It is shown that two specific properties of the unitary matrix $V$ can be
expressed directly in terms of the matrix elements and eigenvalues of the
hermitian matrix $M$ which is diagonalized by $V$. These are the asymmetry
$\Delta(V)= |V_{12}|^2- |V_{21}|^2$, of $V$ with respect to the main diagonal
and the Jarlskog invariant $J(V)= {\rm Im}(V_{11}V_{12}^* V_{21}^* V_{22})$.
These expressions for $\Delta(V)$ and $J(V)$ provide constraints on possible
mass matrices from the available data on $V$.Comment: 5 pages, Late

Kobayashi M.

Maki Z.

S. CHATURVEDI

VIRENDRA GUPTA

English

arXiv

Crossref

CONSTRAINTS ON MASS MATRICES DUE TO MEASURED PROPERTY OF THE MIXING MATRIX

It is shown that two specific properties of the unitary matrix V can be expressed directly in terms of the matrix elements and eigenvalues of the hermitian matrix M which is diagonalized by V. These are the asymmetry Δ(V) = |V12|2-|V21|2, of V with respect to the main diagonal and the Jarlskog invariant J(V ) = Im(V11V ∗12V ∗21V22). These expressions for Δ(V ) and J(V ) provide constraints on possible mass matrices from the available data on V

Chaturvedi, S.

