Maximal CP Violation Maximal CP Violation Hypothesis and Hypothesis and

Phase ConventionPhase Conventionof the CKM Matrixof the CKM Matrix

January 13, 2004, at YITP

Y. Koide (University of Shizuoka)

Based on hep-ph/0411280

(to appear in Phys.Lett.B)

AbstractAbstract 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.

ContentsContents1 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 1 Experimental Status of the Unitary Trianglethe Unitary Triangle

Unitary condition on the CKM matrix

　　　 From decays:

　 From the best fit of the CKM parameters:　　　　

PDG2004

2 Maximal CP violation hypothesis 2 Maximal CP violation hypothesis and the numerical results and the numerical results

-- Examples-- Examples

We define the rotation matrices:

Case of the standard phase Case of the standard phase convention of the CKM matrixconvention 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 3 Why does the shape of the unitary triangle depend on the unitary triangle depend on the

phase convention ?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.

AssumptionsAssumptions

The phase factors in the quark mass matrices Mf (f=u,d) are factorized by the

phase matrices Pf as

where are real matrices and

so that the CKM matrix V is given by

where

(3.4)

• The quark masses mfi 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 4 General expressions of the CKM matrix and the related formulaematrix and the related formulae

Let us define the CKM matrix V(i,k) as

V(i,k) = RiT Pj Rj Rk . (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) 90.0o 23.2o 66.8o

V(3,3) 89.0o 23.2o 67.8o

Experiment

5 Quark mass matrices speculated 5 Quark mass matrices speculated from the F-X phase conventionfrom the F-X phase convention

The successful case

V(3,3) = R3T P1 R1 R3 (5.1)

suggests the following quark mass matrix

structure:

Fritzsch-Xing, PLB413, 396 (1997)

Xing, PRD68, 073008 (2003)

The explicit forms of V and M are given by

If we assume Md11 =0, we obtain

the well-known relation

Also if we assume Mu11 =0, we obtain

which is roughly consistent with

If we assume Mu22=0 together with

sd23=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, 073008 (2003).

6 Application to the lepton sector6 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 Me11=0 predicts

(6.6)

where we have assume s23=/4.

7 Summary7 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:

90o, 23o, 67o(2) The F-X expression V(3,3) suggests a quark mass

matrix structure

which leads to

under

under

Open QuestionsOpen 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!