Recent measurements of the neutrino solar mixing angle and the Cabibbo angle
satisfy the empirical relation theta_{sol} + theta_{C} ~ 45^{o}. This relation
suggests the existence of a correlation between the mixing matrices of leptons
and quarks, the so called quark-lepton complementarity. Here, we examine the
possibility that this correlation originates in the strong hierarchy in the
mass spectra of quarks and charged leptons, and the seesaw mechanism that gives
mass to the Majorana neutrinos. In a unified treatment of quarks and leptons in
which the mass matrices of all fermions have a similar Fritzsch texture, we
calculate the mixing matrices V_{CKM} and U_{MNSP} as functions of quark and
lepton masses and only two free parameters, in very good agreement with the
latest experimental values on masses and mixings. Three essential ingredients
to explain the quark-lepton complementarity relation are identified: the strong
hierarchy in the mass spectra of quarks and charged leptons, the normal seesaw
mechanism and the assumption of maximal CP violation in the lepton sector.Comment: 6 pages, to appear in "Particles and fields: Xth Mexican Workshop on
  Particles and Fields" (Morelia, Mich. Mexico, November 6-12, 2005), Eds. A.
  Bashir and L. Villasenor, AIP Conference proceedings (2006

Gonzalez-Canales, F.

Mondragon, A.

English

arXiv

F. González Canales

A. Mondragón

Crossref

On quark-lepton complementarity

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On quark-lepton complementarity
F. González Canales and A. Mondragón
Instituto de Física, UNAM, 04510, México D.F., MEXICO
Abstract. Recent measurements of the neutrino solar mixing angle and the Cabibbo angle satisfy
the empirical relationθsol+ θC ≃ π4 . This relation suggests the existence of a correlation betwe n
the mixing matrices of leptons and quarks, the so called quark-lepton complementarity. Here, we
examine the possibility that this correlation originates in the strong hierarchy in the mass spectra of
quarks and charged leptons, and the seesaw mechanism that gives mass to the Majorana neutrinos.
In a unified treatment of quarks and leptons in which the mass matrices of all fermions have a similar
Fritzsch texture, we calculate the mixing matricesVCKM andUMNSPas functions of quark and lepton
masses and only two free parameters, in very good agreement with the latest experimental values
on masses and mixings. Three essential ingredients to explain the quark-lepton complementarity
relation are identified: the strong hierarchy in the mass spectra of quarks and charged leptons, the
normal seesaw mechanism and the assumption of maximal CP violation in the lepton sector.
Keywords: Quark and lepton masses and mixings, Neutrino masses and mixings,CKM matrix
PACS: 12.15.Ff, 14.60.Pq, 12.15.Hh, 14.60.St
INTRODUCTION
In the last few years, the neutrino oscillations between different flavour states were
measured in a series of experiments with atmospheric neutrios[1], solar neutrinos[2][3],
neutrinos produced in nuclear reactors [4] and accelerators [5]. As a result, the difference
of the squared neutrino masses and the mixing angles in the lepton mixing matrix,
UMNSP, were determined:
0.34≤ sin2 θ23 ≤ 0.68, 1.4×10−3(eV)2 ≤ ∆m223 ≤ 3.0×10
−3(eV)2, (1)
0.29≤ sin2 θ12 ≤ 0.40, 7.1×10−5(eV)2 ≤ ∆m212 ≤ 8.9×10
−5(eV)2, (2)
sin2 θ13 ≤ 0.046, (3)
at 90% confidence level [6]-[7]. The CHOOZ experiment [8] determined an upper bound
for theθ13 mixing angle. It was soon realized [9] that the solar mixing aleθMNSP12 and
the Cabibbo angleθCKM12 , which is the corresponding angle in the quark sector, satisfiy
an interesting and intriguing numerical relation,
θMNSP12 +θ
CKM
12 = 45
o+1o±2.4o, (4)
with θMNSP12 = 33.9
o±2.4o (1σ ) andθCKM12 = 12.8
o±0.15o. Equation (4) relates the 1-2
mixing angles in the quark and lepton sectors, it is commonlyca led the quark-lepton
complementarity relation (QLC) and, if not accidental, it could imply a quark-lepton
symmetry (for a recent review see [10]) or a quark-lepton unification [11]-[14].
A second QLC relation,θMNSP23 +θ
CKM
23 ≈
π
4 , is also satisfied. However, this is not as
interesting as (4) becauseθCKM23 is only about two degrees, and the corresponding QLC
relation would be satisfied, within the errors, even if the angle θCKM23 had been zero, as
long asθMNSP23 is close to the maximal valueπ/4. A third possible QLC relation is not
realized at all, or at least not realized in the same way, since θCKM13 +θ
MNSP
13 is less than
ten degrees. In this short note we will focus our attention onunderstanding the nature of
the QLC relation shown in equation (4).
UNIVERSAL FRITZSCH TEXTURE OF QUARKS AND LEPTONS
The quark and lepton flavour mixing matrices,UMNSP andVCKM, arise from the mis-
match between diagonalization of the mass matrices ofu and d type quarks and the
diagonalization of the mass matrices of charged leptons andneutrinos,
UMNSP=U
†
l Uν , VCKM =U
†
uUd. (5)
Therefore, to get predictions for the flavour mixing angles and CP violating phases, we
should specify the mass matrices.
In this work, we propose a unified treatment of quarks and leptons. Lepton and quark
mass matrices could have the same mass texture from a universal flavour symmetry
(exact at a certain energy scale). Imposing a flavour symmetry has been successful in
reducing the number of parameters of the Standard Model. In particular, a permutational
S3 flavour symmetry and its sequential explicit breaking, allows us to represent the mass
matrices as a modified Fritzsch texture:
M(F) =