Gupta, Virendra

Indian Academy of Sciences

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Constraints on mass matrices due to measured
property of the mixing matrix
S. Chaturvedi ∗
School of Physics
University of Hyderabad, Hyderabad 500 046, India
Virendra Gupta †
Departamento de Fiśica Aplicada, CINVESTAV-Unidad Mérida
A.P. 73 Cordemex 97310 Mérida, Yucatan, Mexico
February 2, 2008
Abstract
It is shown that two specific properties of the unitary matrix V can
be expressed directly in terms of the matrix elements and eigenvalues
of the hermitian matrix M which is diagonalized by V . These are
the asymmetry ∆(V ) = |V12|
2 − |V21|
2, of V with respect to the main
diagonal and the Jarlskog invariant J(V ) = Im(V11V
∗
12
V
∗
21
V22). These
expressions for ∆(V ) and J(V ) provide constraints on possible mass
matrices from the available data on V .
∗scsp@uohyd.ernet.in
†virendra@aruna.mda.cinvestav.mx
1
1 Introduction
Flavor mixing in both the quarks and lepton sector has been firmly es-
tablished experimentally for a long time, However, there is still no deep
understanding of the observed mixings.
In the Standard Model, the unitary mixing matrices arise from the di-
agonalization of the corresponding hermitian mass matrices. In the lepton
sector one usually works in the basis in which the charged lepton mass ma-
trix is diagonal, so that the neutrino flavor mixing is described by a single
unitary matrix [1] which diagonalizes the neutrino mass matrix. In the quark
sector [2], in the physical basis, the CKM-mixing matrix V = U †U ′, where
the unitary matrices U and U ′ diagonalize the up-quark and down-quark
mass matrices respectively. One can also work in a basis in which the up-
quark ( down-quark ) mass matrix M ( M ′ ) is diagonal. In these bases,
the mixing matrix in the quark sector ( like the neutrino sector ) will come
from a single mass matrix. Clearly, if we knew the mass matrices fully then
the corresponding mixing matrices are completely determined. In practice,
the mass matrices are guessed at, while experiment can only determine the
numerical values of the matrix elements of the mixing matrix. Our objec-
tive is to learn something about the structure of the underlying mass matrix
M from the knowledge of the mixing matrix V which diagonalizes it. Can
a general property of V imply a constraint on M ? In particular we show
that the asymmetry ∆(V ) w.r.t. the main diagonal and, J(V ), the Jarlskog
invariant [3] which is a measure of CP-violation can be directly expressed
in terms of the eigenvalues and matrix elements of M ! These can provide
simple criterion for selecting suitable mass matrices.
2 Derivations of the formulas for ∆(V ) and
J(V )
Consider a 3 × 3 mass matrix M which is diagonalized by the unitary
matrix V , so that
M = V M̂ V †, (1)
where M̂ = diag(m1, m2, m3). We can also write
M = m1 N1 + m2 N2 + m3 N3, (2)
where Nα are the projectors of M . They satisfy,
Nα Nβ = Nα δαβ and (Nα)kℓ = Vkα V
∗
ℓα. (3)
2
Furthermore, in terms of M and its eigenvalues,
Nα =
(mβ − M)(mγ − M)
(mβ − mα)(mγ − mα)
, α 6= β 6= γ, (4)
with α, β, γ taking values from 1 to 3. It is clear from Eq.(3) that
|Vkα|
2 = (Nα)kk. (5)
Through this equation each |Vkα| can be calculated in terms of the eigenval-
ues1 and matrix elements of M .
(a) Formula for ∆(V )
The asymmetry with respect to the main diagonal of V is given by
∆(V ) ≡ |V12|
2 − |V21|
2 = |V23|
2 − |V32|
2 = |V31|
2 − |V13|
2. (6)
The last two equations follow from the unitarity of V , namely, V V † = V †V =
I. Using Eqs.(4, 5), simple algebra gives,
∆(V ) =
1
D(m)
{
∑
k
(
mk (M
2)kk − m
2
k Mkk
)
}, (7)
where
D(m) ≡
∣∣∣∣∣∣∣
1 1 1
m1 m2 m3
m2
1
m2
2
m2
3
∣∣∣∣∣∣∣
= (m2 − m1)(m3 − m1)(m3 − m2). (8)
Our result in Eq.(7) tells us, given mi and M , whether V will be symmetric
(∆(V ) = 0) or not. We note that the asymmetry for the CKM-matrix, ∆(V )
is intriguingly small!
(b) Formula for J(V )
We use the definition
J(V ) = Im(V11V
∗
12
V ∗
21
V22). (9)
The imaginary parts of eight other plaquettes are just ±J(V ) because V is
unitary [3]. To derive our result we note that
M12M23M
∗
13
=
∑
k,ℓ,n
mkmℓmnV1kV
∗
2kV2ℓV
∗
3ℓV
∗
1nV3n, (10)
1Our results require non-degenerate eigenvalues. This is true for the quarks.
3
since from Eq.(1), Mij =
∑
k mkVikV
∗
jk. Now use the unitarity relation
V ∗
1ℓV1n + V
∗
2ℓV2n + V
∗
3ℓV3n = δℓn and take imaginary parts to obtain
Im(M12M23M
∗
13
) =
∑
k,ℓ
mkm
2
ℓ Im(V1kV
∗
2kV2ℓV
∗
1ℓ)
−
[
∑
n
mn(|V1n|
2 + |V2n|
2)
]
·
∑
k,ℓ
mkmℓ Im(V1kV
∗
2kV2ℓV
∗
1ℓ)
(11)
The imaginary parts on the RHS, for various plaquettes of V , yield ±J(V )
for various values of k and ℓ. As a result the second term sums up to zero.
One thus obtains
J(V ) =
Im(M12M23M
∗
13
)
D(m)
. (12)
This remarkable result shows that if M12M23M
∗
13
is real for a given M, then
the Jarlskog invariant for the matrix V which diagonalizes it vanishes. Thus
to obtain CP-violation, the mass matrix for up-quark (down- quark) must
have Im(M12M23M
∗
13
) non-zero in a basis in which down-quark (up-quark)
mass matrix is diagonal. Equivalently, Θ ≡ θ12 + θ23 − θ13 6= nπ (n =
0, 1, 2, · · ·), here θij is the phase of Mij. This is reminiscent of the fact
[4] that physically the relevant phase for CP-violation in CKM-matrix is
Φ ≡ φ12 + φ23 − φ13, where φij is the phase of Vij. The reason is that
Φ is invariant under re-phasing transformations. Clearly, under re-phasing
transformations Θ is also invariant (See Eq.(11)). Furthermore it can be
shown directly that if any one of the three off-diagonal elements of V is zero
then J(V ) vanishes [5]. Thus the appearance of the numerator in Eq.(12) is
understandable.
The use of Eqs.(7) and (12) for the quark sector will be considered else-
where.
Acknowledgments One of us (VG) is grateful to Dr. A. O. Bouzas for
discussions.
References
[1] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 247 (1962);
V. N. Gribov and B. M. Pontecorvo, Phys. Lett. B 28, 493 (1969).
[2] M.Kobayashi and T.Maskawa,Prog.Theor.Phys. D 35, 652 (1973); N.
Cabibbo Phys. Rev. Lett. 10, 531 (1953); L. L. Chau and W.-T. Keung,
Phys. Rev. Lett. 53, 1802 (1984).
4
[3] C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985).
[4] M. Gronau and J. Schechter, Phys. Rev. Lett. 54, 385 (1985); Phys.
Rev. Lett. 54, 2176 (1985). Other references can be found in these.
[5] Let Mij = m1Vi1V
∗
j1 + m2Vi2V
∗
j2 + m3Vi3V
∗
j3 = 0, i 6= j. Multiply by
V ∗i3Vj3 and take imaginary parts to obtain (m1−m2)J(V ) = 0. Similarly,
multiplying by V ∗i2Vj2 and V
∗
i1Vj1 and taking imaginary parts one obtains
(m1−m3)J(V ) = 0 and (m2−m3)J(V ) = 0. Thus either m1 = m2 = m3,
that is, M is trivial or J(V ) = 0.
[6] H. Fritzsch, Phys. Lett. B 70, 436 (1977); Phys. Lett. B 166, 423(1986).
[7] T. Kitazoe and K. Tanaka, Phys. Rev. 18, 3476 (1978); H. Georgi and
D. Nanapoulos, Phys. Lett. B 82, 392 (1979); B. Stech, Phys. Lett. B
130, 189 (1983).
5


