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\title{ A grand unified technicolor theory and the fermion masses of the first and third generation }

\author{A. Doff and A. A. Natale\\ Instituto de F\'{\i}sica Te\'orica, UNESP,
Rua Pamplona 145,
01405-900, S\~ao Paulo, SP,
Brazil\\ E-mail: \email{doff@ift.unesp.br, natale@ift.unesp.br}
}


\abstract{We build a grand unified theory based on the $SU(9)$ group. The theory contains the standard
model and a $SU(4)_{TC}$ technicolor group. The replication of families is introduced through a
horizontal (or family) $SU(3)_H$ symmetry. Assuming a Fritzsh texture for the fermion mass
matrix and with only one hypothesis about the dynamics of the strongly interacting sector
we obtain a fermionic mass spectrum with a reasonable value for the top quark mass as well
as for the fermion masses of the first generation. The $b$ quark and $\tau$ lepton
masses are overestimated and, as a consequence, the $s$ quark and $\mu$ lepton masses
are roughly a factor four larger than the experimental values. Within this model the
origin of the mass scales of the first and third fermionic generation is naturally explained.
The model has no flavor changing neutral currents at unwanted levels. We discuss the masses
of pseudo-goldstone bosons and possible modifications in the model that could promote a
larger splitting between the masses of the third fermionic generation, bringing also the
masses of the second generation to a reasonable value.
}


\keywords{Technicolor and Composite Models, GUT}


\begin{document}

\section{Introduction}


\par The standard model is in excellent agreement with the experimental data, the only still obscure part of
the model is the one responsible for the mass generation, i.e. the Higgs mechanism. In order to make the
mass generation mechanism more natural there are several alternatives, where the most popular ones are supersymmetry
and technicolor. In the first one the mass generation occurs through the existence of non-trivial
vacuum expectation values of fundamental scalar bosons while in the second case the bosons responsible for
the breaking of gauge and chiral symmetry are composite. In both cases it is quite reasonable to expect that these
theories belong to a larger symmetry of a grand unified theory (gut).

In any extension of the standard model it is also very probable
the presence of a horizontal (or family) symmetry to provide the replication of the different fermion
generations. Actually the existence of a family symmetry and grand unification are considered very plausible possibilities
in the current scenario of particle physics. Within this point of view a good hint to start building one
extension of the standard model is provided by the fermionic mass spectrum, which may give us some idea about
the possible horizontal and grand unified symmetries.

A simple and interesting way to describe the fermionic mass spectrum is to suppose that the mechanism behind
mass generation is able to produce a non-diagonal mass matrix with the Fritzsch texture \cite{fritzsch}
\br
 M_f =\left(\begin{array}{ccc} 0 & A & 0\\ A^* & 0 & B \\
0 & B^* & C
\end{array}\right).
\label{e1} \er
This matrix is similar for the charged leptons, $1/3$ and $2/3$ charged quarks. The entry $C$ is proportional
to the mass of the third generation fermion, while the entry $A$ is proportional to the mass
of the lighter first generation. The diagonalization of such mass matrix will determine the
CKM mixing angles and the resulting diagonal mass matrix should reproduce the observed current
fermion masses. There are other possible patterns for the mass matrix and we choose the one of
Eq.(\ref{e1}) just for simplicity.  We call attention to the values of $A$ and $C$. They must be of
order of a few MeV and a hundred GeV respectively. In models with a fundamental Higgs boson
the values of $A$ and $C$ are obtained due to adjusted vacuum expectation values (vev) or Yukawa
couplings. We will discuss the generation of the matrix (\ref{e1}) in the case of dynamical symmetry breaking
where the scalar bosons responsible for the mass generation are composite, as appears in the so called
technicolor (tc) models \cite{weisus,hillsi}, and where we do not have the freedom  to adjust the vevs.

In tc models the fermion masses are functions of the
technifermion self energies. As we pointed out recently \cite{aa}, the most general expression for this
self energy is given by
\be
\S(-p^{2})_{tc} = \mu\(\frac{\mu^2}{-p^2}\)^{\a}[1 + bg^2(\mu^2)ln(-p^2/\mu^2)]^{-\beta{cos(\a\pi)}},
\label{sa}
\ee
where $\a$ and $\beta$ are terms that depend on the choice of the technicolor group and $\m = \m_{tc}$
is the scale of the dynamical technicolor mass. It is known (see, for instance, Ref. \cite{hillsi}) that
the ordinary fermion masses will be roughly of the order $M_f \propto \m_{tc} (\m^2_{tc}/M^2_{etc})^\a$, where
$M_{etc}$ is the mass of a extended technicolor boson \cite{dim}. The different
scales in (\ref{e1}) will appear due to different values of $M_{etc}$. For the heavier fermions (when
$\a \approx 1$) $M_{etc}$ is small and we have the problem of flavor changing neutral currents (fcnc) \cite{dime},
which has been a longstanding problem to the construction of realistic models \cite{hillsi}.
Within the approach described above we see that to explain the fermion mass matrix we need an
extra symmetry (the etc group) and new mass scales (the different etc boson masses). We are basically
transferring one problem, the hierarchy of fermion masses, to another, the hierarchy of etc boson masses.
There is no natural explanation (at least in ordinary models) for the hierarchy of fermion masses and it is necessary to
increase the number of mass scales beyond the ones already existent.

