A complex-angle rotation and geometric complementarity in fermion mixing

Abstract

The mixing among flavors in quarks or leptons in terms of a single rotation
angle is defined such that three flavor eigenvectors are transformed into three
mass eigenvectors by a single rotation about a common axis. We propose that a
geometric complementarity condition exists between the complex angle of quarks
and that of leptons in $\mathbb{C}^2$ space. The complementarity constraint has
its rise in quark-lepton unification and is reduced to the correlation among
$\theta_{12}, \theta_{23}, \theta_{13}$ and the CP phase $\delta$. The CP phase
turns out to have a non-trivial dependence on all the other angles. We will
show that further precise measurements in real angles can narrow down the
allowed region of $\delta$. In comparison with other complementarity schemes,
this geometric one can avoid the problem of the $\theta_{13}$ exception and can
naturally keep the lepton basis being independent of quark basis.

Full text

Page 1

arXiv:0808.0788v1 [hep-ph] 6 Aug 2008
A complex-angle rotation and
geometric complementarity in fermion mixing
Kee-Hwan Nam∗and Kim Siyeon†
Department of Physics, Chung-Ang University, Seoul 156-756, Korea
Seungsu Hwang‡
Department of Mathematics, Chung-Ang University, Seoul 156-756, Korea
(Dated: today)
Abstract
The mixing among flavors in quarks or leptons in terms of a single rotation angle is defined such
that three flavor eigenvectors are transformed into three mass eigenvectors by a single rotation
about a common axis. We propose that a geometric complementarity condition exists between the
complex angle of quarks and that of leptons in C2space. The complementarity constraint has its
rise in quark-lepton unification and is reduced to the correlation among θ12,θ23,θ13and the CP
phase δ. The CP phase turns out to have a non-trivial dependence on all the other angles. We
will show that further precise measurements in real angles can narrow down the allowed region of
δ. In comparison with other complementarity schemes, this geometric one can avoid the problem
of the θ13exception and can naturally keep the lepton basis being independent of quark basis.
PACS numbers: 11.30.Fs, 14.60.Pq, 14.60.St
Keywords: neutrino mixing, quark-lepton complementarity
∗Electronic address: snowall@gmail.com
†Electronic address: siyeon@cau.ac.kr
‡Electronic address: seungsu@cau.ac.kr

Page 2

I. INTRODUCTION
With a number of successful neutrino oscillation experiments, the information on fermion
masses and on the transformation between the mass basis and the weak interaction basis is
getting more balanced between the quark part and the lepton part. The Cabibbo-Kobayashi-
Maskawa(CKM) matrix of quark mixing is just a few steps from the completion. The allowed
ranges of the magnitudes of the CKM elements are narrow, |VCKM| =





0.97383+0.00024
−0.00023
0.2272+0.0010
−0.0010
(3.96+0.09
−0.09) × 10−3
0.2271+0.0010
−0.0010
0.97296+0.00024
−0.00024
(42.21+0.10
−0.80) × 10−3
(8.14+0.32
−0.64) × 10−3(41.61+0.12
−0.78) × 10−3
0.999100+0.000034
−0.000004




,
while the ranges of the magnitudes of Potecorvo-Maki-Nakagawa-maskawa(PMNS) elements
are still broad;
|UPMNS| =





0.79 − 0.86 0.50 − 0.610 − 0.20
0.25 − 0.53 0.47 − 0.73 0.56 − 0.79
0.21 − 0.51 0.42 − 0.69 0.61 − 0.83





(1)
at the 3σ level [1]. The unitary transformations are conventionally described as Euler-type
subsequent operations of three separate rotations,
U(θ23,θ13,θ12,δ) = R(θ23)R(θ13,δ)R(θ12);(2)
that is, through a rotation in the 1−2 plane, another rotation in the 1′−3 plane, and a third
rotation in the 2′−3′plane, the mass basis is switched into the weak basis. From an analysis
of the global data, the three angles in CKM have the best-fit values θq
12= 0.229,θq
13= 0.004,
and θq
23= 0.042, and the angles in PMNS have the best-fit values θl
12= 0.588,θl
13= 0, and
θl
23= 0.756 [2, 3].
Quark-lepton complementarity (QLC) is one of the theoretical frameworks on which
phenomenological data can be naturally connected to the quark-lepton unification. In many
works, the QLC idea was built by the relation [4]
θq
ij+ θl
ij=π
4
(3)
with the mixing angle between i- and j- generations, θij. Only θ12’s and θ23’s satisfy the
above relation such that the complementarity gives rise to bi-maximal mixing.
2

