\documentclass[11pt]{article}
\usepackage{moriond,epsfig}
\usepackage{amsmath,amssymb}
\usepackage{bm}
\usepackage{graphicx, color}
\usepackage{wrapfig}
\bibliographystyle{unsrt}
% for BibTeX - sorted numerical labels by order of
% first citation.
% A useful Journal macro
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}
% Some useful journal names
\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} A}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.} B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}
% Some other macros used in the sample text
\def\st{\scriptstyle}
\def\sst{\scriptscriptstyle}
\def\mco{\multicolumn}
\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
\def\kb{\bar{K^0}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\CPbar{\hbox{{\rm CP}\hskip-1.80em{/}}}
%temp replacement due to no font
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% BEGINNING OF TEXT %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\vspace*{4cm}
\title{Lepton and Slepton mass matrices from $\Delta(54)$ symmetry}
\author{ H. Ishimori}
\address{Graduate~School~of~Science~and~Technology,~Niigata~University, \\
Niigata~950-2181,~Japan }
\maketitle\abstracts{
We present the lepton flavor model with $\Delta (54)$.
Our model reproduces the tri-bimaximal mixing in
the parameter region around
degenerate neutrino masses or two massless neutrinos.
We also study SUSY breaking terms in the slepton sector.
Three families of left-handed and right-handed slepton masses are almost degenerate.
Our model
leads to smaller values of flavor changing neutral currents
than the present experimental bounds.}
\section{Introduction}
Recent experimental data of the neutrino oscillation
indicate the tri-bimaximal form~\cite{HPS} of mixing angles in the
lepton sector within a good accuracy~\cite{Threeflavors}.
Thus, it is a promising step to study how to realize the
tri-bimaximal mixing matrix, in order to understand
the origin of the lepton flavor.
Many authors have been attempting it by using various scenarios.
Non-Abelian discrete flavor symmetries
can provide a natural guidance to constrain
many free parameters in the Yukawa sector.
Actually, several types of models with various non-Abelian discrete flavor
symmetries have been proposed, such as
$S_3$, $D_4$, $Q_4 ,~Q_6$, $A_4$, $T^\prime $,
$S_4$, $\Delta (27)$.
In addition to the above (rather) bottom-up motivation,
we also have a top-down motivation.
Certain classes of non-Abelian flavor
symmetries can be derived from superstring theories.
For example, $D_4$ and $\Delta(54)$ flavor symmetries can be obtained
in heterotic orbifold models
\cite{Kobayashi:2006wq}.
In addition to these flavor symmetries, the $\Delta(27)$ flavor symmetry
can be derived from magnetized/intersecting D-brane models
\cite{Abe:2009vi}.
Thus, it is quite important to study phenomenological aspects
of these non-Abelian flavor symmetries.
Here, we focus on the $\Delta(54)$ discrete symmetry
\cite{Ishimori:2008uc,Ishimori:2009ew}.
Although it includes several interesting
aspects, few authors have considered up to now.
The first aspect is that it consists of
two types of $Z_3$ subgroups and an $S_3$ subgroup.
The $S_3$ group is known as the minimal non-Abelian discrete symmetry,
and the semi-direct product structure of $\Delta(54)$
between $Z_3$ and $S_3$ induces triplet
irreducible representations.
That suggests that the $\Delta(54)$ symmetry could lead to
interesting models.
\section{$\Delta(54)$ flavor model for leptons}
The group $\Delta(54)$ has irreducible representations $1_1$, $1_2$, $2_1$, $2_2$,
$2_3$, $2_4$, $3_1^{(1)}$, $3_1^{(2)}$, $3_2^{(1)}$,
and $3_2^{(2)}$.
There are four triplets and products of
$3_1^{(1)}\times 3_1^{(2)}$ and $3_2^{(1)}\times 3_2^{(2)}$ lead to the trivial singlet.
The relevant multiplication rules are shown
in \cite{Ishimori:2008uc}.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|ccc||c||ccc|}
\hline
&$(l_e,l_\mu,l_\tau)$ & $(e^c,\mu^c,\tau^c)$ &
$(N_e^c,N_\mu^c,N_\tau^c)$ &$h_{u(d)}$ &$ \chi_1 $& $(\chi_{2},\chi_3)$&
$(\chi_{4},\chi_5,\chi_6)$ \\ \hline
$\Delta(54)$ &$3_1^{(1)}$ & $3_2^{(2)}$ & $3_1^{(2)}$
& $1_1$ & $1_2$ & $2_1$ & $3_1^{(2)}$ \\
\hline
\end{tabular}
\end{center}
\caption{Assignments of $\Delta(54)$ representations}
\end{table}
We present the model of the lepton flavor with the $\Delta(54)$
group. The triplet representations of the group
correspond to the three generations of leptons.
