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\preprint{BU-02-27}

\title{Flavor Symmetries in Extra Dimensions}

\author{Alfredo Aranda\footnote{aranda@physics.bu.edu}}

\vskip 0.1in

\address{Department of Physics, Boston University,
590 Commonwealth Ave, Boston, MA 02215 }

\vskip .1in

\author{J. Lorenzo Diaz-Cruz\footnote{ldiaz@sirio.ifuap.buap.mx}}

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\address{Instituto de F\'{\i}sica, BUAP, A.P. J-48, 72570 Puebla,
Pue. M\'{e}xico}

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%\date{\hourmin \ \today}
\date{July, 2002}
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\maketitle
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\begin{abstract}
We present a model of flavor based on a discrete local symmetry that
reproduces all fermion masses and mixing angles both in the
quark and lepton sectors. The particle content of the model is that of 
the standard model plus an additional flavon field. All the fields 
propagate in a fifth universal extra dimension and the flavor
scale is associated with the cutoff of the 5D theory which is
$\sim 10$~TeV. The Yukawa matrices as well as the Majorana mass matrix
for the neutrinos are generated by higher dimension operators involving
the flavon field. When the flavon field acquires a vacuum expectation value
it breaks the flavor symmetry and thus generates the 
Yukawa couplings. The model
is consistent with the nearly bimaximal solution to the solar and atmospheric
neutrino deficits. 
\end{abstract}

\thispagestyle{empty}

\newpage
\setcounter{page}{1}

\section{Introduction} \label{sec:intro}
It is possible that there exist extra dimensions and that they might
even play an important role in electroweak physics. This possibility has led
to an impressive amount of work over the past few years resulting in 
new and exciting ways of tackling (or interpreting) some of 
the main problems in particle physics. For a partial list
of references see~\cite{XD}.
The problem of flavor is one of the pressing problems in particle physics.
In the context of extra dimensions some interesting solutions have been presented. For
example it is possible to generate hierarchies among the fermion masses and mixing 
parameters by restricting them to a brane while imposing some
flavor symmetry broken by the mechanism of shinning~\cite{nima1}. Another way is to localize the
fermion fields in different positions along the extra dimensions, and in this way
generate hierarchies among masses and mixing parameters through wave-function
overlaps~\cite{martin1}. Yet another alternative, similar in 
spirit to the previous ``cartographic'' solution, is to localize the fermion
fields along different points inside a fat brane~\cite{fat} or by placing different matter fields 
in different branes~\cite{dvali1}. In the fat-brane case, the hierarchies and mixing properties are
obtained, as before, through the wave-function overlaps. In the case of several branes this is
accomplished by having the branes intersect in some specific way. 
Other new mechanisms employ warped extra dimensions~\cite{randall} to, for
example, produce small Yukawa couplings and hierarchies, 
as well as small neutrino masses without the need of a seesaw~\cite{flavor}.

In this letter we present a model of flavor in 5D with the following properties: 
\begin{itemize}
\item There is one 
extra spacetime dimension in which all fields propagate, i.e. a universal extra dimension.
This extra dimension is compactified on an $S^1/Z_2$ orbifold with a 
radius of compactification $R$.
\item A discrete local flavor symmetry is added to the standard model 
(SM) gauge group. This symmetry is broken by
the vacuum expectation value (vev) of a single flavon field leading to 
the generation of Yukawa matrix
textures for all matter fields. 
\item The particle content of the model
is that of the SM plus a single additional scalar flavon field. This is the
minimal set we require in order to reproduce the observed hierarchies.
\end{itemize}

We show that given the above conditions, it is possible to generate a viable model
of flavor that reproduces all observed masses and mixing angles, both in the quark and lepton
sectors. Furthermore, this is accomplished with a flavor scale determined by the current 
perturbativity bound on the scale of a single universal extra dimension 
of $\sim 10$~TeV~\cite{dobrescu1}.
The properties of the model are chosen so as to minimize the amount of additions beyond the SM.