If we assume that the standard model is embedded in an unified theory
containing a technicolor group we know that apart from the grand unification scale we have only
two other scales in the theory: $\m_{tc}$ which of the order of TeV and $\m_{qcd}$ of the order
of $250$MeV (the quantum chromodynamics - qcd - dynamical mass). It is interesting to notice that if
we multiply these two scales by a number O($10^-2$) we obtain two new scales of the
order of few MeV and a dozen of GeV, exactly what is expected for $A$ and $C$ in (\ref{e1}). We will
argue that this is the origin for the mass scales of the first and third generation and present a
model where a realistic fermion mass matrix can be obtained.
Our model contains a horizontal symmetry which will make the third generation fermion to couple to technicolor
and the first generation to couple to quantum chromodynamics (qcd). We only introduce one hypothesis
about the strongly interacting sector: the high energy behavior of the chiral symmetry breaking of this sector is
dominated by higher order interactions (like four fermion interactions) that are relevant at or above
the tc or qcd scale of chiral symmetry breaking. This is the point of view described in Ref. \cite{carpenter}
and \cite{soni}, and it is also the one of Ref. \cite{natale} where the factor $\a$ was taken equal to zero
in a basically {\sl ad hoc} way. The reason for this hypothesis is going to be discussed in one appendix
and in the remaining  of this work it will be assumed just as one ansatz, in order to show that a realistic model of
fermion mass generation can be built along this line, leading to a quite natural
explanation of the mass origin for the different fermion generations,  without flavor changing neutral current
problems  or unwanted light pseudo-goldstone bosons.

We consider a grand unified theory based on the gauge group $SU(9)$, we embed in this one the
minimal Georgi-Glashow $SU(5)_{gg}$ group and a technicolor $SU(4)_{tc}$ group stronger than qcd. The
replication of families is obtained through the introduction of a $SU(3)_H$ local horizontal symmetry.
This symmetry is extremely relevant to generate the desired fermion mass matrix. The chiral symmetry of
technicolor and qcd will be broken spontaneously and the respective condensates will transform
under the horizontal symmetry as the {\bf{3}} and $\bar{\bf{6}}$ representations.
This system is equivalent to the existence of two composite Higgs bosons with vacuum expectation
values of O(250)GeV and O(250)MeV. The choice of representations will restrict the
fermions of the first generation to couple only to the composite state formed at the qcd scale, while
the third generation family will couple to the composite boson formed at the technicolor scale.
The mass of the intermediate generation appears in the vacuum alignment of the two strongly
interacting theories under $SU(3)_H$.

The $SU(9) \otimes SU(3)_H$ symmetry breaking will be assumed to happen at the gut scale,
with their gauge bosons acquiring very heavy masses,
therefore we will not have problems of flavor changing neutral currents. These
bosons will play the same role of the extended technicolor theory, where the difference here is that
we assume $\a = 0$ in Eq.(\ref{sa}) and their effects appear only logarithmically in the expressions
of the fermion masses. These masses will be roughly of order $\a_g \times \m$ where $\a_g $ is the gut
coupling constant and $\m$ can be the tc or qcd scale, leading to the desired $A$ and $C$ values discussed before.

The distribution of our work is the following: In Section II we present the anomaly free representations \cite{ref1}
of our model. We show the couplings of the model that will lead to the main diagrams contributing to the
fermion masses of the first and third generation. Section III contains the calculation of the fermion mass, showing
that this one is going to be proportional to  $\a_g \times \m$. In Section IV we compute the mass matrix for
the first and third generation and discuss the origin of the intermediate masses. In Section V we compute the
terms responsible by the mass of the intermediate generation. In Section VI we show the results of the
diagonalized mass matrix as a function of the scales $\m_{tc}$ and $\m_{qcd}$ and the coupling $\a_g$.
Section VII is dedicated to a computation of the pseudo Goldstone boson masses showing that they are
heavy enough to have escaped detection up to now. Section VIII contains our conclusions.  In the appendix
we justify our ansatz for the self-energy.

\section{The model}

\par
We consider a grand unified theory based on the $SU(9)$ gauge group with the following anomaly free
fermionic representations \cite{ref1}

\be 5\otimes[9,8] \oplus 1\otimes [9,2] \ee

\noindent
where the $[\underline{8}]$ and $[\underline{2}]$ are antisymmetric under $SU(9)$. Therefore the fermionic content of these
representation can be decomposed according to the group product $SU(4)_{{\tiny{tc}}}\otimes SU(5)_{{\tiny{gg}}}$ as:
\br
&&\!\!\!\!\!\!\!\!\!\!\!\![\bf{9},\bf{2}]\nonumber\\ \nonumber
&&(1,10) = \left(\begin{array}{ccccc} 0 & \bar{u_{i}}_{B} & - \bar{u_{i}}_{Y} & -{u_{i}}_{R} & -{d_i}_{R}  \\
-\bar{u_i}_{B} & 0 & \bar{u_i}_{R} & -{u_i}_{Y} &  -{d_{i}}_{Y} \\ \bar{u_{i}}_{Y} & -\bar{u_{i}}_{R} & 0
& -{u_i}_{B} &  -{d_{i}}_{B} \\ {u_i}_{R} & {u_i}_{Y} & {u_i}_{B} & 0 & \bar{e_i}\\
{d_i}_{R} & {d_i}_{Y} & {d_i}_{B} & -\bar{e_{i}}  & 0\end{array}\right)\,\,\,,\,\,\,(4,5) = \,\,\,\left(\begin{array}{c} {Q_i}_{R} \\ {Q_i}_{Y} \\ {Q_i}_{B} \
\\ \bar{L_i}\\ \bar{N_i}
\end{array}\right)_{TC}\!\!,\,\,\,\,(\bar{6},1)= N_{i}\nonumber \\
&&\!\!\!\!\!\!\!\!\!\!\!\![\bf{9},\bf{8}] \nonumber\\
\nonumber\\
&&(1,\bar{5}) =\,\,\, \left(\begin{array}{c} \bar{d_i}_{R} \\ \bar{d_i}_{Y} \\ \bar{d_i}_{B} \\
e_i \\ \nu_{e_i}
\end{array}\right)
\,\,\,\,,\,\,\,\,(1,\bar{5}) = \left(\begin{array}{c} \bar{X}_{R_{k}} \\ \bar{X}_{Y_{k}} \\ \bar{X}_{B_{k}} \\
E_{k} \\ N_{E_{k}}
\end{array}\right)_i\,\,\,,\,\,\,(\bar{4},1)= \,\,\,\,\,\bar{Q_i}_{\varepsilon}, L_i ,{N_i}_{L} , \nonumber \er

\noindent where  $\varepsilon = 1..3$ is a color index  and
$k=1..4$ indicates the generation number of exotic fermions that must be introduced in order to render
the model anomaly free. These fermions will acquire masses of the order of the grand unified scale.
We are also indicating a generation (or horizontal) index $i=1..3$, that will appear due to the necessary
replication of families associated to the  $SU(3)_H$ group.

