05 April 2007
CP violation due to multi Froggatt-Nielsen fields
Shinya Kanemura
a,b∗, Koichi Matsudaa,c†, Toshihiko Otad‡
Serguey Petcov
e,f §, Tetsuo Shindoue,f¶, Eiichi Takasugia‖, Koji Tsumuraa∗∗
a)Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
b)Department of Physics, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan
c)Center for High Energy Physics, Tsinghua University, Beijing 100084, China
d)Max–Planck–Institut für Kernphysik, Postfach 10 39 80, 69029 Heidelberg, Germany
e)Scuola Internazionale Superiore di Studi Avanzati, I-34014 Trieste, Italy
f)Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34014 Trieste, Italy
Abstract
We study how to incorporate CP violation in the Froggatt–Nielsen (FN) mechanism. To this
end, we introduce non-renormalizable interactions with a flavor democratic structure to the fermion
mass generation sector. It is found that at least two iso-singlet scalar fields with imposed a discrete
symmetry are necessary to generate CP violation due to the appearance of the relative phase between
their vacuum expectation values. In the simplest model, ratios of quark masses and the Cabibbo-
Kobayashi-Maskawa (CKM) matrix including the CP violating phase are determined by the CKM
element |Vus| and the ratio of two vacuum expectation values R = |R|e
iα (a magnitude and a phase).
It is demonstrated how the angles φi(i = 1-3) of the unitarity triangle and the CKM off-diagonal
elements |Vub| and |Vcb| are predicted as a function of |Vus|, |R| and α. Although the predicted value
of the CP violating phase does not agree with the experimental data within the simplest model, the
basic idea of our scenario would be promising to construct a more realistic model of flavor and CP
violation.
1 Introduction
The Standard Model (SM) of electroweak interactions has been successful. It can explain all experimental
results except for neutrino oscillation phenomena. Masses of quarks and leptons are generated through the
Yukawa interaction after the electroweak symmetry breaking. However, no principle has been established
to determine the flavor structure of the Yukawa couplings, and the origin of fermion mass hierarchy remains
unknown.
There have been many attempts to explain the flavor structure of Yukawa couplings. A promising
approach would be the idea of the flavor symmetry. In models based on the Froggatt–Nielsen (FN)
mechanism[1], the U(1) global symmetry is imposed as a flavor symmetry, in which the vacuum expectation
value (VEV) of an iso-singlet scalar field (FN field) gives a power-like structure of Yukawa couplings due
to the U(1) charge assignment for the relevant fields. Extension to more complicated flavor symmetries
has also been studied; i.e., non-Abelian global symmetries such as U(2)[2], discrete symmetries such as
S3[3], A4[4], D5[5], etc. They have several distinct patterns for the symmetry breaking, and the difference
in VEVs for each scalar field gives a hierarchical structure of the Yukawa matrix. In most of such models,
∗ E-mail: kanemu@sci.u-toyama.ac.jp
† E-mail: matsuda@mail.tsinghua.edu.cn
‡ E-mail: toshi@mpi-hd.mpg.de
§ E-mail: petcov@sissa.it
¶ E-mail: shindou@sissa.it
‖ E-mail: takasugi@het.phys.sci.osaka-u.ac.jp
∗∗ E-mail: ko2@het.phys.sci.osaka-u.ac.jp, Address after April 2007, Theory Group, KEK
http://arxiv.org/abs/0704.0697v1
only the orders of magnitude of the Yukawa matrix elements are estimated, so that O(1) uncertainties
exist in coefficients of the coupling constants between the scalar and matter fields. In this framework, CP
violation (CPV) comes from complex phases in these coefficients.
On the other hand, some kinds of the texture such as a democratic structure[6] have been investigated
for the Yukawa matrix. In the model with the democratic structure, all the elements of the Yukawa matrix
are assumed to have the same value up to the leading order, and mass hierarchy and flavor mixing are
given by diagonalizing these Yukawa matrices. CPV is supposed to appear as a consequence of complex
nature of the terms of tiny breaking of democratic structure. In any case the CP violating phase comes
from complex phases of free parameters so that it is not predictable.
Since both the flavor mixing and the CP violating phase are determined by the Yukawa matrices, it
would be natural to consider that they are given through the same mechanism which is relevant to the
Yukawa interaction. In the scenario of spontaneous CPV[7], the phase is deduced from the relative complex
phase between VEVs of the scalar fields. Combining the spontaneous CPV scenario with the idea of the
flavor symmetry, one can obtain the non-zero complex phase in the Yukawa matrix from the VEVs of
scalar fields. This idea has been developed in several flavor models; e.g., a model with three U(1) scalar
fields[8], spontaneous CPV in non-Abelian flavor symmetry[9], SO(10) model with the complex VEVs of
Higgs field[10], etc.
