CP odd weak basis invariants

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CP-odd Weak Basis Invariants for FGMTextures of Neutrino

Mass Matrices


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Low energy CP-violation in the leptonic sector

can be described using the following CP-odd WB

invariants

where

Low energy CP Invariance

Hl=M

lM

land H=M

M


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The invariant I1 was proposed by Jarlskog [1]as a rephasing invariant measure of Dirac type CPviolation in the quark sector. It, also, describes theCP violation in the leptonic sector and is sensitive

to the Dirac type CP violating phase. The invariantsI2

and I3were proposed by Branco, Lavoura and

Rebelo [2] as the WB invariant measures oMajorana type CP violation.

[1] C.Jarlskog, Phys.Rev.Lett. 55, 1039 (1985)

[2] G.C.Branco, L. Lavoura and M.N. Rebelo,Phys. Lett. B 180 (1986)264.

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The CP violation in the lepton number conserving

(LNC) processes is contained in Jarlskog CP

invariant J which can be calculated from the WB

invariant I1 using the relation

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Where me

, m

and m

are charged leptons masses and

m1, m2 and m3 are the eigenvalues of the complex

neutrino mass matrix M.


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In terms of the charged leptons masses and elements of

the complex neutrino mass matrixI1 can be written as

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Where the quantities Aee, A and A are given by


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The CP violation in lepton number violating (LNV)

processes can be calculated from the WB invariantsI2andI3 which have been given below:

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and


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Where the coefficients Bee, B and B are given by

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From above expressions of I1, I2 and I3 one can

immediately conclude that

A sufficient condition for CP conservation in LNC and

LNV processes is that all the diagonal entries in theneutrino mass matrix vanish. Therefore, the symmetric

neutrino mass matrices having Zee-type structure in the

diagonal charged lepton basis are CP conserving.

A sufficient condition for CP conservation in LNC

processes is that any three independent entries in the

neutrino mass matrix vanish.

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Implications for texture zeros

The seven allowed textures of the neutrino mass matrices

with Frampton, Glashow and Marfatia (FGM) texture zerostructure have been summarized in Table1.


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Class A:

For neutrino mass matrices of type A1 we obtain

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The corresponding WB invariants for type A2 can be obtained

by interchanging mu and tau indices in above equations.


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Class B:

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For neutrino mass matrices of type B1

we obtain

The WB invariants for neutrino mass matrices of type B2

can be obtained by interchanging the and indices in the above relations.


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For neutrino mass matrices of type B3 , we have

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The WB invariants for neutrino mass matrices of type B4

can be obtained by interchanging the

and indices in the above relations.


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Class C:

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For neutrino mass matrices of type C , we have


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Results and Discussion:

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CP invariance condition for A1 is equivalent to the

condition that the neutrino mass matrix M

can be

factorized asPM(r)P, wherePis a diagonal phase matrix

and M(r) is real neutrino mass matrix.


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The CP violation sttructure of Class C is muchdifferent from that of other classes. In class C the

quantityI1 depends upon the difference between

absolute squares of two matrix elements Me and Me

unlike other classes. So,I1 vanish if Me=Me. So,

this special case is - symmetric as M=M in

class C. Hence, neutrino mass matrices with class C

cannot exhibit CP violation in LNC processes.


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The three WB invariants for neutrino mass matrices are

related with one another in each class of neutrino mass

matrices with FGM textures and only one of them isindependent.

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Hence, the three CP violating phases are not

independent. There is only one independent physical

phase in the mass matrix and it contributes to both LNC

and LNV processes. However, such a phase cannot belabelled either Dirac or Majorana unambiguously since it

contributes to the CP violation in both LNC and LNV

processes. Hence, the distinction between Dirac and

Majorana phases cannot be maintained in the presence otwo texture zeros.

CP odd weak basis invariants

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