The model is presented in section \ref{sec:model}. 
There we show how to generate the Yukawa matrices from
operators in the 5D Lagrangian. In Section~\ref{sec:results}, we present the results from a fit 
to the experimental results and comment on future work. 
Finally Section~\ref{sec:conclusion} contains
our conclusions.

\section{A Model} \label{sec:model}

As mentioned in the Introduction, the model consists of the SM fields plus a flavon field all
propagating in a 5D spacetime. We assume the fifth coordinate is compactified
on  an $S^1/Z_2$ orbifold with a radius
of compactification $R = 1/M_c$, where $M_c$ is the compactification scale. The 5D fermion
fields can be decomposed in the usual way as~\cite{dobrescu1}

\begin{eqnarray} \label{decomposition-fermions}
{\cal{Q}}(x^{\mu},y)  = \frac{1}{\sqrt{\pi R}} \left( Q^{(0)}_L(x^{\mu})
+ \sqrt{2} \sum_{n=1}^{\infty}\left[ P_L Q_L^{(n)}(x^{\mu})\cos\left(\frac{ny}{R}\right)
+ P_R Q_R^{(n)}(x^{\mu})\sin\left(\frac{ny}{R}\right) \right] \right) \, , \\ \nonumber \\ 
{\cal{E}}(x^{\mu},y) = \frac{1}{\sqrt{\pi R}} \left( E^{(0)}_R(x^{\mu})
+ \sqrt{2} \sum_{n=1}^{\infty}\left[ P_R E_R^{(n)}(x^{\mu})\cos\left(\frac{ny}{R}\right) 
+ P_L E_L^{(n)}(x^{\mu})\sin\left(\frac{ny}{R}\right) \right] \right) \, ,
\end{eqnarray}
where ${\cal Q}$ and ${\cal E}$ denote ${\rm SU(2)}_W$ doublet and singlet fields 
respectively. $P_L$ and
$P_R$ are the 4D chiral projection operators 
$P_{R,L} = (1 \pm \gamma_5)/2$. This decomposition ensures that the 
zero-modes correspond to SM fields. Since the Higgs and flavon field must
couple to these zero-modes, they must be even under the $Z_2$.
Their decomposition is then given by

\begin{eqnarray} \label{decomposition-scalar}
{\cal{S}}(x^{\mu},y) = \frac{1}{\sqrt{\pi R}} \left( S^{(0)}(x^{\mu})
+ \sqrt{2} \sum_{n=1}^{\infty} S^{(n)}(x^{\mu})\cos\left(\frac{ny}{R}\right) \right) \, .
\end{eqnarray}

Now we can consider the generation of Yukawa matrices.
Once a flavor symmetry is included in the model, all the terms in the Lagrangian 
responsible for the fermion (modulo neutrinos) masses will in general have the form

\begin{eqnarray} \label{general}
{\cal L}^5 \sim \hat{\lambda}_{ab} \bar{{\cal Q}}_{a} i\sigma_{2}{\cal H}^*{\cal U}_{b} 
\frac{\Phi^n}{\Lambda^{3n/2}} + h.c. \, ,
\end{eqnarray}
where $a$ and $b$ are generation indices, 
$\hat{\lambda}_{ab} = \lambda_{ab}\sqrt{\pi R}$, 
$\lambda_{ab}$ is the 4D Yukawa coupling
(which we assume to be a number of order one), 
$\Lambda$ is the cutoff of the theory, and $\Phi$ represents the flavon field. The hierarchies in masses
and mixing angles are thus obtained by assigning different charges to different generations
in such a way as to obtain realistic textures for the Yukawa matrices. To see how this works
in the present model, lets consider Eq.~(\ref{general}) after compactification. We are assuming
that the flavor symmetry is broken by the vev of the flavon field at or very close 
to the scale $\Lambda$. After compactification, we obtain