New interactions occurs among fermions within the same gauge group representation, which, on the basis
of the multiplet structure shown above, lead to the new couplings shown in Fig.(\ref{fig1}), where it is also
shown the couplings generated by the horizontal symmetry.

The couplings shown in Fig.(\ref{fig1}) are some of the ones connecting fermions and technifermions.
We have to remember that the bosons of the grand unified theory eventually carry also quantum
numbers of technicolor and qcd. It is important to note that many of these couplings are not important
in usual technicolor models, because in these models the ordinary fermion masses that are generated through
the exchange of such bosons are given by $M_f \propto \m_{tc}^3 /M^2_X$, where, if $M_X$ is the mass
of the heavy $SU(9)$ or $SU(3)_H$ bosons that we are considering, their contributions would be
totally negligible. As we shall see in the next section the dependence on $M_X$ in our model
will be logarithmic and all the couplings of Fig.(\ref{fig1}) will appear in the diagrams  responsible for
the fermionic masses.
\section{The dynamical mass}
\par The exact expression for the fermionic self energy used to compute the dynamical masses
in technicolor models is the most relevant issue in the model building. It is known that
the self energy given by Eq.(\ref{sa}),  with the parameter $\a = 1$, usually adopted in the
early models lead to flavor changing neutral currents at a level incompatible with experimental
data \cite{dim}. Here we will use Eq.(\ref{sa}) with $\a = 0$ and $\beta = \g$ where
$\g = 3c/16\pi^{2}b$ and $c$ is given by

\be
 c = \frac{1}{2}[C_{2}(R_{1}) +
C_{2}(R_{2})- C_{2}(R_{3}) ].
\label{ce3}
\ee
The coefficient $c g^2$ measures the strength of the interaction between
fermions in the representations $R_{1}$ and $R_{2}$ condensating in the representation
$R_{3}$ and $C_2$ are their respective Casimir operators. Therefore, the self energy used
in our model is giving by

\be \hspace{0.5cm}\S_\chi (-p^{2}) = \mu_\chi [1 + bg_\chi^2(\mu_\chi^2)ln(-p^2/\mu_\chi^2)]^{-\g_\chi} ,
\label{sr} \ee.

There are pros and cons about this expression and it can be used in the sense prescribed in
Ref. \cite{carpenter,soni}. In the appendix we will discuss these points, meanwhile we use it as
one ansatz with the intent to show that it leads to a totally viable model.

\par
\begin{figure}[ht]
\begin{center}
\epsfig{file=acoplamguttcJHEP.eps}
%\includegraphics{acoplamguttc.eps}
\vspace{0.5cm}
\caption{Couplings of ordinary fermions and technifermions to the gauge bosons
of $SU(9)$, $SU(5)_{gg}$ and $SU(3)_{H}$ which are relevant for the generation of fermion masses.}
\label{fig1}
\end{center}
\end{figure}

 \par Using Eq.(\ref{sr}) we can now compute the general diagram responsible for mass feed down shown
 in Fig.(\ref{diagmassa}).
\par
\begin{figure}[htb]
\begin{center}
\epsfig{file=diagmassa.eps}
%\includegraphics{diagmassa.eps}
\vspace{0.5cm}
\caption{Typical diagram contributing to the fermion masses of the first  and third generation. $SU(k)$
indicates the exchange of a $SU(9)$ or $SU(3)_H$ boson, with  coupling $\alpha_k$ to fermions
($f$) or technifermions ($T$). }
\label{diagmassa}
\end{center}
\end{figure}

Assuming the  complete fermion propagator to be given by
$ S^{-1}_{F}= A(p^2){\not{\!\!p}} -  B(p^2)$,  working in the Landau gauge, with $A(p^2)=1$,
$B(p^2)=\S$, performing the angular integration and some simplifications we obtain (in Euclidean
space)

\be \overline{m}_{f} \simeq  \,\frac{3C_{2k}\mu_{\chi}}{16\pi^2}\int\,dq^2q^2\frac{g^{2}_{k}[1 +
b_{\chi}g^2_{\chi}ln(\frac{q^2}{\mu_{\chi}})]^{-\gamma_{\chi}}}{(q^2 + M^{2}_{k})^2},\ee

\noindent where $\overline{m}_{f}$ denotes the nondiagonal fermion mass, the $\chi$ index refers to a technicolor
or qcd self energy and $M_{k}$ is a gut  (or horizontal)
boson mass. The gut ($g_k$) coupling constant is given by
\be
g^2_{k}(p^2) \simeq \frac{g_k^2(M^2_{k})}{(1 + bg^2_{k}(M^2_{k})ln(\frac{p^2}{M^2_{k}}))}.
\label{ggut}
\ee
Putting all this together the expression for the fermion mass is

\be \overline{m}_{f} \simeq \,\,\frac{3C_{2k}g^{2}_{k}\mu_\chi}{16\pi^2}\int_{0}^{\infty}\!\!\!\!\frac{dxx}{(x +
M^{2}_{k})^{2}}\left(1 + bg^2_{k}ln(\frac{x}{M^2_{k}})\right)^{\!\!-1}
 \!\!\left(1 +
bg^{2}_{\chi}ln\frac{x}{\m^2_\chi}\right)^{\!\!-\gamma_\chi}.
\label{mf1}
\ee