In this paper, we introduce a simple model where the FN mechanism works with democratic Yukawa
structure between quarks and FN fields, and show how the CPV can be obtained. As mentioned in [8],
this type of models requires at least more than two FN fields for a successful prediction of physical CP
violating phase.
This paper is organized as follows. In Sec.II, we study generation of CPV for quarks based on the FN
mechanism with the democratic ansatz. In Sec.III, simple models with two FN fields are discussed. We
present analytic expressions for the CKM parameters and numerical evaluations are also shown. Conclu-
sions are given in Sec.IV.
2 CP violation in democratic models
The democratic ansatz for the flavor structure of the Yukawa matrix has been implemented in Refs. [6, 13,
14, 15]. In this framework, the Yukawa matrices for the up- and down-type quarks are simply written as
Yu,d ∝
1 1 1
1 1 1
1 1 1
 . (1)
This flavor structure can be constructed by models with the S3R × S3L permutation symmetry[13]. The
symmetry can be realized in the geometrical origin of the brane-world scenario[14], and also in the strong
dynamics[15]. Two of the three eigenvalues are zero in these matrices in Eq. (1), and no CP violating phase
appears in the S3R×S3L limit 1. It is clear that in order to explain the experimental data this permutation
symmetry must be broken by some small effects. When the small breaking terms for the permutation and
CP symmetries are introduced by hand, the mass splitting between the 1st- and 2nd-generation quarks,
the mixing angles and the CP violating phase are explained.
1With keeping the S3R × S3L symmetry, the complex Yukawa matrices Yu,d in Eq. (1) can be re-expressed, for example,
by an appropriate unitary transformation as
Yu,d ∝
ω ω2 1
ω2 1 ω
1 ω ω2
A , (2)
where ω = ei2π/3 is the cube root of one. However the complex phase in this matrix is unphysical, because it is rephased out
by the redefinition of quark fields.
The FN mechanism is a simple idea to generate the mass hierarchy of quarks and leptons. In the
simplest FN model[1], an iso-singlet scalar field, Θ, is introduced with the U(1)FN flavor symmetry in
order to discriminate the fermion flavor by the U(1)FN charge.
The U(1)FN charge assigned for Θ is taken to be fΘ = −1 without loss of generality. Under the U(1)FN
symmetry, non-renormalizable interactions relevant to the quark mass matrix can be written as
LFN = −(CU )ij ŪiQj ·Hu
+fQj+fHu
− (CD)ijD̄iQj ·Hd
+fQj+fHd
, (3)
where Hu and Hd are iso-doublet fields (the Higgs fields) with their hypercharge to be −1/2 and +1/2,
respectively2, Qi is the left-handed quark doublet, Ui and Di are right-handed up- and down-type quarks
in the i-th generation, and CU and CD are coupling constants of order one. The U(1)FN charge of the
field X is expressed by fX .The cut-off scale is given by Λ, which describes the mass scale of new physics
dynamics. The coefficients (CU )ij and (CD)ij are generally complex numbers.
After U(1)FN is broken by the VEV of Θ,
〈Θ〉 = λΛ , (4)
where λ is a small dimensionless parameter, the quark Yukawa matrices are obtained as
(YU,D)ij = (CU,D)ijλ
+fQj+fHu,d . (5)
With the assignment of U(1)FN charges[16] as
(fQ1 , fUc1 ), (fQ2 , fU
), (fQ3 , fUc3 )
= (3, 2, 0) , (fDc
, fDc
, fDc
) = (2, 1, 1),
(fHu , fHd) = (0, 0), (6)
observed quark mass hierarchy and the CKM mixings can be derived by assuming λ to be close to the
Cabibbo angle sin θc = 0.22. At the leading order, induced masses for quarks are mu ∼ λ6〈Hu〉, mc ∼
λ4〈Hu〉, mt ∼ 〈Hu〉, md ∼ λ5〈Hd〉, ms ∼ λ3〈Hd〉 and mb ∼ λ〈Hd〉, and the CKM matrix is given by
UCKM ∼
1 λ λ3
λ 1 λ2
λ3 λ2 1
 . (7)
Now we consider the possibility of the spontaneous CPV due to the complex phase of 〈Θ〉. We assume
that CU and CD in Eq. (3) have the democratic structure, i.e.,
CU,D = αU,D
1 1 1
1 1 1
1 1 1
 , (8)
There is no CPV in the model with only one FN field. Although complex phases may be obtained in
the mass matrices by introducing the complex VEV of Θ, such phases are rotated away by the phase
redefinition of quark fields. Hence the model should have at least two FN fields Θ1,2 in order to obtain
non-vanishing CP violating phase. We start from the following Lagrangian with two FN fields,
LFN2 =−
Ūi(CU )ijQj ·Hu
D̄i(CD)ijQj ·Hd
, (9)
2In the SM, Hd = iσ2H
u is satisfied.
where n
1 and n
2 run from zero to n
ij ≡ fUci ,Dci +fQj+fHu,d with satisfying the constraint n
ij . After the U(1)FN symmetry is broken, the Yukawa couplings in the SM are given by
(YU,D)ij = (CU,D)ijλ
Rk , (10)
where
, R =
≡ |R|eiα . (11)
Therefore, physical CP violating phase can be obtained from the relative phase α between 〈Θ1〉 and 〈Θ2〉.