The left-handed leptons $(l_e,l_\mu,l_\tau)$,
the right-handed charged leptons $(e^c,\mu^c,\tau^c)$
and the right-handed neutrinos $(N_e^c,N_\mu^c,N_\tau^c)$
are assigned to $3_1^{(1)}$,
$3_2^{(2)}$, and $3_1^{(2)}$, respectively.
New scalars are supposed to be $SU(2)$ gauge singlets.
$\chi_1$, $(\chi_2, \chi_3)$ and
$(\chi_4, \chi_5, \chi_6)$ are assigned to
$1_2$, $2_1$, and $3_1^{(2)}$ of the $\Delta(54)$ representations,
respectively.
The particle assignments of $\Delta(54)$ are summarized in Table 1.
The usual Higgs doublets $h_u$ and $h_d$ are assigned to
the trivial singlet $1_1$ of $\Delta(54)$.
We assume that the scalar fields, $h_{u,d}$ and $\chi_i$, develop
their vacuum expectation values (VEVs) as follows:
\begin{eqnarray}
\left=v_u, \ \left=v_d,
\quad
\left<\chi_i\right>=\alpha_i\Lambda,
\end{eqnarray}
where $i=1,\cdots 6$ and $\Lambda$ is the cutoff scale.
We obtain the diagonal matrix for charged leptons
\begin{eqnarray}
M_l
=
y_1^lv_d
\begin{pmatrix}\alpha_1 & 0 & 0 \\
0 & \alpha_1 & 0 \\
0 & 0 & \alpha_1 \\
\end{pmatrix}
+y_2^l v_d
\begin{pmatrix} \omega\alpha_2-\alpha_3 & 0 & 0 \\
0 & \omega^2\alpha_2-\omega^2\alpha_3 & 0 \\
0 & 0 & \alpha_2-\omega\alpha_3 \\
\end{pmatrix},
\label{ME}
\end{eqnarray}
By using the seesaw mechanism $M_\nu = M_D^T M_N^{-1} M_D$, the neutrino
mass matrix can be obtained.
In our model,
the lepton mixing comes from the structure of the neutrino mass matrix.
In order to reproduce the maximal mixing between
$\nu_\mu$ and $\nu_\tau$, we take $\alpha_5=\alpha_6$, and then
we have
\begin{eqnarray}
\label{mass}
M_\nu
&=&
\frac{y_D^2v_u^2}{ \Lambda d}
\begin{pmatrix}y_1^2\alpha_5^2-y_2^2\alpha_4^2 & -y_1y_2
\alpha_5^2+y_2^2\alpha_4\alpha_5 & -y_1y_2 \alpha_5^2+y_2^2\alpha_4\alpha_5 \\
-y_1y_2 \alpha_5^2+y_2^2\alpha_4\alpha_5 &
y_1^2\alpha_4\alpha_5-y_2^2\alpha_5^2 & -y_1y_2 \alpha_4^2+y_2^2 \alpha_5^2 \\
-y_1y_2 \alpha_5^2+y_2^2\alpha_4\alpha_5 & -y_1y_2
\alpha_4^2+y_2^2 \alpha_5^2 & y_1^2\alpha_4\alpha_5-y_2^2\alpha_5^2 \\
\end{pmatrix},
\label{neutri}
\end{eqnarray}
where $ d=y_1^3\alpha_4\alpha_5\alpha_6-y_1y_2^2\alpha_4^3-
y_1y_2^2\alpha_5^3-y_1y_2^2\alpha_6^3
+2y_2^3\alpha_4\alpha_5\alpha_6$. From now on, we denote
$y_D$ as Yukawa coupling of Dirac neutrino and $y_1$, $y_2$ of
Majorana neutrino.
Above mass matrix indicates $\theta_{23}=45^\circ$, $\theta_{13}=0$ and
\begin{eqnarray}
\theta_{12}
=\frac12\arctan\frac{2\sqrt2 y_2\alpha_5}
{y_1\alpha_5+y_2\alpha_4-y_1 \alpha_4} \qquad (y_2\alpha_4\not
=y_1\alpha_5).