\noindent

With the substitution $x\rightarrow \frac{M^{2}_{k}y}{\m^2_\chi }$ and using

\be
  g^2_{\chi}(M^2_{k})=\frac{g^2_{\chi}(\m^2_\chi)}
 {(1 + bg^2_{\chi}(\m^2_\chi)ln(\frac{M^2_{k}}{\m^2_\chi}))},
\ee




\noindent  Eq.(\ref{mf1}) can  be cast in the following form

\be \overline{m}_{f} \simeq  \,\frac{3C_{2k}g^{2}_{k}\mu_\chi}{16\pi^2}\left[1 + bg^2_{\chi}ln\frac{M^2_{k}}{\m^2_\chi}\right]^{-\gamma_\chi}
\int_{0}^{\infty}\!\!\!\!\!\!\frac{dyy}{(y + \m^2_\chi)^{2}}\left(1 +
bg^{2}_{k}ln\frac{y}{\m^2_\chi}\right)^{\!\!-\gamma_\chi-1}\!\!\!\!\!.
\label{mf2}
\ee

 \noindent To perform the integral in Eq.(\ref{mf2}) we can use the following Mellin transform

 \be
\left[ 1 + \kappa \ln \frac{x}{\mu^2} \right]^{-\epsilon} \!\!\!\!= \frac{1}{\Gamma(\epsilon)}\int_0^\infty
d\sigma \, e^{-\sigma} \left( \frac{x}{\mu^2} \right)^{-\sigma \kappa}\!\!\!\!\!\! \sigma^{\epsilon - 1},
\label{mt} \ee

\noindent in order to obtain

\be \overline{m}_{f} \simeq\ \,\frac{3C_{2k}g^{2}_{k}\mu_\chi}{16\pi^2}\left[1 + bg^2_{\chi}ln\frac{M^2_{k}}{\m^2_\chi}\right]^{-\gamma_\chi}\int_{0}^{\infty}\!\!\!d{\sigma}\sigma^{\epsilon
-1}e^{-\sigma}\left(\frac{1}{\m^2_\chi}\right)^{-\kappa\sigma}\int_{0}^{\infty}
 \frac{dyy^{1-\sigma\kappa}}{{(y +
\m^{2}_\chi)^{2}}},
 \ee

\noindent where we defined $\epsilon\equiv \gamma_\chi  + 1$, $\kappa = bg^2_{k}$. After integration
in $y$ and using the inverse transformation  of Eq.(\ref{mt}) we obtain

\br \overline{m}_{f}\simeq \,\, &&\frac{C_{2k}\alpha_{k}\mu_k}
{\alpha_{\chi}c_\chi}\left[1 + bg^2_{\chi}ln\frac{M^2_{k}}{\m^2_\chi}\right]^{-\gamma_\chi + 1}\!\!\!\!\!\!\!\!\!\!.
 \label{massge}
 \er

\noindent where  in  the last equation $\alpha_{\chi,k}=g_{\chi,k}^2/4\pi$.

 Eq.(\ref{massge}) is a quite general expression for the fermion mass that we will use in the sequence
to calculate specific diagrams. We will assume  $\alpha_{k} \sim \frac{1}{45}$ at the gut scale, which is of
the expected order of the coupling constant, as long as we do not have too much interactions
between the tc and the gut scales. The
index $k$ may refer to the exchange of $SU(5)_{gg}$  ($k=5$) or $SU(9)$ ($k=9$) bosons,
and computing the quadratic Casimir operators for $SU(5)_{gg}$ and $SU(9)$ gauge groups, we have two
classes of diagrams or expressions for the fermion mass (depending on the internal fermion
if it is a quark or a technifermion):

\par \textit{For fermions acquiring masses due to the technicolor condensate}

\be
\overline{m}_f \simeq \frac{\mu_{tc}}{10 c\alpha_{tc}}\left[ 1 + bg^2_{tc}ln\frac{M^2_{9}}{\m^2_{tc}}\right]^{-\gamma_{tc} + 1}
\label{mftc}
\ee

\par \textit{ For fermions acquiring masses due to the qcd condensate}

\be
\overline{m}_{f} \simeq  \frac{\mu_{qcd}}{20 c\alpha_{qcd}}\left[ 1 + bg^2_{qcd}ln\frac{M^2_{5}}{\m^2_{qcd}}\right]^{-\gamma_{qcd} + 1}
\label{mfqcd}
\ee

Note that in both expressions the values at the right hand side of $\frac{\m_\chi}{c\alpha_{\chi}}$ will depend on
the parameters $b$, $c$ and $\gamma$. We will also assume in  the next sections,  where we compute the mass matrix,
that  $M_{k} \sim 10^{16}GeV$ is a quite reasonable value for the gut scale. The higher
is the gut scale the smaller are the terms on the right hand side of these two last equations.
The effect of this mass scale appears only logarithmically and we expect that the
variation in the coefficients of Eqs.(\ref{mftc}) and (\ref{mfqcd}) with $M_k$ is quite small.