3 Examples for the model with two Froggatt-Nielsen fields
In this section, we show how to generate CPV from two FN fields by considering simple models. In order
to generate the quark mass hierarchy, we employ the U(1)FN charge assignment for matter fields given in
Eq. (6). In general, U(1)FN charges for the FN fields Θ1 and Θ2 can be different with each other. We here
assume that the both have the same U(1)FN charge for simplicity; (fΘ1 , fΘ2) = (−1,−1).
(a) The simplest toy model
First of all, we discuss the naive model defined in Eq. (9). The mass matrices Mu,d for up- and
down-type quarks are given by
(Mu)ij = αU 〈Hu〉AnU
ij , (Md)ij = αD〈Hd〉AnD
ij , (12)
where An =
k=0 R
k. This model predicts m2s/m
b = O(λ6) and |Vus| = O(λ). Only one of the two
experimental values can be adjusted. Furthermore, each mass matrix gives one massless eigenstate because
of the conditions detMu = detMd = 0
In order to avoid these difficulties, the following possibilities can be considered: (i) introducing an
additional symmetry, (ii) throwing away the democratic ansatz given in Eq. (8), etc. In the next model,
we explore the possibility of keeping the democratic structure for CU,D.
(b) The extended models with the Z2 symmetry
We try to construct more realistic model. The U(1)FN charges are assigned again as in Eq. (6). In
order to obtain the observed value of m2s/m
b by setting λ ∼ sin θc, we impose the Z2 symmetry under the
transformation of Θ1 → Θ1 and Θ2 → −Θ2. For Z2 parity assignment for quark fields, there are a lot of
choices. If we consider the scenario associated with the grand unified theory (GUT), it would be natural
that the Z2 parity for Qi and U
i is common and that for Hu is set to be + for the prediction of a large
top-quark mass. Because Hd always couples to D
i , we can set the Z2 parity for Hd to be + without loss
of generality. Then, there are 64 possibilities on parity assignment for quarks. However, it turns out that
most of them cannot give correct numbers of m2c/m
t and m
b . Consequently, only the following sets
of Z2 parity assignment are enough to be discussed;
• Type I-a,
((Q1, U
1), (Q2, U
2), (Q3, U
3 )) = (+,+,−) , (Dc1, Dc2, Dc3) = (+,+,−) . (13)
• Type I-b,
((Q1, U
1), (Q2, U
2), (Q3, U
3 )) = (+,+,−) , (Dc1, Dc2, Dc3) = (−,+,−) . (14)
3Even when the u- and d-quarks are massless at the electroweak scale, their finite masses would be generated at lower
energy scales due to the strong dynamics[17].
• Type II-a,
((Q1, U
1), (Q2, U
2), (Q3, U
3 )) = (+,−,+) , (Dc1, Dc2, Dc3) = (+,−,+) . (15)
• Type II-b,
((Q1, U
1), (Q2, U
2), (Q3, U
3 )) = (+,−,+) , (Dc1, Dc2, Dc3) = (−,−,+) . (16)
Type I-a gives the mass matrices as
Mu(I-a) =
6 B4λ
5 RB2λ
5 B4λ
4 Rλ2
3 Rλ2 1
αU 〈Hu〉 , Md(I-a) =
5 B4λ
4 Rλ2
4 B2λ
4 RB2λ
αD〈Hd〉 , (17)
where B2n =
k=0 R
2k. Diagonalizing above matrices, we obtain mass ratios m2c/m
t , m
, the
CKM mixing angles (absolute values of CKM matrix elements), and the Kobayashi-Maskawa phase
φ3 ≡ arg(V ∗ubVud/V ∗cbVcd) at the leading order as
= |1 +R4|2λ8 , m
|1−R4|2
(1 + |R|2)2
|Vus| =
|R|8 + |R|−8 − 2(2 cos2 4α− 1)
|Vub| =
|R|||R| − |R|−1|
(|R|+ |R|−1)
|R|4 + |R|−4 + 2 cos 4α
|Vcb| =
|R|2 + |R|−2 + 2 cos 4α
|R|+ |R|−1
λ2 , (18)
φ1 = arg
|R|4 + 1
+ (|R|2 + 1) cos 4α− i(|R|2 − 1) sin 4α
φ2 = arg
(1− |R|2)
− |R|4 + 2i sin 4α
φ3 = arg
(1− |R|2)
|R|4 −
+ (|R|2 − 1) cos 4α+ i(|R|2 + 1) sin 4α
Let us discuss appropriate values of |R|, cos 4α and sin 4α. We first expect that λ ∼ sin θc. Then, |R| > 1
is needed to obtain the reasonable value of |Vub|. However, |R| cannot be much greater than unity, because
m2c/m
t exceeds the experimentally acceptable value. In addition, cos 4α < 0 is necessary for |R| > 1 to
explain the data of |Vcb|. Finally, sin 4α < 0 is required for φ1 to be in the first quadrant. In this case,
however, it turns out that φ3 cannot be in the first quadrant simultaneously.