0 A 0
A∗ B C
0 C D

 i = u,d, l ,ν. (6)
Some reasons to propose the validity of the modified Fritzschtexture as a universal mass
texture for all fermions in the theory are the following:
1. The idea ofS3 flavour symmetry and its explicit breaking has been realizedas a
modified Fritzsch texture in the quark sector to interpret thstrong mass hierarchy
of up and down type quarks [15].
2. The quark mixing angles and the CP violating phase appearing in theVCKM mixing
matrix were computed as explicit, exact functions of the four q ark mass ratios
(mu/mt ,mc/mt ,md/mb,ms/mb), one symmetry breaking parameterZ1/2 =
(81
32
)1/2
and one CP violating phaseφu−d = 90o, in very good agreement with experiment
[16].
3. Since the mass spectrum of the charged leptons exhibits a simil r hierarchy to the
quark’s one, it would be natural to consider the sameS3 symmetry and its explicit
breaking for the charged lepton mass matrix.
4. As for the Dirac neutrinos, we have no direct information about the absolute
values or the relative values of the neutrino masses, but theFritzsch texture can
be incorporated in aSO(10) neutrino model [17]. Therefore it would be sensible
to assume that the Dirac neutrinos have a mass hierarchy similar to that of the u-
quarks and it would be natural to take for the Dirac neutrino mass matrix also a
modified Fritzsch texture.
5. The left handed Majorana neutrinos naturally acquire their mass through an effec-
tive seesaw mechanism of the form
MνL = MνDM
−1
R M
T
νD, (7)
whereMνD and MR denote the Dirac and right handed Majorana neutrino mass
matrices. From our conjecture of a universalS3 flavour symmetry it follows that
MR could have the same texture as that ofMνD andMl . Then, it is straightforward
to show thatMνL has the same modified Fritzsch texture [18].
MIXING MATRICES AS FUNCTIONS OF THE FERMION
MASSES
When the unitary matrices that diagonalize the mass matrices M(F)i are written in polar
form,Ui = PiOi andM
(F)
i = P
†
i M̄Pi, the expressions (5) for the mixing matrices take the
form
UMNSP= O
T
l P
(l−ν)OνK, VCKM = O
T
d P
(u−d)Ou, (8)
whereOi , i = u,d,ν, l , are the orthogonal matrices that diagonalize the real symmetric
mass matricesM̄(F)i and P
(u−d) = diag
[
1,eiφ ,eiφ
]
, P(l−ν) = diag
[
1,eiΦ,eiΦ
]
, where
φ = φu−φd, andΦ=Φl −Φν , are the matrices of the Dirac phases andK is the diagonal
matrix of the Majorana phases.
We reparametrized the matrices̄Mi in terms of their eigenvalues. The orthogonal
matrices are then expressed in terms of the mass eigenvaluesof Mi :
Oi =


[
m̃i2 fi1
Di1
] 1
2
−
[
m̃i1 fi2
Di2
] 1
2
[
m̃i1m̃i2 fi3
Di3
] 1
2
[
m̃i1(1−δi) fi1
Di1
] 1
2
[
m̃i2(1−δi) fi2
Di2
] 1
2
[
(1−δi) fi3
Di3
] 1
2
−
[
m̃i1 fi2 fi3
Di1
] 1
2
−
[
m̃i2 fi1 fi3
Di2
] 1
2
[
fi1 fi2
Di3
] 1
2