\end{eqnarray}
%where we exclude the case $$.
Neutrino masses are given by
\begin{eqnarray}
m_1&=& \frac{y_D^2v_u^2}{ \Lambda d}
[y_1^2\alpha_5^2-y_2^2\alpha_4^2 -\sqrt2 (-y_1y_2
\alpha_5^2+y_2^2\alpha_4\alpha_5)\tan\theta_{12}],
\nonumber\\
m_2&=& \frac{y_D^2v_u^2}{ \Lambda d}
[y_1^2 \alpha_4\alpha_5-y_1y_2 \alpha_4^2 + {\sqrt2} (-y_1y_2
\alpha_5^2+y_2^2\alpha_4\alpha_5) \tan\theta_{12}],
\nonumber\\
m_3&=&\ \frac{y_D^2v_u^2}{ \Lambda d}
[y_1^2 \alpha_4\alpha_5+y_1y_2 \alpha_4^2-2y_2^2\alpha_5^2],
\label{m3}
\end{eqnarray}
which are reconciled with the normal hierarchy of neutrino masses
in the case of $y_1\alpha_5\simeq y_2\alpha_4$.
Now, we can estimate magnitudes of $\alpha_i(i=4,5,6)$
by using Eq.(\ref{m3}) and assuming $\alpha_4\simeq \alpha_5=\alpha_6$.
If we take all Yukawa couplings to be order one,
Eq.(\ref{m3}) turns to be
$v_u^2\sim\Lambda \alpha_4 m_3$ because of $d\sim \alpha_4^3$.
Putting $v_u\simeq 165$GeV ($\tan\beta=3$),
$m_3\simeq \sqrt{\Delta m_{\rm atm}^2}\simeq 0.05$eV,
and $\Lambda = 2.43\times 10^{18}$GeV, we obtain
$\alpha_4={\cal O}(10^{-4}-10^{-3})$.
Thus, values of $\alpha_i (i=4,5,6)$ are enough suppressed
to discuss perturbative series of higher mass operators.
\section{Numerical result}
We show our numerical analysis of neutrino masses and mixing angles
in the normal mass hierarchy.
Neglecting higher order corrections of mass matrices,
we obtain the allowed region of parameters and predictions of neutrino masses and mixing angles. Here, we neglect the renormalization effect of the
neutrino mass matrix because we suppose the normal hierarchy of
neutrino masses and take $\tan\beta = 3$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Input data of masses and mixing angles are taken in the region of
3$\sigma$ of the experimental data \cite{Threeflavors}:
\begin{eqnarray}
&&\Delta m_{\rm atm}^2=(2.07\sim 2.75)\times 10^{-3}
{\rm eV}^2 \ ,
\quad \Delta m_{\rm sol}^2= (7.05\sim 8.34) \times 10^{-5} {\rm eV}^2 \ ,
\nonumber \\
&& \sin^2 \theta_{\rm atm}=0.36\sim 0.67 \ ,
\quad \sin^2 \theta_{\rm sol}=0.25 \sim 0.37 \ , \quad
\sin^2 \theta_{\rm reactor} \leq 0.056\ ,
%\label{}
\end{eqnarray}
and $\Lambda=2.43 \times 10^{18}$GeV is taken.
We fix $y_D=y_1=1$ as a convention, and vary $y_2/y_1$.
The change of $y_D$ and $y_1$ is absorbed into the change of
$\alpha_i(i=4,5,6)$.
If we take a smaller value of $y_1$, values of $\alpha_i$ scale up.
On the other hand,
if we take a smaller value of $y_D$, the magnitude of $\alpha_i$ scale down.
As expected in the discussion of previous section,
the experimentally allowed values are reproduced
around $\alpha_4=\alpha_5=\alpha_6$.
We can predict the deviation from the tri-bimaximal mixing.
The remarkable prediction is given in the magnitude of $\sin^2\theta_{13}$.
In Figures 1 (a) and (b), we plot the allowed region of mixing angles
in planes of $\sin^2\theta_{12}$-$\sin^2\theta_{13}$ and
$\sin^2\theta_{23}$-$\sin^2\theta_{13}$, respectively.
It is found that the upper bound of $\sin^2\theta_{13}$ is $0.01$.
It is also found the strong correlation between
$\sin^2\theta_{23}$ and $\sin^2\theta_{13}$.