For numerical evaluation, we take |R| =
3/2, cos 4α = −3/4 and sin 4α = −
7/4. This parameter
set determines λ = 0.25 under the experimental value of |Vus|(= 0.22). The matrices Mu,d(I-a) are
diagonalized, and we obtain
|Vub| = 0.0028 , |Vcb| = 0.032 , (19)
which are in excellent agreement with the CKM mixing angles at the GUT scale, |Vcb| = 0.029–0.039 and
|Vub| = 0.0024–0.0038 which are evaluated from renormalization group method with the experimental data
at low energies[18]. However, quark mass ratios are predicted as
= 0.0061 ,
= 0.073 , (20)
which are about twice as large as the expected values at the GUT scale; mc/mt ∼ 0.0032 ± 0.0007 and
ms/mb = 0.036± 0.005. Finally, we find
φ1 = 12
◦ , φ2 = 31
◦ , φ3 = 137
◦ . (21)
Although the predictions cannot explain all the data simultaneously, it would be amazing to observe that
this model can reproduce most of them in a considerable extent. We also find that for Type I-b, the mixing
angles, mass ratios between 2nd- and 3rd-generations, and the CKM phase are completely the same at
leading order as in Eq. (18).
For Type II-a, the mass matrices are
Mu(II-a) =
6 RB4λ
5 B2λ
5 B4λ
4 Rλ2
3 Rλ2 1
αU 〈Hu〉 , Md(II-a) =
5 RB2λ
4 B2λ
4 B2λ
4 RB2λ
αD〈Hd〉 .
The resulting mass ratios, the CKM parameters and phases φ1, φ2 and φ3 are the same as those in Type
I except for |Vus| and |Vub|, which are
|Vus| =
|R|8 + |R|−8 − 2(2 cos2 4α− 1)
|Vub| =
|R|2||R| − |R|−1|
(|R|+ |R|−1)
|R|4 + |R|−4 + 2 cos 4α
λ3 . (23)
These expressions are different from those in Type I by the multiplication factor |R|.
In this case, we take |R| =
3/2 and cos 4α = −1/2 in order to compensate the effect of the extra factor
|R| in |Vus| in comparison with the that in Type I. In addition, we take sin 4α = −
3/2 and λ = 0.23. We
obtain the following results;
|Vus| = 0.22 , |Vub| = 0.0038 , |Vcb| = 0.035 ,
= 0.0060 ,
= 0.059 ,
φ1 = 18
◦, φ2 = 38
◦, φ3 = 123
◦. (24)
Although the size of ms/mb becomes smaller than that of the Type I, it is still too large to be phenomeno-
logically acceptable. Moreover, mc/mt remains two times greater than the expected value, and φ3 is in
the 2nd quadrant. Type II-b gives almost the same results for the CKM parameters and the mass ratios
between 2nd- and 3rd- generations.
4 Conclusion
We have studied possibility of incorporating CPV by using the FN mechanism in the context of democratic
flavor FN couplings. We have considered models with two FN fields, in which the relative phase of their
VEVs plays as the origin of CP violating phase at low energies. In the scenario with the Z2 symmetry, the
relationship among ratios between quark masses, the absolute values of CKM matrix elements and the CP
violating phase has been examined in several simplest models. We have found that the predictions have
been in good agreement with most of the data. However, the CP violating phase φ3 has been predicted to
be around 130◦, so that the models we have examined are not acceptable. It may be oversimplification to
assume the flavor blind couplings. We expect that the small modification for the democratic assumption
would cure this phenomenological problem.
We have demonstrated the way how introduce the CPV to the FN model and have shown that the
scenario with two FN fields would be promising. An application of our scenario to the lepton sector
including neutrinos is under way and will appear in our future publications. If this will be successfully
achieved, we would obtain a model which gives the unified description of CP phases for quarks and leptons.