Constraints on mass matrices due to measured property of the mixing matrix

Publications of the IAS Fellows

ar
X
iv
:h
ep
-p
h/
05
09
02
5v
1 
 3
 S
ep
 2
00
5
Constraints on mass matrices due to measured
property of the mixing matrix
S. Chaturvedi ∗
School of Physics
University of Hyderabad, Hyderabad 500 046, India
Virendra Gupta †
Departamento de Fis´ica Aplicada, CINVESTAV-Unidad Me´rida
A.P. 73 Cordemex 97310 Me´rida, Yucatan, Mexico
February 2, 2008
Abstract
It is shown that two specific properties of the unitary matrix V can
be expressed directly in terms of the matrix elements and eigenvalues
of the hermitian matrix M which is diagonalized by V . These are
the asymmetry ∆(V ) = |V12|
2 − |V21|
2, of V with respect to the main
diagonal and the Jarlskog invariant J(V ) = Im(V11V
∗
12
V
∗
21
V22). These
expressions for ∆(V ) and J(V ) provide constraints on possible mass
matrices from the available data on V .
∗scsp@uohyd.ernet.in
†virendra@aruna.mda.cinvestav.mx
1
1 Introduction
Flavor mixing in both the quarks and lepton sector has been firmly es-
tablished experimentally for a long time, However, there is still no deep
understanding of the observed mixings.
In the Standard Model, the unitary mixing matrices arise from the di-
agonalization of the corresponding hermitian mass matrices. In the lepton
sector one usually works in the basis in which the charged lepton mass ma-
trix is diagonal, so that the neutrino flavor mixing is described by a single
unitary matrix [1] which diagonalizes the neutrino mass matrix. In the quark
sector [2], in the physical basis, the CKM-mixing matrix V = U †U ′, where
the unitary matrices U and U ′ diagonalize the up-quark and down-quark
mass matrices respectively. One can also work in a basis in which the up-
quark ( down-quark ) mass matrix M ( M ′ ) is diagonal. In these bases,
the mixing matrix in the quark sector ( like the neutrino sector ) will come
from a single mass matrix. Clearly, if we knew the mass matrices fully then
the corresponding mixing matrices are completely determined. In practice,
the mass matrices are guessed at, while experiment can only determine the
numerical values of the matrix elements of the mixing matrix. Our objec-
tive is to learn something about the structure of the underlying mass matrix
M from the knowledge of the mixing matrix V which diagonalizes it. Can
a general property of V imply a constraint on M ? In particular we show
that the asymmetry ∆(V ) w.r.t. the main diagonal and, J(V ), the Jarlskog
invariant [3] which is a measure of CP-violation can be directly expressed
in terms of the eigenvalues and matrix elements of M ! These can provide
simple criterion for selecting suitable mass matrices.
2 Derivations of the formulas for ∆(V ) and
J(V )
Consider a 3 × 3 mass matrix M which is diagonalized by the unitary
matrix V , so that
M = V M̂ V †, (1)
where M̂ = diag(m1, m2, m3). We can also write
M = m1 N1 +m2 N2 +m3 N3, (2)
where Nα are the projectors of M . They satisfy,
Nα Nβ = Nα δαβ and (Nα)kℓ = Vkα V
∗
ℓα. (3)
2
Furthermore, in terms of M and its eigenvalues,
Nα =
(mβ −M)(mγ −M)
(mβ −mα)(mγ −mα)
, α 6= β 6= γ, (4)
with α, β, γ taking values from 1 to 3. It is clear from Eq.(3) that
|Vkα|
2 = (Nα)kk. (5)
Through this equation each |Vkα| can be calculated in terms of the eigenval-
ues1 and matrix elements of M .
(a) Formula for ∆(V )
The asymmetry with respect to the main diagonal of V is given by
∆(V ) ≡ |V12|
2 − |V21|
2 = |V23|
2 − |V32|
2 = |V31|
2 − |V13|
2. (6)
