Discrete Symmetry and CP Phase of the Quark Mixing Matrix

A simple specific pattern of the two 3 X 3 quark mass matrices is proposed, resulting in a prediction of the CP phase of the charged-current mixing matrix V_CKM, i.e. sin 2 phi_1 (beta) = 0.733, which is in remarkable agreement with data, i.e. sin 2 phi_1 = 0.728 +/- 0.056 +/- 0.023 from Belle and sin 2 beta = 0.722 +/- 0.040 +/- 0.023 from Babar. This pattern can be maintained by a discrete family symmetry, an example of which is D_7, the symmetry group of the heptagon.

The three families of quarks have very different masses and mix with one another in the charged-current mixing matrix V CKM in a nontrivial manner. This 3 × 3 matrix has three angles and one phase, the latter being the source of CP nonconservation in the Standard Model (SM) of particle interactions. In the context of the SM, this phase is now measured with some precision, i.e. sin 2φ 1 = 0.728 ± 0.056 ± 0.023 (1) from Belle [1], and sin 2β = 0.722 ± 0.040 ± 0.023 (2) from Babar [2], where φ 1 (also known as β) is defined as the phase of the element V td , i.e.
Together with |V us |, |V cb |, and |V ub |, the entire V CKM matrix can now be fixed, up to sign and phase conventions. Given the experimentally measured values of these parameters, is there a pattern to be recognized? The answer is not obvious, because the relevant physics comes from the structure of the two 3 × 3 quark mass matrices from which the observed quark mixing matrix is obtained: A theoretically consistent approach to understanding M u and M d is to extend the Lagrangian of the SM to support a family symmetry in such a way that the forms of these mass matrices are restricted with fewer parameters than are observed, thus making one or more predictions. Because of complex phases, this is often not a straightforward proposition. In this paper, we advocate a simple specific pattern, i.e. M u is diagonal, whereas M d is of the which was first proposed by one of us long ago [3]. The difference here is that whereas |ξ| was fixed at m u /m c in that model, it is now a free parameter. The family symmetry used previously was S 3 × Z 3 , which still works, but with different Z 3 assignments and a larger Higgs sector. As a more elegant example for our discussion, we choose instead D 7 , the symmetry group of the heptagon [4]. A recent proposal [5] based on Q 6 has both M u and M d of the form of Eq. (7), but with ξ = 0. To maintain this latter condition consistently, an extra Z 12 symmetry has to be assumed. Here ξ is simply another parameter, equal to the ratio of two arbitrary vacuum expectation values.
The group D 7 has 14 elements, 5 equivalence classes, and 5 irreducible representations.
Its character table is given by class n h χ 1 χ 2 χ 3 χ 4 χ 5 Here n is the number of elements and h is the order of each element. The numbers a k are given by a k = 2 cos(2kπ/7). The character of each representation is its trace and must satisfy the following two orthogonality conditions: where n = i n i is the total number of elements. The number of irreducible representations must be equal to the number of equivalence classes.
The three irreducible two-dimensional representations of D 7 may be chosen as follows.
For D n with n prime, there are 2n elements divided into (n + 3)/2 equivalence classes: C 1 contains just the identity, C 2 has the n reflections, C k from k = 3 to (n + 3)/2 has 2 elements each of order n. There are 2 one-dimensional representations and (n − 1)/2 two-dimensional ones. For D 3 = S 3 , the above reduces to the "complex" representation with ω = exp(2πi/3) discussed in a recent review [6].
The group multiplication rules of D 7 are: where 2 4,5 means 2 1,2 . In particular, let (a 1 , a 2 ), (b 1 , b 2 ) ∼ 2 1 , then In the decomposition of 2 1 × 2 2 , we have instead To arrive at our proposed pattern, let where the scalar fields φ d,u i are distinguished by an extra symmetry such as supersymmetry so that they couple only to d c , u c respectively. Using the multiplication rules listed above, we see that M u is indeed diagonal, and M d is of the form of Eq. (7), with a, d coming from φ d 3 and (b, ξb), (c, ξc) from φ d 1,2 respectively. These latter are distinct from φ u 1,2 , so that the constraint |ξ| = m u /m c in Ref. [3] no longer applies.
As in Ref. [3], we can redefine the phases of M d so that a, b, c, d are real, but ξ is complex.
Assuming that a 2 << b 2 and |ξ| 2 << 1, then to a very good approximation, After rotating the phase of V us to make it real to conform to the standard convention, we then predict sin 2φ 1 = 0.733, in remarkable agreement with experiment, i.e. Eqs. (1) and (2). The three angles φ 1 (β), In the future, these input parameters will be determined with more precision and our model will be more severely tested.
Flavor-changing neutral-current interactions are mediated by the three neutral Higgs bosons in the d sector with Yuakawa couplings given by where q i , q c j are the basis states of the mass matrix M d of Eq. (7). Let then V = V CKM and V c is its analog for the charge-conjugate states. In this model, they are approximately given by where ξ = v 2 /v 1 , and Using q i = V iα d α and q c j = V c jβ d c β , we can rewrite the couplings of φ 0 1,2,3 in terms of the quark mass eigenstates and evaluate their contributions to flavor-changing processes such as K −K and B − B mixings, etc.
An important point to notice [9] is that if φ 1,2 are replaced by φ 3 in the Yukawa sector, then there would be no flavor-changing interactions at all. Hence all such effects are contained in the terms φ 0 Whereas the mass of the SM combination (v 1 φ 0 be of order the electroweak breaking scale, the two orthogonal combinations contained in the above are allowed to be much heavier, say a few TeV.
The K L − K S mass difference gets its main contribution from the (q 1 q c Using f K = 114 MeV, B K = 0.4, v 1 = 100 GeV, and m 2 = 7 TeV, we find this contribution to be 2.5 × 10 −17 , well below the experimental value of 7.0 × 10 −15 . Similarly, and as advocated in Ref. [4], which is a successful description of neutrino oscillation phenomena.
This hints at the intriguing possibility that despite their outward dissimilarities, both quark and lepton family structures may actually come from the same mold.
In conclusion, we have pointed out that the M d of Eq. (7) predicts the correct value of the CP phase of the quark mixing matrix. Its form is derivable from a discrete family symmetry such as D 7 , which also works for leptons as previously shown. Extra Higgs doublets are