 We assume that the gauge symmetry breaking of $SU(9) \otimes SU(3)_H$ occurs near the Planck scale, and we
cannot discard that it is promoted by the vacuum expectation values of fundamental scalars or other
mechanisms. Note also that if we choose other gut or horizontal symmetry we can also have a different number
of diagrams contributing to the fermion masses. Therefore, the models that we can build along this line may
vary due to differences in the coefficients of Eqs.(\ref{mftc}) and (\ref{mfqcd}), in the choice
of the horizontal symmetry and its fermionic representations (because this will lead to different
textures of the mass matrix), and in the number of diagrams contributing to each entry of the mass
matrix what is connected to their group structure. However the expressions of Eqs.(\ref{mftc}) and
(\ref{mfqcd}) come out solely from the dynamics and are not going to be modified. As one example,
the diagrams contributing to the charge $2/3$ quark masses are shown in Fig.(\ref{f11}),
where the diagrams in $(a)$, $(b)$ and $(c)$ contribute respectively to the terms $A$, $B$ and
$C$ of Eq.(\ref{e1}).

\begin{figure}[ht]
\begin{center}
\epsfig{file=listfig.eps}
%\includegraphics{listfig.eps}
\vspace{0.5cm}
\caption{Diagrams contributing to the charge {2}/{3} quark masses}
\label{f11}
\end{center}
\end{figure}


\section{The mass matrix}

\par In the previous section we have seen how to generate the masses of lighter and heavier fermions. These
masses are consequence of the qcd and technicolor condensation or the effective theories generated by these
fundamental interactions. From Eqs.(\ref{mftc}) and (\ref{mfqcd}) we see that we can generate
masses of the order $\alpha \mu_{tc}$ and $\alpha\mu_{qcd}$, where $\alpha$ is  $O(1/10)$ and the masses
come out of the order of the first and third generation.  Only these two mass scales appear and it remains
to determine the masses of the second generation. Before we compute the intermediate fermion masses, it is
convenient to discuss in more detail how a mass matrix similar to Eq.(\ref{e1}) can be formed, i.e.  how
another mass scale, the one of the second family, can be generated. We will make use of effective lagrangians
to show that an intermediate mass scale is quite natural in such system of composite fields. The calculation of the full mass matrix will be left for the next section.

The mass matrix of Eq.(\ref{e1}) will be formed according to the representations of the strongly interacting
fermions of the theory under the $SU(3)_H$ group. The technifermions form a quartet under  $SU(4)_{tc}$
and the quarks are triplets of qcd. The technicolor and color condensates will be formed at the scales
$\mu_{tc}$ and $\mu_{qcd}$ in the most attractive
channel (mac) \cite{suscor} of the products  ${\bf \bar{4}\otimes 4}$ and ${\bf \bar{3}\otimes 3}$ of each strongly interacting theory.
We assign the horizontal quantum numbers to technifermions and quarks such that these same products
can be decomposed in the following representations of  $SU(3)_H$:  ${\bf \overline{6}}$ in the case of the technicolor
condensate, and  ${\bf 3}$ in the case of the qcd condensate. For this it is enough that the standard
left-handed (right-handed) fermions transform as triplets (antitriplets) under $SU(3)_H$, assuming that the tc and qcd
condensates are formed in the ${\bf \overline{6}}$ and in the ${\bf 3}$ of  the $SU(3)_H$ group. This is consistent
with the mac hypothesis \cite{suscor} although a complete analysis of this problem is out
of the scope of this work.

The above choice for the condensation channels is crucial for our model, because the condensate in
the representation  ${\bf \overline{6}}$  (of $SU(3)_H$) will interact only with the third fermionic generation while the  ${\bf 3}$
only with the first generation. In this way we can generate the coefficients $C$ and $A$ respectively of Eq.(\ref{e1}),
because when we add these condensates (vevs) and write them as a $3 \times 3$ matrix  we will end up with
 \br
 M_f =\left(\begin{array}{ccc} 0 & A & 0\\ A^* & 0 & 0 \\
0 & 0 & C
\end{array}\right).
\label{e12} \er
This problem is very similar to the one proposed by Berezhiani and Gelmini {\it et al.} \cite{ref3} where the condensates were given by fundamental scalars.

We can now go back to our problem that was the mass generation of
the intermediate fermionic family. It is clear that the problem of the symmetry breaking
of two strongly interacting theories and their alignment with the  $SU(3)_H$ interaction (and the others
interactions of the standard model) is not an easy one. However, we will make one discussion in terms of effective
lagrangians and show that an intermediate vev (or mass term) will appear in the alignment of these
theories. After this we present a dynamical determination of this value.

We expect that the composite Higgs system can naturally be described by the following
potential

\be
 V(\eta,\varphi) = \m^2_{\eta}\eta^{\dagger}\eta + \lambda_{\eta}(\eta^{\dagger}\eta)^2 +
\m^2_{\varphi}\varphi^{\dagger}\varphi + \lambda_{\varphi}(\varphi^{\dagger}\varphi)^2,
\label{vh}
\ee
in such a way that we can identify the vevs to the ratio of masses and couplings

\be
v^{2}_{\eta}=-\frac{\m^2_{\eta}}{\lambda_{\eta}}\,\,\,,\,\,\,v^{2}_{\varphi}
=-\frac{\m^2_{\varphi}}{\lambda_{\varphi}} ,
\label{mc}
\ee
where the bosons  represented by $\eta$ and $\varphi$, respectively, are related to the system of
composite  Higgs bosons formed in the representations ${\bf 3}$ and $\overline{\bf {6}}$ of the horizontal group.
Such supposition is quite plausible if we consider the results of Ref.\cite{carpenter,soni}, where it was
shown that the interactions of a composite Higgs boson is very similar to the ones of a fundamental boson.
Our intention is to show that such system leads to an intermediate mass scale and to a mass matrix identical
to Eq.(\ref{e1}).

The bosons $\eta$ and $\varphi$ in the representations ${\bf 3}$ and $\overline{\bf {6}}$ of $SU(3)_H$
will have the following Yukawa couplings \cite{fritzsch,ref3}


\be {\cal{L}}_{Y} = a\bar{\Psi}^{i}_{L\theta}\eta^{k}_{\theta}U^{j}_{R}\epsilon_{ijk} +
b\bar{\Psi}^{i}_{L\theta}\varphi^{ij}U^{j}_{R} ,
\label{y1} \ee
where $\theta$ is a weak hypercharge ($SU(2)_{w}$) index,
for instance, $\theta = 1$ represents charge $2/3$ quarks and $\theta  = 2$ correspond to the charge $1/3$
quarks, $i,j$ e $k$ indicate the components of the composite scalar bosons of
the representations ${\bf 3}$ and $\overline{\bf {6}}$ of $SU(3)_{H}$
and $a$ and $b$ are the coupling constants. Ultimately
the terms of Eq.(\ref{y1}) are going to be related to the fermionic masses.

