Maximal CP Violation Hypothesis and Phase Convention of the CKM Matrix January 13, 2004, at YITP Y. Koide (University of Shizuoka) Based on hep-ph/ (to appear in Phys.Lett.B)

Abstract The maximal CP violation hypothesis depends on the phase convention of the Cabibbo-Kobayashi-Maskawa matrix. Phase conventions which lead to successful prediction under the maximal CP violation hypothesis are only two: the original K-M phase convention and the F-X phase convention. Thereby, possible structures of the quark mass matrices are speculated.

Contents 1 Experimental status of the unitary triangle 2 Maximal CP violation hypothesis and the numerical results -- Examples 3 Why does the shape of the unitary triangle depend on the phase convention ? 4 General expressions of the CKM matrix and the related formulae 5 Quark mass matrices speculated from the Fritzsch-Xing phase convention 6 Application to the lepton sector 7 Summary

1 Experimental Status of the Unitary Triangle Unitary condition on the CKM matrix

From decays: From the best fit of the CKM parameters: PDG2004

2 Maximal CP violation hypothesis and the numerical results -- Examples We define the rotation matrices:

Case of the standard phase convention of the CKM matrix where

Rephasing invariant quantity J Maximal CP Violation Hypothesis The CP violation phase is chosen so that J is maximal. The predicted value of  is favorable, but the value of  is in disagreement.

Case of the original KM phase convention Max CPV hypothesis predicts in good agreement with experiments

3 Why does the shape of the unitary triangle depend on the phase convention ? The CKM matrix is rephasing invariant. Should the shape of the unitary triangle be independent of the phase convention? 

Note that in the present maximal CPV hypothesis we have assumed that only free parameter is a CPV phase  and the rotation angles are fixed.

Assumptions The phase factors in the quark mass matrices M f (f=u,d) are factorized by the phase matrices P f as where are real matrices and so that the CKM matrix V is given by where (3.4)

The quark masses m fi are only determined by. In other words, the rotation parameters are given only in terms of the quark mass ratios, and independent of the CPV phase. In such a scenario, the maximal CPV hypothesis means that the CPV phase  takes its maximum value without changing the quark mass values.

4 General expressions of the CKM matrix and the related formulae Let us define the CKM matrix V(i,k) as V(i,k) = R i T P j R j R k. (4.1) Then, for the 9 cases of V(i,k), the rephasing invariant quantity J is given by (4.2) Also see, Fritzsch-Xing, PRD57, 594 (1998).

The angles (l=1,2,3) in the unitary triangle are also given by (4.3) where (l,m,n) is a cyclic permutation of (1,2,3), and. Note that the magnitudes are independent of the phase .

Under the approximation we obtain the following 4 types of J: (A) (4.4) for V(1,2), V(1,3), V(2,1) and V(2,3), (B) (4.5) for V(1,1) and V(3,3), (C) (4.6) for V(3,1) and V(3,2), and (D) (4.7) for V(2,2).

Under the maximal CPV hypothesis, only two cases can give the observed shape of the CKM matrix and value of J: V(1,1): the original Kobayashi-Maskawa phase convention [PTP 49, 652 (1973)] V(3,3): the Fritzsch-Xing phase convention [PLB413, 396 (1997)]  V(1,1) 90.0 o 23.2 o 66.8 o V(3,3) 89.0 o 23.2 o 67.8 o Experiment

5 Quark mass matrices speculated from the F-X phase convention The successful case V(3,3) = R 3 T P 1 R 1 R 3 (5.1) suggests the following quark mass matrix structure: F ritzsch-Xing, PLB413, 396 (1997) Xing, PRD68, (2003)

The explicit forms of V and M are given by

If we assume M d 11 =0, we obtain the well-known relation Also if we assume M u 11 =0, we obtain which is roughly consistent with

If we assume M u 22 =0 together with s d 23 =0, we obtain (5.8) For a further detailed phenomenological study of the V(3,3) model with the renormalization group effects, see Xing, PRD68, (2003).

6 Application to the lepton sector From the observed fact (6.1)

We can classify the prediction of J into the following three types: (A) (6.2) for V(1,3), V(2,3), V(1,2), V(1,1), and V(3,3), (B) (6.3) for V(3,1) and V(2,1), and (C) (6.4) for V(3,2) and V(2,2).

From the analogy to the quark sector, we consider that the lepton mixing matrix U is also given by V(3,3). Then, the maximal CPV hypothesis predicts (6.5) The requirement M e 11 =0 predicts (6.6) where we have assume s 23 =  /4.

7 Summary (1) Under the maximal CPV hypothesis, only two expressions V(1,1) and V(3,3) can give the successful predictions for unitary triangle:  90 o,  23 o,  67 o  (2) The F-X expression V(3,3) suggests a quark mass matrix structure which leads to under 

Open Questions (1) What mechanism can cause such a maximal CP violation? (2) What mechanism can give the successful quark mass matrix structure, ? (3) Is there a simple ansatz for the mixing angle  23 ? Especially,  23 =  /4 for the lepton sector.

Phenomenology of the unitary triangle will provide a promising clue to the unified understanding of the quark and lepton mass matrices. Thank you!