form is broken. Thus, we expect two parameters 1 (1 ± O(x s )). Since x s is small, we neglect this S 2 breaking correction. Similar comments apply to 2 and 3 .
where θ i represent the orthogonal matrix angles θ O,i . Here, O is taken as a real KS parametrization, where the angles are in the 1st quadrant. Angles given in (22) matches to Eq. (18). Thus, there is one angle parameter in V KS real , which is taken as However, because of the S 2 breaking effect as commented above, V which does not depend on V (u) R . The matrix elements and Q em = 2 3 quark masses have the following relations Thus, the mass matrix elements in the weak basis are where angles in V  (12) and (21) elements to be O(λ 4 ). By the same argument, we require both (13) and (31) elements to be O(λ 3 ). Thus, we require (11) O(λ 4 )m t (28) where we used m t = 173.21, m c = 1.275, and λ = sin θ 1 cos θ 3 = 0.2253. The determinant can be m u m c m t with (31), (22), and (13) elements for the orders given above. So, we take (11), (12), and (21) elements with inequality signs. Before presenting a numerical study, let us check that solutions suggested in Eqs. (28)(29)(30)(31)(32)(33)(34)(35)(36) are possible. From the (23) element, we restrict s 2 and s 3 at order λ 2 , Then, we satisfy (32) and (22) Now, the (12) element restricts s 1 at order λ 2 , Then, the (11) element is very small, O(λ 5 ). The remaining (21), (13), and (31) elements are where we considered m c = O(λ 2 )m t . Here the rough bound of Eqs. (28)(29)(30)(31)(32)(33)(34)(35)(36) are satisfied except in Eq. (41). But, m c is between O(λ 2 )m t and O(λ 3 )m t and Eqs. (41) is acceptable in our rough estimation. In our order of estimation, δ CKM is not restricted. 5 Therefore, the mass matrices Eqs. (17) and (18) obtained from anti-SU(7) UGUTF leads to a reasonable CKM matrix. Similarly, one can consider the lepton mixing angles which however need singlet contributions. Since there will appear additional parameters for the unkown heavy neutral lepton masses, there will be more freedom fitting for a reasonable PMNS matrix [37]. R . The color code is: the projection on θ 2 versus θ 1 for all allowed θ 3 and δ CKM as blue, and δ CKM versus θ 3 for all allwed θ 1 and θ 2 as red. We allowed the 1 σ for θ 1 , θ 2 , θ 3 and δ CKM in V . For the right-hand sides (of equality or inequalities) in Eqs. , the expansion parameter λ n is varied in the region a n ≤ λ n ≤ b n . In Fig. 2 (a), we choose a = 2 3 sin θ C and b = 3 2 sin θ C . In Fig. 2 (b), we choose a = 1 1.2 sin θ C and b = 1.2 sin θ C . From Fig. 2, we conclude that the mass matrices Eqs. (17) and (18), suggested from the UGUTF anti-SU (7), are phenomenologically allowed.

V. CONCLUSION
We presented bounds on the mixing angles of the right-handed currents, diagonalizing the quark mass matrices, suggested from a recently proposed families unified GUT model based on anti-SU(7) [1]. The investigation suggests that quark mass matrices suggested in [1] are phenomenologically allowable, and a numerical search is presented in figures on four mixing angles of V (u) R within the 1σ bounds of the CKM parameters, θ 1 , θ 2 , θ 3 , and δ CKM . Along the way, we also commented on some aspects of the Jarlskog determinant. In particular, the currently allowed CKM parametrization falls into three classes for choosing δ CKM = α, β, or γ of the PDG book. The Kobayashi-Maskawa and Kim-Seo parametrization choose δ CKM = α and Chau-Keung-Maiani parametrization chooses δ CKM = γ. It suggests that with three real CKM angles fixed, the Jarlskog determinant is maximum with α = π 2 .