Page 3

The QLC represented by Eq. (3) has a few points unexplained so far. First, like bi-
maximal mixing, the QLC is obliged to keep the exception with small θ13’s of quarks and
leptons, which cannot make the sum maximal. Second, the relation implies that the angles
θq
ijand θl
ijare in a plane. In low-energy theory where the quark-lepton symmetry is broken,
the common plane including those two angles requires a strong generic connection between
quark bases and lepton bases in process of symmetry breaking, but there is no supporting
theory.
Here, we propose a model that accommodates small θ13’s and that allows θq
ijand θl
ijto
belong to independent planes. In this attempt, the transformation from the weak basis to
the mass basis can be obtained by a single complex-angle rotation about a properly defined
axis [5]. In other words, by a single rotation about an axis, the weak eigenstates (νe,νµ,ντ) or
(d,s,b) are switched into (ν1,ν2,ν3) or (d1,d2,d3), respectively. Thus, there are two complex
angles, one corresponding to quark mixing and the other corresponding to lepton mixing.
We introduce the complementarity by a relation of those two complex angles such that
Θ2
L− Θ2
Q=
?π
4
?2
, (4)
where they are the orthogonal components of a hyperbola of radius π/4. With such a
geometric constraint, the model can protect the complementarity from the θ13 exception.
Furthermore, the quark basis and the lepton basis are independent of each others as implied
by completely broken quark-lepton symmetry.
In Section II, the definition of the complex angle and the axis to represent the transfor-
mation will be introduced. In Section III, using the hyperbolic condition, the allowed range
of Dirac Charge-Parity violating(CP) phase will be predicted. It will be shown that the
more precise measurements in other angles in the future can test the model itself, as well
as the CP violation testable in neutrino oscillation. Only the CKM-type matrix without
Majorana phases is considered as the PMNS matrix. An extended work with more details,
including Majorana phases and the physical implication relevant to them, is in progress in
other work. A brief on the convention to deal with complex angles is attached.
3

Page 4

FIG. 1: The rotation of Θ about |n > transforms |ei>’s into|fα>’s.
II. A SINGLE COMPLEX-ANGLE ROTATION
In both quarks and leptons, a unitary transformation between flavor eigenstates |fα? and
mass eigenstates |ei? consists minimally of three angles and a CP phase:
|fα? = U(θ12,θ13,θ23,δ)|ei?. (5)
If the weak interaction basis is properly chosen such that the mass matrix of up-type quarks
and that of charged leptons are diagonal, |fα? and U in Eq. (5) represent either down-type
quarks and the CKM matrix in the quark sector or neutrinos and the PMNS matrix in
the lepton sector. Suppose, in a representation, that |e1? = (1,0,0)T,|e2? = (0,1,0)T, and
|e3? = (0,0,1)T. Then |fα? = (Ufα1,Ufα2,Ufα3)T, for fα= d,s,b or fα= e,µ,τ. The Ufαiis
an element of the transformation matrix in Eq. (5). The components of fαare denoted by
(f1,f2,f3) if the specification of ‘quark or lepton’ is not necessary. In a real vector space,
an orthogonal set of three vectors can fit into another orthogonal set of three vectors simply
by rotating the original set about a common axis, which can be found to be invariant under
the rotation. Likewise, the unitary transformation in Eq. (5) can be replaced by a rotation
with a single complex angle.
If one constructs a vector |n? as the axis of the rotation, the rotation angle of a vector
|ei? about |n? is the same as the angle between the following two vectors |e⊥
i? and |f⊥
α? that
are orthogonal to |n?:
cosΘ =
?e⊥
i|f⊥
i???f⊥
α?
??e⊥
?ei|fα? − ?ei|n??n|fα?
??e⊥
i|e⊥
α|f⊥
α?
=
i|e⊥
i???f⊥
α|f⊥
α?
, (6)
4

Page 5

where
|f⊥
α? = |fα? − |n??n|fα?,
|e⊥
i? = |ei? − |n??n|ei?.
The rotation axis |n? = (nx,ny,nz)Thas the same components on the mass basis |ei? as on
the flavor basis |fα?, that is,
nx|e1? + ny|e2? + nz|e3? = nx|f1? + ny|f2? + nz|f3?, (7)
which is a normalized vector with |nx|2+ |ny|2+ |nz|2= 1. Substituting Eq. (5) into Eq.
(7) results in the following combined equations:
c13c12nx− c13s12ny+ s13e−iδnz= nx,
(−c23s12− s23s13c12eiδ)nx+ (c23c12− s23s13s12eiδ)ny+ s23c13nz= ny, (8)
(s23s12− c23s13c12eiδ)nx+ (−s23c12− c23s13s12eiδ)ny+ s23c13nz= nz.
It is possible to express |n? immediately in terms of mixing parameters. For instance,
|e⊥
1?,|f⊥
1?, the projected vectors of |e1?,|f1? on the plane perpendicular to |n?, are, according
to Eq. (6),
|e⊥
1? =










1 − |nx|2
−n∗
xny
−n∗
xnz




,(9)
|f⊥
1? =
U11
U12
U13




− (n∗
xU11+ n∗
yU12+ n∗
zU13)





nx
ny
nz




.
Then, the cosΘ in Eq. (6) reduces to
cosΘ =
U11− nx(n∗
xU11+ n∗
yU12+ n∗
zU13)
?
(1 − |nx|2)(1 − |n∗
xU11+ n∗
yU12+ n∗
zU13|2)
, (10)
because (nx,ny,nz) is obtained in terms of mixing angles and a phase, as is cosΘ.
The four physical parameters in the unitary mixing in Eq. (5) are now all embedded in
the direction of |n?. For a complex vector, one can remove the imaginary phase in one of the
5