What does the Observed What does the Observed Shape of the Unitary Shape of the Unitary

Triangle Tell Us? Triangle Tell Us?

February 15, 2005, at Sendai

Y. Koide (University of Shizuoka)

Based on hep-ph/0502054 and

PLB607, 123 (2005) (hep-ph/0411280)

AbstractAbstract

We assume that, of 4 parameters in the CKM matrix (3 rotation angles and 1 phase parameter the 3 rotation angles are completely fixed by the quark mass ratios, while the CP violating phase parameter is independent of the rotation angles (i.e. quark masses). Then the shape of the unitary triangle depends on the phase conventions of the CKM matrix. Therefore, we investigate what phase conventions are in favor of the observed unitary triangle. Thereby, possible structures of the quark and lepton mass matrices are speculated.

ContentsContents1 Experimental status of the unitary triangle

2 Phase conventions of the CKM matrix and the related formulae

3 Why do we pay attention to the phase conventions ?

4 Maximal CP violation hypothesis

5 Minimal circumscribed-circle hypothesis

6 Application to the lepton sector

7 Quark-to-lepton correspondence

8 Concluding remarks

1 Experimental status of 1 Experimental status of the unitary trianglethe unitary triangle

Unitary condition on the CKM matrix is given by

　　　 From decays:

　 From the best fit of the CKM parameters:　　　　

PDG2004

2 Phase conventions of the CKM 2 Phase conventions of the CKM matrix and the related formulaematrix and the related formulae

We define the rotation matrices:

Then, the general expressions of the CKM matrix V are given by

V(i,k) = RiT PjRj Rk , (2.4)

where Pj are phase matrices

and so on.• V(1,3): the standard phase convention

• V(1,1): the original KM phase convention

• V(3,3): the Fritzsch-Xing phase convention

Then, for the 9 cases of V(i,k), the rephasing invariant quantity J is given by

(2.6)

The angles (l=1,2,3) in the unitary triangle are

also given by

(2.7)

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) (2.8)

for V(1,2), V(1,3), V(2,1) and V(2,3),

(B) (2.9)

for V(1,1) and V(3,3),

(C) (2.10)

for V(3,1) and V(3,2), and

(D) (2.11)

for V(2,2).

Thus, the predictions are dependent on the phase conventions.

33 Why do we pay attention Why do we pay attention to the phase conventions?to the phase conventions?

The shape of the unitary triangle is dependent on the phase conventions

when we assume that only free parameter is a CPV phase parameter and the rotation angles are fixed.

Let us investigate the relations between the phase conventions V(i,k) and qaurk mass matrix structues.

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)

Thus, the rotation angles in the CKM matrix are determined only by , i.e. only by the quark mass ratios, and the CP violating parameter is independent of the fermion mass ratios.

For example, the phase convention V(1,3)

suggests the following quark mass matrix structure:

4 Maximal CP violation hypothesis4 Maximal CP violation hypothesis

Y.K., Phys.Lett.B607, 123 (2005):hep-ph/0411280

We assume that

only the free parameter is a CPV phase and the rotation angles are fixed.

Maximal CP Violation Hypothesis

The CP violation phase is chosen so that J is maximal, where rotation angles are fixed, i.e. ittakes the value

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

Especially, we notice the phenomenological success of the phase convention V(3,3).

The successful case

V(3,3) = R3T P1 R1 R3 (4.1)

suggests the following quark mass matrix

structure:

(4.2)

Fritzsch-Xing, PLB413, 396 (1997)

Xing, PRD68, 073008 (2003)

If we assume Md11 =0, we obtain

the well-known relation

　　　　　　　　　　　　　　　　　　　　　　 (4.3)

Also if we assume Mu11 =0, we obtain

(4.4)

which is somewhat deviated from

(4.5)

5 Minimal circumscribed-5 Minimal circumscribed-circle hypothesis circle hypothesis

　　　　　　　　　　　　 hep-ph/0502054

We consider that the discrepancy of the prediction (4.4) from the observed value (4.5) is serious. Therefore, in this section, we seek for another phenomenological ansatz for the unitary triangle.

We assume that the radius R() of the circle circumscribed about the unitary triangle takes its minimum value.

The radius R() is given by the sine rule

(5.1)

where

(5.2)

and .

Note that the side ri is independent of the

parameter in the expression V(i,k) .

Therefore, the minimum of the radius R() means the maximum of sini( .

We put a further assumption:

the phase parameter takes own value so that sin i(takes its maximal value,

i.e. sin i(

Then, only the two cases V(2,3) and V(2,1) can give reasonable predictions

(5.3)

Quark mass matrix forms Quark mass matrix forms suggested by V(2,3)suggested by V(2,3)

We notice the successful phase convention V(2,3):

(5.4)

Then, the form V(2,3) suggests the following quark mass matrix structures

(5.5)

Now we put simple requirements on the quark mass matrix textures:

Mu11=0 :

(5.6)

Md11=0 :

(5.7)

which are in excellent agreement with

(5.8)

(5.9)

so that we obtain

(5.10)

(5.11)

(5.12)

(5.13)

which is in excellent agreement with

. (5.14)

Now, we put further phenomenological constraints

Notice the appearance of the factor 2 in Eq.(5.11).

However, at present, those requirements

Mu11=0 , Md

11=0 ,

are only phenomenological ones.

Although the origins of those requirements are unknown, in the next section, we will put similar requirements on the lepton sector, and we will obtain successful predictions.

6 Application to the lepton sector6 Application to the lepton sector

We assume

M11=0

Me11=0

We assume that the lepton mixing matrix U is also given by the type V(2,3).

We obtain

Notice that only when the neutrino mass hierarchy is a normal type, we can naturally obtain sin22atm=1 from the relation

,

where the existence of the factor 2 is essential.

7 Quark-to-lepton 7 Quark-to-lepton correspondencecorrespondence

Since we assume that

and are given by the type V(2,3), the mass matrix structures ostensibly look like an inverse quark-to-lepton correspondence

However, the normal correspondence

is recovered by the constraints on the mass matrix elements given in Sec.5 and Sec.6

as you see below:

The final forms under those phenomenological constraints are given by

(7.1)

for Mu and M, and

(7.2)

for Md and Me.

Thus, the normal correspondence is recovered.

Here we read (7.1) and (7.2) as follows:

= c23/s23 > 1, b > 0 for Mu,

- = s23/c23 < 1, b < 0 for M,

7 Concluding remarks7 Concluding remarks(1) Under the maximal CP violation hypothesis, only

two expressions V(1,1) and V(3,3) can give the

successful predictions for unitary triangle:

90o, 23o, 67o Also, under the minimal circumscribed-circle

hypothesis, only two expressions V(2,1) and V(2,3) can give the same results 90o, 23o, 67o

(2) From the phenomenological point of view, the case V(2,3) is the most attractive one:

(3) Under some phenomenological constraints on the mass matrix elements, we obtain simple structures of quark and lepton mass matrices:

for Mu , M,

for Md , Me.

Those mass matrices successfully predict

sin22atm=1

Open QuestionsOpen Questions

(1) What mechanism can cause such a minimal circumscribed-circle?

(2) What mechanism (symmetries) can give the successful quark and lepton mass matrix structures?

In the present work, the following assumption is essential:

the CP violation parameter is independent of the fermion mass ratios (i.e. the rotation angles).

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

Thank you!