, (9)
fi1 = 1− m̃i1−δi , fi2 = 1+ m̃i2−δi , fi3 = δi , (10)
Di1 = (1−δi)(m̃i1+ m̃i2)(1− m̃i1), (11)
Di2 = (1−δi)(m̃i1+ m̃i2)(1+ m̃i2), (12)
Di3 = (1−δi)(1− m̃i1)(1+ m̃i2), (13)
the small parametersδi are also functions of the mass ratios and the symmetry breaking
parameterZ1/2 = (81/32)1/2.
Substitution of the expressions (9) and (10)-(13) in (8) allows us to express the mixing
matricesUMNSPandVCKM as explicit functions of the quark and lepton masses.
QUARK-LEPTON COMPLEMENTARITY
The resulting theoretical expression for the Cabibbo anglewritten to first order inmu/mc
andmd/ms, is
sin2 θ thC =|V
th
us |
2≈
m̃d
m̃s
+ m̃um̃c −2
√
m̃u
m̃c
m̃d
m̃s
cosφ
(
1+ m̃um̃c
)(
1+ m̃dm̃s
) . (14)
Taking for the quark masses the valuesmu = 2.75MeV,mc = 1310MeV,md = 6.0MeV,
ms = 120MeV and maximal CP violation,φ = 90o [16], we reproduce the numerical
value of the Cabibbo angle
sinθ thc = 0.225 or θc = 12.8
o, (15)
in very good agreement with the latest analysis of the experimental data [19].
The theoretical expression for the solar mixing angle is derived in a similar way. From
| (UMNSP)12 |2 / | (UMNSP)11 |2= tan2θ th12 , we obtain
tan2θ th12 =
m̃ν1
m̃ν2
+ m̃em̃µ −2
√
m̃ν1
m̃ν2
m̃e
m̃µ
cosΦ
1+
m̃ν1
m̃ν2
m̃e
m̃µ
+2
√
m̃ν1
m̃ν2
m̃e
m̃µ
cosΦ
. (16)
In the absence of experimental information, we assumed thatCP violation is also max-
imal in the lepton sectori.e. Φ = 90o. Taking for the masses of the left handed Majo-
rana neutrinos a normal hierarchy with the numerical valuesmν1 = 4.4×10
−3eV and
mν2 = 1.0× 10
−2eV, and for the charged lepton masses the valuesme = 0.5109MeV,
mµ = 105.685MeV andmτ = 1776.99GeV, we obtain the following numerical value for
the solar mixing angle
tan2θ th12 = 0.45 or θ
th
12 = 33.9
o. (17)
We may now address the question of the meaning of the quark-lepton complementar-
ity relation as expressed in eq(4). The previous theoretical an lysed allows us to calcu-
late,
tan
(
θ thc +θ th12
)
= 1+∆th, (18)
∆th=
(
m̃d
m̃s
+ m̃um̃c
) 1
2
[(
m̃ν1
m̃ν2
+ m̃em̃µ
) 1
2
+
(
1+
m̃ν1
m̃ν2
m̃e
m̃µ
) 1
2
]
+
(
1+ m̃dm̃s
m̃u
m̃c
) 1
2
[(
m̃ν1
m̃ν2
+ m̃em̃µ
) 1
2
−
(
1+
m̃ν1
m̃ν2
m̃e
m̃µ
) 1
2
]
(
1+
m̃ν1
m̃ν2
m̃e
m̃µ
) 1
2
(
1+ m̃dm̃s
m̃u
m̃c
) 1
2
−
(
m̃d
m̃s
+ m̃um̃c
) 1
2
(
m̃ν1
m̃ν2
+ m̃em̃µ
) 1
2
.
(19)
After substitution of the numerical value of the mass ratiosof quarks and leptons in
(19), we obtain,
∆th = 0.061 and θ thc +θ
th
12 = 45
o+1.7o. (20)
in very good agreement with the experimental value.
CONCLUSIONS
In this short communication, we outlined a unified treatmentof masses and mixing of
quarks and leptons in which the left handed Majorana neutrinos acquire their masses via
the seesaw mechanism, and the mass matrices of all fermions have a similar Fritzsch
texture and a normal hierarchy. In this scheme, we derived exact, explicit expressions
for the Cabibbo and solar mixing angles as functions of the quark and lepton masses.
The quark-lepton complementarity relation takes the form,
θCKM12 +θ
MNSP
12 = 45
o+δ12. (21)
The correction term,δ12, is an explicit function of the ratios of quark and lepton masses,
given in eq.(19), which reproduces the experimentally determined value,δ12 ≈ 1.7o,
when the numerical values of the quark and lepton masses are substituted in (19) and
maximal violation of CP in the lepton sector is assumed.
Three essential ingredients are needed to explain the correlations implicit in the small
numerical value ofδ12:
1. The strong hierarchy in the mass spectra of the quarks and charged leptons, realized
in our scheme through the explicit breaking of theS3 flavour symmetry in the
Fritzsch mass texture, explains the resulting small or verysmall quark mixing
angles, the very small charged lepton mass ratios explain the very smallθMNSP13
which, in our scheme, is independent of the neutrino masses.
2. The normal seesaw mechanism that gives very small masses to the left handed
Majorana neutrinos with relatively large values of the neutrino mass ratiomν1/mν2
and allows for largeθMNSP12 andθ
MNSP
23 mixing angles.
3. The assumption of maximal CP violation in the lepton sector.
A more complete and detailed version of this work will be presented in a forthcoming
publication [20]
ACKNOWLEDGEMENTS
We thank Dr. M. Mondragón for many inspiring discussions on this exciting problem
and for a critical reading of the manuscript.
This work was partially supported by CONACyT Mexico under Contract No. 42026F,
and DGAPA-UNAM Contract No. PAPIIT IN116202.
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