Let us, for simplicity, consider only the charge $2/3$ quarks, i.e. $\theta =1$, and write the Yukawa lagrangian
in terms of its components

\br
 {\cal{L}}_{Y} = \,\,&&a\bar{\Psi}^{1}_{L1}\eta^{2}_{1}U^{3}_{R} - a\bar{\Psi}^{2}_{L1}\eta^{1}_{1}U^{2}_{R} +
 a\bar{\Psi}^{2}_{L1}\eta^{3}_{1}U^{1}_{R} - a\bar{\Psi}^{3}_{L1}\eta^{2}_{1}U^{1}_{R} + a\bar{\Psi}^{3}_{L1}\eta^{1}_{1}U^{2}_{R} -
 a\bar{\Psi}^{1}_{L1}\eta^{3}_{1}U^{2}_{R}\nonumber\\
 &&\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+\, b\bar{\Psi}^{1}_{L1}\varphi^{11}U^{1}_{R} + b\bar{\Psi}^{1}_{L1}\varphi^{12}U^{2}_{R} +
 b\bar{\Psi}^{1}_{L1}\varphi^{13}U^{3}_{R} + b\bar{\Psi}^{2}_{L1}\varphi^{21}U^{1}_{R} + b\bar{\Psi}^{2}_{L1}\varphi^{22}U^{2}_{R} +
 b\bar{\Psi}^{2}_{L1}\varphi^{23}U^{3}_{R}\nonumber\\
 &&\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+\, b\bar{\Psi}^{3}_{L1}\varphi^{31}U^{1}_{R} + b\bar{\Psi}^{3}_{L1}\varphi^{32}U^{2}_{R} +
 b\bar{\Psi}^{3}_{L1}\varphi^{33}U^{3}_{R},
\er

\noindent where we can identify $U^{1}_{R}=u_{R}$, $U^{2}_{R}=c_{R}$
and $U^{3}_{R}=t_{R}$, and where we also have
$\Psi^{1}_{L}= u_{L}$, $\Psi^{2}_{L}= c_{L}$ and $\Psi^{3}_{L}= t_{L}$, $\eta^{i}$ and $\varphi^{ij}$ are
the components of the composite Higgs multiplets corresponding to ${\bf{3}}$ and $\overline{\bf{6}}$
respectively.

\par The vevs of qcd and technicolor, due to the horizontal symmetry, can be written
respectively in the following form \cite{ref3}

\be
 <\eta> \sim \left(\begin{array}{c} 0 \\ 0 \\  v_{\eta}
\end{array}\right)\,\, \,,\,\,\, <\varphi> \sim \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0
& v_{\varphi} \end{array}\right).
\label{veta}
\ee


\noindent Therefore, substituting these vevs in the Yukawa lagrangian, we obtain

\be {\cal{L}}_{Y} = a\bar{c}_{L}v_{\eta}u_{R} - a\bar{u}_{L}v_\eta{c}_{R} + b\bar{t}_{L}v_{\varphi}t_{R} ,\ee
and the mass matrix in the $(u\,,c\,,t)$ base is given by

\be \overline{m}^{\frac{2}{3}} = \left(\begin{array}{ccc} 0 & -av_{\eta} & 0\\ av_{\eta} & 0 & 0 \\ 0 & 0 & bv_{\varphi}
\end{array}\right). \ee

The main point of the model is that the fermions of the third generation obtain large masses because they couple
directly to technifermions, while the ones of the first generation obtain masses originated by the ordinary
condensation of QCD quarks.

\par It is clear that the $SU(3)_H$ symmetry is broken and there is no way to prevent the coupling of
such composite bosons at higher order. Examples of such couplings are shown in Fig.(\ref{f3})

\begin{figure}[ht]
\begin{center}
\epsfig{file=figura2artigo2.eps,width=0.7\textwidth}
%\includegraphics{figura2artigo2.eps}
\caption{Higher order corrections coupling the $\eta$ and $\varphi$ composite bosons.}
\label{f3}
\end{center}
\end{figure}

\par The diagrams of Fig.(\ref{f3}) will produce new terms for the effective potential of our composite
 system, therefore we must add to Eq.(\ref{vh}) the following terms

\be
V_{2}(\eta,\varphi) =  \Pi\eta^{\dagger}\eta\varphi^{\dagger}\varphi +
\delta\eta^{\dagger}\varphi\eta\varphi^{\dagger} + ...
\label{vh2}
\ee
The introduction of this expression in the potential of Eq.(\ref{vh}) will shift the vevs generated
by the effective fields $\eta$ and  $\varphi$, and the vev associated to the field $\eta$ will be shifted to

\be
<\eta> \sim \left(\begin{array}{c} \varepsilon \\ 0 \\  v_{\eta}
\end{array}\right) .
\label{veshi}
\ee

\noindent We do not include the shift in the vev of $\varphi$, because $v_{\eta}\ll v_{\varphi}$
and the modification is negligible. With this shift the Yukawa lagrangian in terms of the new vevs can
be written as

\be {\cal{L}}_{Y} = \,\, a\bar{c}_{L}v_{\eta}u_{R} - a\bar{u}_{L}v_\eta{c}_{R} + b\bar{t}_{L}v_{\varphi}t_{R} - a\bar{c}_{L}\varepsilon t_{R} +
a\bar{t}_{L}\varepsilon c_{R}.
 \ee
Therefore, in the base $(u\,,c\,,t)$, the structure of the mass matrix now is

\be \overline{m}^{\frac{2}{3}} = \left(\begin{array}{ccc} 0 & -av_{\eta} & 0\\ av_{\eta} & 0 & a\varepsilon \\ 0 &
-a\varepsilon & bv_{\varphi}
\end{array}\right) . \ee

We have shown that in this model we can easily generate the mass of the first and third
generation. Through the use of effective theories we have also demonstrated how the mass of the
second generation will also appear. It results from the vacuum alignment of the two strongly interacting
systems (qcd and technicolor) contained in $SU(9)$,  which belong to different representations of
the horizontal symmetry but interact among themselves as shown in Fig.(\ref{f3}).
Of course, we must now compute the precise value of the intermediate mass
(i.e. the $B$ entry of Eq.(\ref{e1})), what is going to be done in the following section.