Unless $\theta_{23}$ is deviated from the maximal mixing considerably,
$\theta_{13}$ remains to be tiny. This is a testable relation in this model.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=7 cm]{theta12-13.eps}
\quad
\includegraphics[width=7 cm]{theta23-13.eps}
\quad
\caption{Prediction of the upper bound of $\sin^2\theta_{13}$
on (a) $\sin^2\theta_{12}-\sin^2\theta_{13}$ and
(b) $\sin^2\theta_{23}-\sin^2\theta_{13}$ planes.}
\end{center}
\end{figure}
\section{SUSY breaking terms}
In this section, we study SUSY breaking terms, which
are predicted in our $\Delta(54)$ model.
We consider the gravity mediation within the
framework of supergravity theory.
Let us study soft scalar masses.
Within the framework of supergravity theory,
soft scalar mass squared is obtained as \cite{Kaplunovsky:1993rd}
\begin{eqnarray}
m^2_{\bar{I}J} K_{{\bar{I}J}}= m_{3/2}^2K_{{\bar{I}J}}
+ |F^{\Phi_k}|^2 \partial_{\Phi_k}
\partial_{ \bar{\Phi_k} } K_{\bar{I}J}-
|F^{\Phi_k}|^2 \partial_{\bar{\Phi_k}} K_{\bar{I}L} \partial_{\Phi_k}
K_{\bar{M} J} K^{L \bar{M}},
\label{eq:scalar}
\end{eqnarray}
where $K$ denotes the K\"ahler potential, $K_{\bar{I}J}$ denotes
second derivatives by fields,
i.e. $K_{\bar{I}J}={\partial}_{\bar{I}} \partial_J K$
and $K^{\bar{I}J}$ is its inverse.
The invariance under the $\Delta(54)$ flavor symmetry
as well as the gauge invariance requires the following form
of the K\"ahler potential of $l_I$ and $e_I$ $(I=e,\mu,\tau)$
\begin{equation}
\label{eq:Kahler}
K = Z^{(L)}(Z)\sum_{I=e,\mu,\tau} |l_I|^2 +
Z^{(R)}(Z)\sum_{I=e,\mu,\tau} |e_I|^2 ,
\end{equation}
at the lowest level, where $Z^{(L)}(Z)$ and $Z^{(R)}(Z)$ are
arbitrary functions of the singlet fields $Z$.
Both matrices are proportional
to the $(3\times 3)$ identity matrix.
This form would be obvious because
$(l_e,l_\mu,l_\tau)$ and $(e^c,\mu^c,\tau^c)$
are $\Delta(54)$ triplets.
At any rate, it is the prediction of our model that
three families of left-handed and right-handed masses
are degenerate.
Let us estimate corrections including
$\chi_i \chi_k$ as well as $\chi_i \chi_k^*$ for $i,k=1,2,3,4,5,6$.
The $\Delta(54)$ flavor symmetric invariance allows
only the terms such as $\chi_i \chi_k^*$ for $i,k=4,5,6$ to appear
in off-diagonal entries of the K\"ahler metric of
$(l_e,l_\mu,l_\tau)$.
When we take into account the corrections from $\chi_i \chi_k^*$ for
$i,k=4,5,6$ to the K\"ahler potential,
the soft scalar masses squared for left-handed charged sleptons
have the following corrections,
\begin{eqnarray}
\begin{split}
&(m_{\tilde L}^2)_{IJ} = m_{L}^2
\left(
\begin{array}{ccc}
1 + {\cal O}(\tilde\alpha^2) & {\cal O}(\alpha_4^2) & {\cal O}(\alpha_4^2)
\\
{\cal O}(\alpha_4^2) & 1+ {\cal O}(\tilde\alpha^2) & {\cal O}(\alpha_4^2)
\\
{\cal O}(\alpha_4^2) & {\cal O}(\alpha_4^2) & 1 + {\cal O}(\tilde\alpha^2)
\\
\end{array} \right),
\\
&(m_{\tilde R}^2)_{IJ} = m_{R}^2
\left(
\begin{array}{ccc}
1 + {\cal O}(\tilde\alpha^2) & {\cal O}(\alpha_4^2) & {\cal O}(\alpha_4^2)
\\
{\cal O}(\alpha_4^2) & 1+ {\cal O}(\tilde\alpha^2) & {\cal O}(\alpha_4^2)
\\
{\cal O}(\alpha_4^2) & {\cal O}(\alpha_4^2) & 1 + {\cal O}(\tilde\alpha^2)
\\
\end{array} \right),
\label{eq:soft-mass-2-R}
\end{split}
\end{eqnarray}
where $\tilde\alpha$ is the averaged value of
$\alpha_{1-6}$.