The last two equations follow from the unitarity of V , namely, V V † = V †V =
I. Using Eqs.(4, 5), simple algebra gives,
∆(V ) =
1
D(m)
{
∑
k
(
mk (M
2)kk −m
2
k Mkk
)
}, (7)
where
D(m) ≡
∣∣∣∣∣∣∣
1 1 1
m1 m2 m3
m2
1
m2
2
m2
3
∣∣∣∣∣∣∣
= (m2 −m1)(m3 −m1)(m3 −m2). (8)
Our result in Eq.(7) tells us, given mi and M , whether V will be symmetric
(∆(V ) = 0) or not. We note that the asymmetry for the CKM-matrix, ∆(V )
is intriguingly small!
(b) Formula for J(V )
We use the definition
J(V ) = Im(V11V
∗
12
V ∗
21
V22). (9)
The imaginary parts of eight other plaquettes are just ±J(V ) because V is
unitary [3]. To derive our result we note that
M12M23M
∗
13
=
∑
k,ℓ,n
mkmℓmnV1kV
∗
2kV2ℓV
∗
3ℓV
∗
1nV3n, (10)
1Our results require non-degenerate eigenvalues. This is true for the quarks.
3
since from Eq.(1), Mij =
∑
k mkVikV
∗
jk. Now use the unitarity relation
V ∗
1ℓV1n + V
∗
2ℓV2n + V
∗
3ℓV3n = δℓn and take imaginary parts to obtain
Im(M12M23M
∗
13
) =
∑
k,ℓ
mkm
2
ℓ Im(V1kV
∗
2kV2ℓV
∗
1ℓ)
−
[∑
n
mn(|V1n|
2 + |V2n|
2)
]
·
∑
k,ℓ
mkmℓ Im(V1kV
∗
2kV2ℓV
∗
1ℓ)
(11)
The imaginary parts on the RHS, for various plaquettes of V , yield ±J(V )
for various values of k and ℓ. As a result the second term sums up to zero.
One thus obtains
J(V ) =
Im(M12M23M
∗
13
)
D(m)
. (12)
This remarkable result shows that if M12M23M
∗
13
is real for a given M, then
the Jarlskog invariant for the matrix V which diagonalizes it vanishes. Thus
to obtain CP-violation, the mass matrix for up-quark (down- quark) must
have Im(M12M23M
∗
13
) non-zero in a basis in which down-quark (up-quark)
mass matrix is diagonal. Equivalently, Θ ≡ θ12 + θ23 − θ13 6= nπ (n =
0, 1, 2, · · ·), here θij is the phase of Mij. This is reminiscent of the fact
[4] that physically the relevant phase for CP-violation in CKM-matrix is
Φ ≡ φ12 + φ23 − φ13, where φij is the phase of Vij. The reason is that
Φ is invariant under re-phasing transformations. Clearly, under re-phasing
transformations Θ is also invariant (See Eq.(11)). Furthermore it can be
shown directly that if any one of the three off-diagonal elements of V is zero
then J(V ) vanishes [5]. Thus the appearance of the numerator in Eq.(12) is
understandable.
The use of Eqs.(7) and (12) for the quark sector will be considered else-
where.
Acknowledgments One of us (VG) is grateful to Dr. A. O. Bouzas for
discussions.
References
[1] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 247 (1962);
V. N. Gribov and B. M. Pontecorvo, Phys. Lett. B 28, 493 (1969).
[2] M.Kobayashi and T.Maskawa,Prog.Theor.Phys. D 35, 652 (1973); N.
Cabibbo Phys. Rev. Lett. 10, 531 (1953); L. L. Chau and W.-T. Keung,
Phys. Rev. Lett. 53, 1802 (1984).
4
[3] C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985).
[4] M. Gronau and J. Schechter, Phys. Rev. Lett. 54, 385 (1985); Phys.
Rev. Lett. 54, 2176 (1985). Other references can be found in these.
[5] Let Mij = m1Vi1V
∗
j1 + m2Vi2V
∗
j2 + m3Vi3V
∗
j3 = 0, i 6= j. Multiply by
V ∗i3Vj3 and take imaginary parts to obtain (m1−m2)J(V ) = 0. Similarly,
multiplying by V ∗i2Vj2 and V
∗
i1Vj1 and taking imaginary parts one obtains
(m1−m3)J(V ) = 0 and (m2−m3)J(V ) = 0. Thus eitherm1 = m2 = m3,
that is, M is trivial or J(V ) = 0.
[6] H. Fritzsch, Phys. Lett. B 70, 436 (1977); Phys. Lett. B 166, 423(1986).
[7] T. Kitazoe and K. Tanaka, Phys. Rev. 18, 3476 (1978); H. Georgi and
D. Nanapoulos, Phys. Lett. B 82, 392 (1979); B. Stech, Phys. Lett. B
130, 189 (1983).
5


Constraints on mass matrices due to measured property of the mixing matrix

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