\section{The mass of the intermediate generation}

\par As we have seen in the previous section, fermions of the first generation couple to the qcd composite bosons, while the
fermions of the third generation couple to the composite bosons generated by technicolor. The fermions
of the second generation do not couple directly to technifermions, but they couple only at higher
order due to diagrams like the one depicted in Fig.(\ref{f4}).



\begin{figure}[ht]
\begin{center}
\epsfig{file=figura1artigo2.eps}
%\includegraphics{figura1artigo2.eps}
\caption{Diagram responsible for the c quark mass generation}
\label{f4}
\end{center}
\end{figure}

The $c$ quark can also obtain mass coupling to the ordinary quarks, but this contribution is
negligible when compared to the one of Fig.(\ref{f4}). This is one way to compute the intermediate entry ($B$) in Eq.(\ref{e1}).

\par The second procedure is through the determination of the coefficients of the effective potential
discussed in section IV (i.e. the couplings of Eq.(\ref{vh2})). Strictly speaking we have to study
how the different low energy effective theories for qcd and technicolor mix among themselves.
Suppose that the full potential is given by

\be V(\eta,\varphi) = \,\,\m^2_{\eta}\eta^{\dagger}\eta + \lambda_{\eta}(\eta^{\dagger}\eta)^2 +
\m^2_{\varphi}\varphi^{\dagger}\varphi + \lambda_{\varphi}(\varphi^{\dagger}\varphi)^2 + \Pi\eta^{\dagger}\eta\varphi^{\dagger}\varphi + \delta\eta^{\dagger}\varphi\eta\varphi^{\dagger}.
\label{vcalc}
\ee

\noindent The ratios $\m_{\chi}^2/\lambda_{\chi}$ is fixed by the vacuum expectation values. Our intention is
to determine the small coefficients $\Pi$ and $\delta$ that appear in the mixing terms. Knowing these coefficients and minimizing the potential we can
determine the intermediate vev. After this we need only to diagonalize the mass matrix to obtain the mass spectrum of the three generations model.
\par We start considering the second possibility described above, which is the determination of the
coupling between the different scalars. We follow a model developed by Carpenter et al. \cite{carpenter,soni} some
years ago, where it is discussed the coupling of composite scalars to ordinary
fermions and gauge bosons. The vertex that we are going to need is the one shown in Fig.(\ref{f5}),


\begin{figure}[ht]
\begin{center}
\epsfig{file=figura4artigo2.eps}
%\includegraphics{figura4artigo2.eps}
\caption{Vertex coupling a scalar composite boson to ordinary fermions}
\label{f5}
\end{center}
\end{figure}



\noindent which will allow to determine the coupling between two gauge bosons and two scalar composites,
as shown in Fig(\ref{f6}). $\Sigma$ in Fig. (\ref{f5}) is the self energy of Eq.(\ref{sr}). This coupling
results from the use of the Ward identities and was determined in Ref.\cite{soni}.

\begin{figure}[ht]
\begin{center}
\epsfig{file=figura5artigo2.eps}
%\includegraphics{figura5artigo2.eps}
\caption{Diagram leading to the coupling between two composite scalar bosons and two gauge bosons}
\label{f6}
\end{center}
\end{figure}


 \noindent According to the standard Feynman rules and using the coupling of Fig.(\ref{f5})
the effective coupling  of Fig.(\ref{f6}) between the scalars bosons and the charged weak bosons  is given by

$$
i\Pi_{\phi\phi{WW}} =
-\frac{g^4_{W}\delta^{ab}}{4M^2_{W}}\frac{1}{16\pi^4}\int\!d^4q\frac{\Sigma^2_\chi Tr[{\not{\!\!q}}^2\gamma^{\m}\not{\!\!q}\gamma^{\nu}\not{\!\!q}]}{q^8},
 $$
\noindent computing the trace and writing the equation in the Euclidean space, we obtain
\be
\Pi_{\phi\phi{WW}} = -\frac{g^4_{W}\delta^{ab}}{M^2_{W}}\frac{g^{\m\nu}}{32\pi^2}\int\!d^2q\frac{\Sigma^2_\chi}{q^2}.
\label{c22}
\ee

\noindent Following closely the procedure adopted by Carpenter et al. \cite{soni}
we may approximate the self energy by $\Sigma_\chi \sim \m_{\chi} \left(\frac{q^2}{\m_\chi^2}\right)^{-\epsilon}\!\!,$
where $\epsilon=\frac{3C_{2k}g^2_{\chi}}{16\pi^2}$ ,
to obtain the following coupling between two composite scalars and the intermediate gauge bosons of the weak interaction

\be
\Pi_{\phi\phi{WW}} = -\frac{M^2_{W}\delta^{ab}}{2\pi^2}\frac{G^{2}_{F}\m^2_{\chi}}{\epsilon_{\chi}}g^{\m\nu}\!\!.
\label{cf22}
\ee