\begin{eqnarray} \label{general4D}
{\cal L}^4 \sim \lambda_{ab} \bar{Q}_a i \sigma_2 H^*  U_b \left[
\left(\frac{M_c}{\pi \Lambda}\right)^{n/2}\right] + h.c. \, ,
\end{eqnarray}
where all fields now correspond to the zero-modes of those in Eq.~(\ref{general}), and where 
the vev of $\Phi$ has been set equal to $\Lambda$. We now define a new 4D Yukawa coupling given by
\begin{eqnarray} \label{yukawa4D}
\lambda_{ab}^{\prime} = \lambda_{ab}\left(\frac{M_c}{\pi \Lambda}\right)^{n/2} =
{\rm O(1)} \left(\frac{M_c}{\pi \Lambda}\right)^{n/2} = {\rm O(1)} \epsilon^{n} \, ,
\end{eqnarray}
where as mentioned before $\lambda_{ab} = $~O(1). Using 
the values for $M_c = 0.3$~TeV, and $\Lambda = 10$~TeV from 
Ref.~\cite{dobrescu1}, we obtain
that $\epsilon \approx 0.1$.
This is the prescription we use to generate the Yukawa matrices for the quarks and charged leptons. 

In the case of the neutrino mass matrix, we use the following operator~\cite{weinberg}

\begin{eqnarray} \label{neutrino5D}
{\cal L}^5 \sim l_{ab} \bar{{\cal L}}^c_{ai} {\cal L}_{bj} 
{\cal H}_k {\cal H}_m \frac{\Phi^n}{\Lambda^{3n/2+2}}
\epsilon_{ik}\epsilon_{jm} \, 
\end{eqnarray}
where $i$, $j$, $k$, and $m$ are SU(2) indices; $a$ and $b$ are generation
indices, and $l_{ab}$ is an O(1) parameter. After compactification this operator becomes
\begin{eqnarray} \label{neutrino4D}
{\cal L}^4 \sim l_{ab}^{\prime} \bar{L}^c_{ai} L_{bj} H_k H_m 
\epsilon_{ik} \epsilon_{jm}\, ,
\end{eqnarray}
where
\begin{eqnarray} \label{lprime}
l_{ab}^{\prime} = \frac{l_{ab}}{\Lambda}\left(\frac{M_c}{\pi \Lambda}\right)^{1/2(n+2)}
= {\rm O(1)}\frac{\epsilon^{(n+2)}}{\Lambda} \, .
\end{eqnarray}