Acknowledgements. S. K. was supported, in part, by the Grant-in-Aid of the Ministry of Education,
Culture, Sports, Science and Technology, Government of Japan, Grant Nos.17043008 and 18034004. S. P.
and T. S. were supported in part by the Italian MIUR and INFN under the programs “Fisica Astroparti-
cellare”.
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	Introduction
	CP violation in democratic models
	Examples for the model with two Froggatt-Nielsen fields
	Conclusion

  We study how to incorporate CP violation in the Froggatt--Nielsen (FN)
mechanism. To this end, we introduce non-renormalizable interactions with a
flavor democratic structure to the fermion mass generation sector. It is found
that at least two iso-singlet scalar fields with imposed a discrete symmetry
are necessary to generate CP violation due to the appearance of the relative
phase between their vacuum expectation values.
  In the simplest model, ratios of quark masses and the
Cabibbo-Kobayashi-Maskawa (CKM) matrix including the CP violating phase are
determined by the CKM element |V_{us}| and the ratio of two vacuum expectation
values R=|R|e^{i*alpha} (a magnitude and a phase). It is demonstrated how the
angles phi_i (i=1--3) of the unitarity triangle and the CKM off-diagonal
elements |V_{ub}| and |V_{cb}| are predicted as a function of |V_{us}|, |R| and
\alpha. Although the predicted value of the CP violating phase does not agree
with the experimental data within the simplest model, the basic idea of our
scenario would be promising to construct a more realistic model of flavor and
CP violation.

Introduction
The Standard Model (SM) of electroweak interactions has been successful. It can explain all experimental
results except for neutrino oscillation phenomena. Masses of quarks and leptons are generated through the
Yukawa interaction after the electroweak symmetry breaking. However, no principle has been established
to determine the flavor structure of the Yukawa couplings, and the origin of fermion mass hierarchy remains
unknown.
There have been many attempts to explain the flavor structure of Yukawa couplings. A promising
approach would be the idea of the flavor symmetry. In models based on the Froggatt–Nielsen (FN)
mechanism[1], the U(1) global symmetry is imposed as a flavor symmetry, in which the vacuum expectation
value (VEV) of an iso-singlet scalar field (FN field) gives a power-like structure of Yukawa couplings due
to the U(1) charge assignment for the relevant fields. Extension to more complicated flavor symmetries
has also been studied; i.e., non-Abelian global symmetries such as U(2)[2], discrete symmetries such as
S3[3], A4[4], D5[5], etc. They have several distinct patterns for the symmetry breaking, and the difference
in VEVs for each scalar field gives a hierarchical structure of the Yukawa matrix. In most of such models,
∗ E-mail: kanemu@sci.u-toyama.ac.jp
† E-mail: matsuda@mail.tsinghua.edu.cn
‡ E-mail: toshi@mpi-hd.mpg.de
§ E-mail: petcov@sissa.it
¶ E-mail: shindou@sissa.it
‖ E-mail: takasugi@het.phys.sci.osaka-u.ac.jp
∗∗ E-mail: ko2@het.phys.sci.osaka-u.ac.jp, Address after April 2007, Theory Group, KEK
http://arxiv.org/abs/0704.0697v1
only the orders of magnitude of the Yukawa matrix elements are estimated, so that O(1) uncertainties
exist in coefficients of the coupling constants between the scalar and matter fields. In this framework, CP
violation (CPV) comes from complex phases in these coefficients.
On the other hand, some kinds of the texture such as a democratic structure[6] have been investigated
for the Yukawa matrix. In the model with the democratic structure, all the elements of the Yukawa matrix
are assumed to have the same value up to the leading order, and mass hierarchy and flavor mixing are
given by diagonalizing these Yukawa matrices. CPV is supposed to appear as a consequence of complex
nature of the terms of tiny breaking of democratic structure. In any case the CP violating phase comes
from complex phases of free parameters so that it is not predictable.
Since both the flavor mixing and the CP violating phase are determined by the Yukawa matrices, it
would be natural to consider that they are given through the same mechanism which is relevant to the
Yukawa interaction. In the scenario of spontaneous CPV[7], the phase is deduced from the relative complex
phase between VEVs of the scalar fields. Combining the spontaneous CPV scenario with the idea of the
flavor symmetry, one can obtain the non-zero complex phase in the Yukawa matrix from the VEVs of
scalar fields. This idea has been developed in several flavor models; e.g., a model with three U(1) scalar
fields[8], spontaneous CPV in non-Abelian flavor symmetry[9], SO(10) model with the complex VEVs of
Higgs field[10], etc.
In this paper, we introduce a simple model where the FN mechanism works with democratic Yukawa
structure between quarks and FN fields, and show how the CPV can be obtained. As mentioned in [8],
this type of models requires at least more than two FN fields for a successful prediction of physical CP
violating phase.