These deviations may not be important for
direct measurement of slepton masses.
However, the off-diagonal entries in the SCKM basis
are constrained by the FCNC experiments.
Our model predicts
\begin{eqnarray}
(\Delta_{LL})_{12} = \frac{(m_L^2)_{12}^{(SCKM)}}{(m_L^2)_{11}}= {\cal
O}(\alpha_4^2),
\qquad
(\Delta_{RR})_{12} = \frac{(m_R^2)_{12}^{(SCKM)}}{(m_R^2)_{11}}= {\cal
O}(\alpha_4^2).
\end{eqnarray}
Recall that the diagonalizing matrices of left-handed and right-handed
fermions are almost the identity matrix.
The $\mu \rightarrow e \gamma$ experiments constrain
these values as $(\Delta_{LL,RR})_{12} \leq {\cal
O}(10^{-3})$, when $m_{L,R} = 100$ GeV.
On the other hand, the parameter space in the numerical result
corresponds to $\alpha_4 \leq 10^{-2}$ and
leads to $(\Delta_{LL,RR})_{12} \leq {\cal O}(10^{-4})$.
Thus, our parameter region would be favorable also from
the viewpoint of the FCNC constraints.
\section*{Acknowledgments}
The work is done in the collaboration with
T. Kobayashi, H. Okada, Y. Shimizu, and M. Tanimoto,
and has been supported by Grand-in-Aid for Scientific Research,
No.21.5817 from the Japan Society of Promotion of Science.
\section*{References}
\begin{thebibliography}{99}
\bibitem{HPS}
P.F. Harrison, D.H. Perkins, and W.G. Scott,
Phys. Lett. B {\bf 530}, 167 (2002);\\
%%CITATION = PHLTA,B530,167;%%
P.F. Harrison and W.G. Scott,
Phys. Lett. B {\bf 535}, 163 (2002).
%%CITATION = PHLTA,B535,163;%%
\bibitem{Threeflavors}
T.~Schwetz, M.~Tortola, and J.W.F.~Valle, arXiv:0808.2016;\\
G.L. Fogli, E. Lisi, A. Marrone, A. Palazzo, and A.M. Rotunno,
Phys. Rev. Lett. {\bf 101} 141801 (2008), arXiv:0806.2649.
\bibitem{Kobayashi:2006wq}
T.~Kobayashi, H.~P.~Nilles, F.~Ploger, S.~Raby and M.~Ratz,
Nucl.\ Phys.\ B {\bf 768}, 135 (2007), arXiv:hep-ph/0611020.
%%CITATION = NUPHA,B768,135;%%
\bibitem{Abe:2009vi}
H.~Abe, K.~S.~Choi, T.~Kobayashi and H.~Ohki,
%``Non-Abelian Discrete Flavor Symmetries from Magnetized/Intersecting Brane
%Models,''
Nucl.\ Phys.\ B {\bf 820}, 317 (2009)
[arXiv:0904.2631 [hep-ph]].
%%CITATION = NUPHA,B820,317;%%
\bibitem{Ishimori:2008uc}
H.~Ishimori, T.~Kobayashi, H.~Okada, Y.~Shimizu and M.~Tanimoto,
%``Lepton Flavor Model from Delta(54) Symmetry,''
JHEP {\bf 0904}, 011 (2009)
[arXiv:0811.4683 [hep-ph]].
%%CITATION = JHEPA,0904,011;%%
%\cite{Ishimori:2009ew}
\bibitem{Ishimori:2009ew}
H.~Ishimori, T.~Kobayashi, H.~Okada, Y.~Shimizu and M.~Tanimoto,
%``Delta(54) Flavor Model for Leptons and Sleptons,''
JHEP {\bf 0912}, 054 (2009)
[arXiv:0907.2006 [hep-ph]].
%%CITATION = JHEPA,0912,054;%%
\bibitem{Kaplunovsky:1993rd}
V.~S.~Kaplunovsky and J.~Louis,
%``Model independent analysis of soft terms in effective supergravity and
in
%string theory,''
Phys.\ Lett.\ B {\bf 306}, 269 (1993), arXiv:hep-th/9303040.
%%CITATION = PHLTA,B306,269;%%
\end{thebibliography}
\end{document}