We are now in a position to present the model. It is based on a $Z_{10}$ local symmetry whose
anomalies are assumed to be canceled by a Green-Schwarz
mechanism~\cite{green} (For a discussion
on discrete gauge symmetries see~\cite{discrete}). The charge assignments 
for the matter fields are given by 
\begin{eqnarray} \label{charges}
\nonumber
{\cal Q} \sim (0,9,8) \rightarrow \bar{{\cal Q}} \sim (0,1,2) \,\, , \\ \nonumber
{\cal L} \sim (1,0,9) \rightarrow \bar{{\cal L}} \sim (9,0,1) \,\, , \\ 
{\cal U} \sim (6,7,8) \,\, , \,\, {\cal D} \sim (5,6,6) \,\, , \,\, {\cal E} \sim (6,7,7) \, ,
\end{eqnarray}
where the numbers in parenthesis correspond to the charges of each
generation and add mod $10$. 
The charges for the
Higgs (${\cal H}$) and flavon field ($\Phi$) are 
\begin{eqnarray} \label{chargesscalars}
{\cal H} \sim 0 \,\, , \,\, \Phi \sim 1 \rightarrow \Phi^* \sim 9 .
\end{eqnarray}
Using these assignments together with Eqs.~(\ref{general4D}) and~(\ref{yukawa4D}) to compute the
Yukawa matrices we obtain
\begin{eqnarray} \label{yukawau}
\lambda_U \sim 
\left(
\begin{array}{ccc}
\phi^4 & \phi^3 & \phi^2 \\
\phi^3 & \phi^2 & \phi \\
\phi^2 & \phi & 1
\end{array}
\right) \rightarrow
\left(
\begin{array}{ccc}
\epsilon^4 & \epsilon^3 & \epsilon^2 \\
\epsilon^3 & \epsilon^2 & \epsilon \\
\epsilon^2 & \epsilon & 1 
\end{array}
\right) \,\, , \\ \nonumber \\
\lambda_D \sim 
\left(
\begin{array}{ccc}
\phi^5 & \phi^4 & \phi^4 \\
\phi^4 & \phi^3 & \phi^3 \\
\phi^3 & \phi^2 & \phi^2
\end{array}
\right) \rightarrow
\left(
\begin{array}{ccc}
\epsilon^5 & \epsilon^4 & \epsilon^4 \\
\epsilon^4 & \epsilon^3 & \epsilon^3 \\
\epsilon^3 & \epsilon^2 & \epsilon^2 
\end{array}
\right) \,\, , \\ \nonumber \\
\lambda_E \sim 
\left(
\begin{array}{ccc}
\phi^5 & \phi^4 & \phi^4 \\
\phi^4 & \phi^3 & \phi^3 \\
\phi^3 & \phi^2 & \phi^2
\end{array}
\right) \rightarrow
\left(
\begin{array}{ccc}
\epsilon^5 & \epsilon^4 & \epsilon^4 \\
\epsilon^4 & \epsilon^3 & \epsilon^3 \\
\epsilon^3 & \epsilon^2 & \epsilon^2 
\end{array}
\right) \,\, , 
\end{eqnarray}
where O(1) coefficients have been omitted, and only the powers of $\phi$ are 
shown here for clarity. 
$\phi$ represents the zero-mode of $\Phi$. 
In the next section we show that these textures
reproduce the observed masses and mixing angles. 
In order to do that we also need the neutrino mass matrix
which, by Eqs.~(\ref{neutrino4D}) and~(\ref{lprime}), is given by 
\begin{eqnarray}\label{mll}
M_{\nu} \sim \left(
\begin{array}{ccc}
(\phi^{*})^2 & \phi^{*} & 1 \\
\phi^{*} & 1 & \phi \\
1 & \phi & \phi^2
\end{array}\right) \rightarrow 
\left(\begin{array}{ccc}
\epsilon^4 & \epsilon^3 & \epsilon^2 \\
\epsilon^3 & \epsilon^2 & \epsilon^3 \\
\epsilon^2 & \epsilon^3 & \epsilon^4 \end{array} \right)\, ,
\end{eqnarray}
Note that by Eq.~(\ref{lprime}) if there are $n$ flavon fields participating in a
given entry, their contribution goes as $\epsilon^{n+2}$.

\section{Results} \label{sec:results}
Here we show that the textures obtained in the previous section reproduce the observed
mass patterns and mixing angles in both the quark and lepton sectors. To do this, we
need to include O(1) coefficients in the entries of the Yukawa matrices and the neutrino
mass matrix. These coefficients are determined by performing a fit to the observables.
We emphasize that the hierarchies among the masses and mixing angles are determined by
the textures and not by the coefficients. In order
to determine that this is the case, we perform several fits starting from randomly selected
initial sets of parameters. An important property of the fits is that the O(1) parameters are 
not treated freely and they are
allowed to vary only within a range, say between $1/3$ and $3$. They are then 
included into a $\chi^2$
function and thus treated as additional pieces of data. The particular range is of course arbitrary
and is meant only to determine what we explicitly mean by O(1). For details about the fit and
how to treat the O(1) coefficients see Ref.~\cite{aranda}.