This paper is organized as follows. In Sec.II, we study generation of CPV for quarks based on the FN
mechanism with the democratic ansatz. In Sec.III, simple models with two FN fields are discussed. We
present analytic expressions for the CKM parameters and numerical evaluations are also shown. Conclu-
sions are given in Sec.IV.
2 CP violation in democratic models
The democratic ansatz for the flavor structure of the Yukawa matrix has been implemented in Refs. [6, 13,
14, 15]. In this framework, the Yukawa matrices for the up- and down-type quarks are simply written as
Yu,d ∝
1 1 1
1 1 1
1 1 1
 . (1)
This flavor structure can be constructed by models with the S3R × S3L permutation symmetry[13]. The
symmetry can be realized in the geometrical origin of the brane-world scenario[14], and also in the strong
dynamics[15]. Two of the three eigenvalues are zero in these matrices in Eq. (1), and no CP violating phase
appears in the S3R×S3L limit 1. It is clear that in order to explain the experimental data this permutation
symmetry must be broken by some small effects. When the small breaking terms for the permutation and
CP symmetries are introduced by hand, the mass splitting between the 1st- and 2nd-generation quarks,
the mixing angles and the CP violating phase are explained.
1With keeping the S3R × S3L symmetry, the complex Yukawa matrices Yu,d in Eq. (1) can be re-expressed, for example,
by an appropriate unitary transformation as
Yu,d ∝
ω ω2 1
ω2 1 ω
1 ω ω2
A , (2)
where ω = ei2π/3 is the cube root of one. However the complex phase in this matrix is unphysical, because it is rephased out
by the redefinition of quark fields.
The FN mechanism is a simple idea to generate the mass hierarchy of quarks and leptons. In the
simplest FN model[1], an iso-singlet scalar field, Θ, is introduced with the U(1)FN flavor symmetry in
order to discriminate the fermion flavor by the U(1)FN charge.
The U(1)FN charge assigned for Θ is taken to be fΘ = −1 without loss of generality. Under the U(1)FN
symmetry, non-renormalizable interactions relevant to the quark mass matrix can be written as
LFN = −(CU )ij ŪiQj ·Hu
+fQj+fHu
− (CD)ijD̄iQj ·Hd
+fQj+fHd
, (3)
where Hu and Hd are iso-doublet fields (the Higgs fields) with their hypercharge to be −1/2 and +1/2,
respectively2, Qi is the left-handed quark doublet, Ui and Di are right-handed up- and down-type quarks
in the i-th generation, and CU and CD are coupling constants of order one. The U(1)FN charge of the
field X is expressed by fX .The cut-off scale is given by Λ, which describes the mass scale of new physics
dynamics. The coefficients (CU )ij and (CD)ij are generally complex numbers.
After U(1)FN is broken by the VEV of Θ,
〈Θ〉 = λΛ , (4)
where λ is a small dimensionless parameter, the quark Yukawa matrices are obtained as
(YU,D)ij = (CU,D)ijλ
+fQj+fHu,d . (5)
With the assignment of U(1)FN charges[16] as
(fQ1 , fUc1 ), (fQ2 , fU
), (fQ3 , fUc3 )
= (3, 2, 0) , (fDc
, fDc
, fDc
) = (2, 1, 1),
(fHu , fHd) = (0, 0), (6)
observed quark mass hierarchy and the CKM mixings can be derived by assuming λ to be close to the
Cabibbo angle sin θc = 0.22. At the leading order, induced masses for quarks are mu ∼ λ6〈Hu〉, mc ∼
λ4〈Hu〉, mt ∼ 〈Hu〉, md ∼ λ5〈Hd〉, ms ∼ λ3〈Hd〉 and mb ∼ λ〈Hd〉, and the CKM matrix is given by
UCKM ∼
1 λ λ3
λ 1 λ2
λ3 λ2 1
 . (7)
Now we consider the possibility of the spontaneous CPV due to the complex phase of 〈Θ〉. We assume
that CU and CD in Eq. (3) have the democratic structure, i.e.,
CU,D = αU,D
1 1 1
1 1 1
1 1 1
 , (8)
There is no CPV in the model with only one FN field. Although complex phases may be obtained in
the mass matrices by introducing the complex VEV of Θ, such phases are rotated away by the phase
redefinition of quark fields. Hence the model should have at least two FN fields Θ1,2 in order to obtain
non-vanishing CP violating phase. We start from the following Lagrangian with two FN fields,
LFN2 =−
Ūi(CU )ijQj ·Hu
D̄i(CD)ijQj ·Hd
, (9)
2In the SM, Hd = iσ2H
u is satisfied.
where n
1 and n
2 run from zero to n
ij ≡ fUci ,Dci +fQj+fHu,d with satisfying the constraint n
ij . After the U(1)FN symmetry is broken, the Yukawa couplings in the SM are given by
(YU,D)ij = (CU,D)ijλ
Rk , (10)
where
, R =
≡ |R|eiα . (11)
Therefore, physical CP violating phase can be obtained from the relative phase α between 〈Θ1〉 and 〈Θ2〉.
