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Quark mass hierarchies and CP violation in A4 × A4 × A4 modular symmetric flavor models
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Quark mass hierarchies and CP violation in A4 × A4 × A4 modular symmetric flavor models
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Journal Article
Quark mass hierarchies and CP violation in A4 × A4 × A4 modular symmetric flavor models
2023
Overview
A
bstract
We study
A
4
× A
4
× A
4
modular symmetric flavor models to realize quark mass hierarchies and mixing angles without fine-tuning. Mass matrices are written in terms of modular forms. At modular fixed points
τ
=
i∞
and
ω
,
A
4
is broken to
Z
3
residual symmetry. When the modulus
τ
is deviated from the fixed points, modular forms show hierarchies depending on their residual charges. Thus, we obtain hierarchical structures in mass matrices. Since we begin with
A
4
× A
4
× A
4
, the residual symmetry is
Z
3
× Z
3
× Z
3
which can generate sufficient hierarchies to realize quark mass ratios and absolute values of the CKM matrix
|V
CKM
|
without fine-tuning. Furthermore, CP violation is studied. We present necessary conditions for CP violation caused by the value of
τ
. We also show possibilities to realize observed values of the Jarlskog invariant
J
CP
, quark mass ratios and CKM matrix
|V
CKM
|
simultaneously, if (10) adjustments in coefficients of Yukawa couplings are allowed or moduli values are non-universal.
Publisher
Springer Berlin Heidelberg,Springer Nature B.V