In the quark sector we fit to the six quark masses and three CKM-angles 
(CP violation is neglected)
whereas in the charged lepton sector we use the $e$, $\mu$, and $\tau$ masses.
The experimental uncertainties on the observables (or estimates in the case of the quark masses)
used in the fits are either those of Ref~\cite{PDG} or $1\%$ 
of the central value, whichever is larger.
The values used in the neutrino sector are those given by Ref.~\cite{superk}
for the solar neutrinos where 
$\Delta m^2_{12} \approx 3 - 19 \times 10^{-5} {\rm eV}^2$ and 
$\tan^2\theta_{12} \approx 0.25 - 0.65$ and in the case of atmospheric neutrinos we use
the values in Ref.~\cite{atmos}: $\Delta m^2_{23} \approx 1.5 - 5 \times 10^{-3} {\rm eV}^2$ and 
$\sin^22\theta_{23} > 0.88$. These values are implemented into the fit 
by using the following ranges:
\begin{eqnarray} \label{neutrinoranges}
\nonumber
\ln \left(\frac{\Delta m_{23}^2}{\Delta m_{12}^2}\right) 
& = & 1.56 \pm 0.33 \, , \\ \nonumber
\sin^22\theta_{23} 
& = & 0.94 \pm 0.03 \, , \\
\tan ^2\theta_{12}
& = & 0.55 \pm 0.18 \, .
\end{eqnarray}
The logarithm is taken for the ratio of $\Delta m^2$'s in order to take into 
account that the lower bound is
much smaller than the upper bound. 
We find very good fits for a large number of initial points and conclude that the textures 
presented in the previous section do reproduce the observed patterns. Lets
consider the lepton sector. One of the fits obtained by using the following parametrization
\\
\begin{eqnarray} \label{parametrization}
\lambda_E = \left(
\begin{array}{ccc}
l_1 \epsilon^5 & l_2 \epsilon^4 & l_3 \epsilon^4 \\
l_4 \epsilon^4 & l_5 \epsilon^3 & l_6 \epsilon^3 \\
l_7 \epsilon^3 & l_8 \epsilon^2 & l_9 \epsilon^2 
\end{array}
\right) \,\, , \, \,
M_{\nu} \sim \left(\begin{array}{ccc}
n_1 \epsilon^4 & n_2 \epsilon^3 & n_3 \epsilon^2 \\
n_2 \epsilon^3 & n_4 \epsilon^2 & n_5 \epsilon^3 \\
n_3 \epsilon^2 & n_5 \epsilon^3 & n_6 \epsilon^4 \end{array} \right) \, 
\end{eqnarray}
\\
led to the results presented in Tables~\ref{fit} and~\ref{data}.
We will present a complete analysis involving a large number of fits in a
longer version of this letter~\cite{progress}. 
Here we just mention that in addition to the
values quoted above, there are other constraints that must be met by the model. One of them
is the bound $\sin^22\theta_{13} < 0.1 -0.3$~\cite{13}. Most of the fits fall well below
this bound (see Table~\ref{data}). Another constraint results from  
an upper limit on the total neutrino mass coming from
cosmology where 
the present bound is $m_{\nu ,tot} < 2.2$~eV~\cite{cosmo}. 
Also, when the two ranges for $\Delta m^2$  are combined with the LEP
result that the number of neutrinos is $N_{\nu} = 2.9841 \pm
0.0083$~\cite{LEP} 
one finds a lower limit
on the total neutrino mass of $m_{\nu ,tot} > 0.04$~eV. In summary, we need to satisfy
\begin{eqnarray} \label{neutrinomassrange}
0.04 {\rm eV} \le m_{\nu ,tot} \le 2.2 {\rm eV} \, .
\end{eqnarray}
We find that in most fits the model satisfies this bound with a 
total mass that tends to be in the range
$1.8 - 2.2$~eV (see Table~\ref{data}). We note that since in this 
model the scale of neutrino masses is not free,
we could have incorporated these bounds to the fit instead of the $\Delta m^2$
ratio. However, by fitting to the 
ratio only and obtaining the right mass scale for many 
different fits, we verify that the neutrino
mass scale is insensitive to the O(1) parameters.