3 Examples for the model with two Froggatt-Nielsen fields
In this section, we show how to generate CPV from two FN fields by considering simple models. In order
to generate the quark mass hierarchy, we employ the U(1)FN charge assignment for matter fields given in
Eq. (6). In general, U(1)FN charges for the FN fields Θ1 and Θ2 can be different with each other. We here
assume that the both have the same U(1)FN charge for simplicity; (fΘ1 , fΘ2) = (−1,−1).
(a) The simplest toy model
First of all, we discuss the naive model defined in Eq. (9). The mass matrices Mu,d for up- and
down-type quarks are given by
(Mu)ij = αU 〈Hu〉AnU
ij , (Md)ij = αD〈Hd〉AnD
ij , (12)
where An =
k=0 R
k. This model predicts m2s/m
b = O(λ6) and |Vus| = O(λ). Only one of the two
experimental values can be adjusted. Furthermore, each mass matrix gives one massless eigenstate because
of the conditions detMu = detMd = 0
In order to avoid these difficulties, the following possibilities can be considered: (i) introducing an
additional symmetry, (ii) throwing away the democratic ansatz given in Eq. (8), etc. In the next model,
we explore the possibility of keeping the democratic structure for CU,D.
(b) The extended models with the Z2 symmetry
We try to construct more realistic model. The U(1)FN charges are assigned again as in Eq. (6). In
order to obtain the observed value of m2s/m
b by setting λ ∼ sin θc, we impose the Z2 symmetry under the
transformation of Θ1 → Θ1 and Θ2 → −Θ2. For Z2 parity assignment for quark fields, there are a lot of
choices. If we consider the scenario associated with the grand unified theory (GUT), it would be natural
that the Z2 parity for Qi and U
i is common and that for Hu is set to be + for the prediction of a large
top-quark mass. Because Hd always couples to D
i , we can set the Z2 parity for Hd to be + without loss
of generality. Then, there are 64 possibilities on parity assignment for quarks. However, it turns out that
most of them cannot give correct numbers of m2c/m
t and m
b . Consequently, only the following sets
of Z2 parity assignment are enough to be discussed;
• Type I-a,
((Q1, U
1), (Q2, U
2), (Q3, U
3 )) = (+,+,−) , (Dc1, Dc2, Dc3) = (+,+,−) . (13)
• Type I-b,
((Q1, U
1), (Q2, U
2), (Q3, U
3 )) = (+,+,−) , (Dc1, Dc2, Dc3) = (−,+,−) . (14)
3Even when the u- and d-quarks are massless at the electroweak scale, their finite masses would be generated at lower
energy scales due to the strong dynamics[17].
• Type II-a,
((Q1, U
1), (Q2, U
2), (Q3, U
3 )) = (+,−,+) , (Dc1, Dc2, Dc3) = (+,−,+) . (15)
• Type II-b,
((Q1, U
1), (Q2, U
2), (Q3, U
3 )) = (+,−,+) , (Dc1, Dc2, Dc3) = (−,−,+) . (16)
Type I-a gives the mass matrices as
Mu(I-a) =
6 B4λ
5 RB2λ
5 B4λ
4 Rλ2
3 Rλ2 1
αU 〈Hu〉 , Md(I-a) =
5 B4λ
4 Rλ2
4 B2λ
4 RB2λ
αD〈Hd〉 , (17)
where B2n =
k=0 R
2k. Diagonalizing above matrices, we obtain mass ratios m2c/m
t , m
, the
CKM mixing angles (absolute values of CKM matrix elements), and the Kobayashi-Maskawa phase
φ3 ≡ arg(V ∗ubVud/V ∗cbVcd) at the leading order as
= |1 +R4|2λ8 , m
|1−R4|2
(1 + |R|2)2
|Vus| =
|R|8 + |R|−8 − 2(2 cos2 4α− 1)
|Vub| =
|R|||R| − |R|−1|
(|R|+ |R|−1)
|R|4 + |R|−4 + 2 cos 4α
|Vcb| =
|R|2 + |R|−2 + 2 cos 4α
|R|+ |R|−1
λ2 , (18)
φ1 = arg
|R|4 + 1
+ (|R|2 + 1) cos 4α− i(|R|2 − 1) sin 4α
φ2 = arg
(1− |R|2)
− |R|4 + 2i sin 4α
φ3 = arg
(1− |R|2)
|R|4 −
+ (|R|2 − 1) cos 4α+ i(|R|2 + 1) sin 4α
Let us discuss appropriate values of |R|, cos 4α and sin 4α. We first expect that λ ∼ sin θc. Then, |R| > 1
is needed to obtain the reasonable value of |Vub|. However, |R| cannot be much greater than unity, because
m2c/m
t exceeds the experimentally acceptable value. In addition, cos 4α < 0 is necessary for |R| > 1 to
explain the data of |Vcb|. Finally, sin 4α < 0 is required for φ1 to be in the first quadrant. In this case,
however, it turns out that φ3 cannot be in the first quadrant simultaneously.
