January 13, 2004, at YITP
Maximal CP Violation
Hypothesis and
Phase Convention
of the CKM Matrix
Y. Koide (University of Shizuoka)
Based on hep-ph/0411280
(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
PDG2004
From
decays:
From the best fit of the CKM parameters:
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 b is favorable,
but the value of g 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 d
and the rotation angles are
fixed.
Assumptions
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 d
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) = 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 d .
.
Under the approximation
we obtain the following 4 types of J:
(A)
for V(1,2), V(1,3), V(2,1) and V(2,3),
(B)
for V(1,1) and V(3,3),
(C)
for V(3,1) and V(3,2), and
(D)
for V(2,2).
(4.4)
(4.5)
(4.6)
(4.7)
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)]
a
b
g
90.0o
23.2o
66.8o
V(3,3)
89.0o
Experiment
23.2o
67.8o
V(1,1)
5 Quark mass matrices speculated
from 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 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)
for V(3,1) and V(2,1), and
(6.3)
(C)
for V(3,2) and V(2,2).
(6.4)
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=p/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:
a= 90o, b=23o, g=67o .
(2) The F-X expression V(3,3) suggests a quark mass
matrix structure
which leads to
under
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 q23?
Especially, q23=p/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!