Another comment is that a renormalization group analysis 
should also be incorporated into the fit. For the present model, since 
the flavon field has
a mass of O($\Lambda$), it does not participate in the running and the hierarchies 
are affected only by scale-independent factors of O(1)~\cite{dienes}. 
The running might change the overall
scales, and thus it might be necessary to modify the overall scale of
the Yukawa matrices. This can be easily done by changing either the charge
assignments or by enlarging the flavor group to a $Z_{11}$ for example. 
Also, we don't view this
model as unique, in fact, it might be possible to create a model with a discrete Non-Abelian gauge
symmetry that can be broken sequentially. In this case, depending on the scales, the flavon fields
can participate in non-trivial ways through the RGE analysis. 
We are currently exploring these possibilities.

\section{Conclusion} \label{sec:conclusion}
We have presented a model of flavor based on a $Z_{10}$ local symmetry whose
particle content is that of the SM plus a single additional flavon
field. There is one universal extra dimension compactified on an $S^1/Z_2$
orbifold with radius of compactification 
$R = 1/Mc = 3.33 \,{\rm TeV}^{-1}$, and with
a high energy cutoff of $\Lambda = 10$~TeV 
which is also the flavor scale.
When the flavon field acquires a vacuum expectation value it generates
Yukawa matrices and a Majorana mass matrix which are then used to
perform fits to the observables in both the quark and lepton sectors.
The model successfully accommodates all the the data, and is consistent
with the nearly bimaximal solution to the solar and atmospheric 
neutrino deficits.

\begin{center}
{\bf Acknowledgments}
\end{center}We would like to thank Qaisar Shafi, Danny Marfatia, and Chris Carone for
helpful conversations and comments. We also thank Paolo Amore for his
comments and for reading the manuscript. 
A.A.'s work was supported by the Department of Energy under grant 
DE-FG02-91ER40676, J.L. D.-C. thanks the support of CONACYT and SNI (Mexico).


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%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  TABLES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{tabular}{c|c|c|c|c}
\multicolumn{5}{c}{$\epsilon = 0.1$ \,\,\, $\chi^2 = 2.02$ } \\ \hline
$l_1 = +1.17$ & $l_4 = -1.01$ & $l_7 = -1.14$ & $n_1 = +0.95$ & $n_4 = -0.94$ \\
$l_2 = -1.27$ & $l_5 = -1.97$ & $l_8 = -0.52$ & $n_2 = -0.80$ & $n_5 = -1.96$ \\
$l_3 = +0.83$ & $l_6 = +2.05$ & $l_9 = +0.83$ & $n_3 = -0.68$ & $n_6 = -1.04$
\end{tabular}
\caption{One fit lepton parameters for the $Z_{10}$ model.}
\label{fit}
\end{table}

\begin{table}
\begin{tabular}{lll}
Observable & Expt. value & Fit value \\ \hline
$m_e$ & $( 5.11 \pm 1\% ) \times 10^{-4}$ & $5.11 \times 10^{-4}$ \\
$m_\mu$ & $0.106 \pm 1\%$ & 0.106 \\
$m_\tau$ & $1.78 \pm 1\%$ & 1.78 \\
$\Delta m_{23}^2 / \Delta m_{12}^2$ & 8 -- 167 & 34 \\
$\ln \left( \Delta m_{23}^2 / \Delta m_{12}^2 \right)$ & $1.56 \pm
0.33$ & 1.53 \\
$\tan^2 \theta_{12}$ & $0.55 \pm  0.18$ & $0.61$ \\
$\sin^2 2\theta_{23}$ & $> 0.88$  & 0.93 \\
$\sin^2 2\theta_{13}$ & --- & $0.17$ \\
$m_{\nu , {\rm tot}}$ & --- & $2.1$~eV
\end{tabular}
\caption{Experimental values versus fit central values for observables
using the inputs of Table~\ref{fit} in the lepton sector.  
Masses are in GeV (except for $m_{\nu ,
  {\rm tot}}$) and all other
quantities are dimensionless. Error bars indicate the larger of 
experimental or 1\% theoretical uncertainties, as described in
the text.}
\label{data}
\end{table}
\end{document}