For numerical evaluation, we take |R| =
3/2, cos 4α = −3/4 and sin 4α = −
7/4. This parameter
set determines λ = 0.25 under the experimental value of |Vus|(= 0.22). The matrices Mu,d(I-a) are
diagonalized, and we obtain
|Vub| = 0.0028 , |Vcb| = 0.032 , (19)
which are in excellent agreement with the CKM mixing angles at the GUT scale, |Vcb| = 0.029–0.039 and
|Vub| = 0.0024–0.0038 which are evaluated from renormalization group method with the experimental data
at low energies[18]. However, quark mass ratios are predicted as
= 0.0061 ,
= 0.073 , (20)
which are about twice as large as the expected values at the GUT scale; mc/mt ∼ 0.0032 ± 0.0007 and
ms/mb = 0.036± 0.005. Finally, we find
φ1 = 12
◦ , φ2 = 31
◦ , φ3 = 137
◦ . (21)
Although the predictions cannot explain all the data simultaneously, it would be amazing to observe that
this model can reproduce most of them in a considerable extent. We also find that for Type I-b, the mixing
angles, mass ratios between 2nd- and 3rd-generations, and the CKM phase are completely the same at
leading order as in Eq. (18).
For Type II-a, the mass matrices are
Mu(II-a) =
6 RB4λ
5 B2λ
5 B4λ
4 Rλ2
3 Rλ2 1
αU 〈Hu〉 , Md(II-a) =
5 RB2λ
4 B2λ
4 B2λ
4 RB2λ
αD〈Hd〉 .
The resulting mass ratios, the CKM parameters and phases φ1, φ2 and φ3 are the same as those in Type
I except for |Vus| and |Vub|, which are
|Vus| =
|R|8 + |R|−8 − 2(2 cos2 4α− 1)
|Vub| =
|R|2||R| − |R|−1|
(|R|+ |R|−1)
|R|4 + |R|−4 + 2 cos 4α
λ3 . (23)
These expressions are different from those in Type I by the multiplication factor |R|.
In this case, we take |R| =
3/2 and cos 4α = −1/2 in order to compensate the effect of the extra factor
|R| in |Vus| in comparison with the that in Type I. In addition, we take sin 4α = −
3/2 and λ = 0.23. We
obtain the following results;
|Vus| = 0.22 , |Vub| = 0.0038 , |Vcb| = 0.035 ,
= 0.0060 ,
= 0.059 ,
φ1 = 18
◦, φ2 = 38
◦, φ3 = 123
◦. (24)
Although the size of ms/mb becomes smaller than that of the Type I, it is still too large to be phenomeno-
logically acceptable. Moreover, mc/mt remains two times greater than the expected value, and φ3 is in
the 2nd quadrant. Type II-b gives almost the same results for the CKM parameters and the mass ratios
between 2nd- and 3rd- generations.
4 Conclusion
We have studied possibility of incorporating CPV by using the FN mechanism in the context of democratic
flavor FN couplings. We have considered models with two FN fields, in which the relative phase of their
VEVs plays as the origin of CP violating phase at low energies. In the scenario with the Z2 symmetry, the
relationship among ratios between quark masses, the absolute values of CKM matrix elements and the CP
violating phase has been examined in several simplest models. We have found that the predictions have
been in good agreement with most of the data. However, the CP violating phase φ3 has been predicted to
be around 130◦, so that the models we have examined are not acceptable. It may be oversimplification to
assume the flavor blind couplings. We expect that the small modification for the democratic assumption
would cure this phenomenological problem.
We have demonstrated the way how introduce the CPV to the FN model and have shown that the
scenario with two FN fields would be promising. An application of our scenario to the lepton sector
including neutrinos is under way and will appear in our future publications. If this will be successfully
achieved, we would obtain a model which gives the unified description of CP phases for quarks and leptons.
Acknowledgements. S. K. was supported, in part, by the Grant-in-Aid of the Ministry of Education,
Culture, Sports, Science and Technology, Government of Japan, Grant Nos.17043008 and 18034004. S. P.
and T. S. were supported in part by the Italian MIUR and INFN under the programs “Fisica Astroparti-
cellare”.
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	Introduction
	CP violation in democratic models
	Examples for the model with two Froggatt-Nielsen fields
	Conclusion