\noindent Where in this expression the index $\chi$ differentiates technicolor and qcd composites and we made use of the
relation $\frac{G_{F}}{\sqrt{2}}=\frac{g^2_{W}}{8M^{2}_{W}}$. Note that the coupling between scalars
and gauge bosons is dominated by the ultraviolet limit, where the approximation for the self energy
discussed above is also valid. As our interest is in the quartic coupling between qcd and technicolor
composite scalar bosons that are shown in Fig.(\ref{f3}), we can compute the first diagram of this figure,
which should be equivalent to the coupling $\Pi$ in Eq.(\ref{vcalc}). The result is

\be \Pi_{\eta\eta\varphi\varphi} = \frac{M^4_{W}{G^4_{F}}\m^2_{tc}\m^2_{qcd}}{32\pi^8\epsilon_{tc}
\epsilon_{qcd}}.
\label{pia}
\ee

\par Now we can approximately determine the value of $\varepsilon$ assuming that the potential
of Eq.(\ref{vh}) has a minimum described by the vevs $<\varphi>$, Eq.(\ref{veta}), and $<\eta>$, Eq.(\ref{veshi}), what lead us to  the following value of the potential at minimum

\be
V(\eta,\varphi)|_{min}=\m^2_{\eta}v^2_{\eta}+ \lambda_{\eta}v^4_{\eta} + \m^2_{\varphi}v^2_{\varphi}+
\lambda_{\varphi}v^4_{\varphi} + \lambda_{\eta}\varepsilon^4.
\ee
We then compare the minimum of this potential with the one obtained from Eq.(\ref{vcalc}),
where the term proportional to $\delta$ is neglected in comparison to the one proportional
to  $\Pi$. This is equivalent to say that the second diagram of Fig.(\ref{f3}) is much smaller
then the first diagram, and the vevs entering in  Eq.(\ref{vcalc}) are the unperturbed ones
because the perturbation will enter through the  $\Pi$ term. Finally, assuming
that the coefficient describing the coupling between four scalar bosons
that are formed in the chiral symmetry breaking of QCD is given by\cite{soni}

\be
\lambda_{\eta} = \frac{G^2_{F}\m^4_{qcd}c\alpha_{qcd}}{\pi},
\ee
we conclude that

\be \varepsilon \sim  B  \sim \left(\frac{M^4_{W}G^2_{F}\m^4_{tc}}{18\pi^3c\alpha_{tc}}\right)^{\frac{1}{4}}GeV \sim 16.8 GeV.
\label{mc12} \ee

\par Following an alternative procedure we can also compute $\overline{m}_{c}$, or the entry B in the
mass matrix, through the diagram of Fig.(\ref{f4}) and verify that the
result is not essentially different from the one of Eq.(\ref{mc12}). From Fig.(\ref{f4}) with
the standard Feynman rules we can write the following expression

\be
S^{-1}_{c}=\int\frac{d^4q}{(2\pi)^4}\int\frac{d^4k}{(2\pi)^4}F(k,q),
\ee

\noindent where

$$
F(k,q)=\gamma^{\m}T^{a}g_{h}S_{F}(q)\gamma^{\rho}t^{c}g_{gut}S_{F}(k)\gamma^{\lambda}t^{d}g_{gut}D_{\rho\lambda}(q-k)S_{F}(q)\gamma^{\nu}T^{b}g_{h}D_{\m\nu}(p-q).
$$

\par With the fermion propagator written as a function of the scalar self energies and using Landau
gauge the equation for $\overline{m}_{c}$ entails


\be
 \overline{m}_{c} \simeq  \,\,9c_{g}c_{h}\int\frac{d^4q}{(2\pi)^4}\frac{g^2_{h}}{(q^2 +
m^2_{t})^2}\frac{q^2}{(q^2 + M^2_{h})}\int\frac{d^4k}{(2\pi)^4}\frac{\Sigma_{TC}(k^2)}{(k^2 +
m^2_{Q})}\frac{g^2_{gut}}{(k^2 + M^2_{gut})},
\ee

\noindent where $c_{g}$ and $c_{h}$ are Casimir operators related to the gut and horizontal symmetries.
As we know that $m_{t}$ is given by

\be
m_{t} \simeq \frac{3c_{g}}{16\pi^2}\int\,dk^2k^2\frac{g^{2}_{gut}\Sigma_{TC}(k^2)}{(k^2 + M^{2}_{gut})^2},
\label{mst}
\ee

\noindent we can write for the $c$ quark

\be \overline{m}_{c} \simeq \frac{3c_{h}m_{t}}{16\pi^2}\int\,dq^2\frac{g^{2}_{h}}{(q^2 + M^{2}_{h})}.
\label{msc}
\ee

For simplicity we neglected the momentum dependence in Eq.(\ref{mst}), and will also neglect this
dependence in the coupling constant in Eq.(\ref{msc}).  With these approximations Eq.(\ref{msc})
is divergent and it will be regularized with a cutoff (the full calculation is lengthy and the result will
not differ appreciably). Therefore, the integration gives

\be
 \overline{m}_{c} \simeq \frac{3c_{h}\alpha_{h}m_{t}}{4\pi}\left[\ln(M^2_{h} + \Lambda^2) - \ln(M^2_{h})\right],
 \ee

\noindent where $\Lambda$ is the ultraviolet cutoff and can naturally be taken as $\Lambda^2\sim M^2_{gut}$,
leading to

\br B\, = \,\overline{m}_{c} &&\,\,\sim \frac{3c_{h}\alpha_{h}m_{t}}{4\pi}\ln(1 + \frac{M^2_{Gut}}{M^2_{h}}).
\label{mch1} \er

\noindent Note that we have made several simplifications until we came to Eq.(\ref{mch1}),
the final result should also depend on  $m_{t}$ and $ m_{Q}$. To estimate the nondiagonal $c$ quark
mass  ($\overline{m}_{c}$ or $B$)  we will suppose that the horizontal symmetry boson mass scale
is  about $M_{h} \sim 10^{13}$ GeV \cite{ref3}  in order to obtain

\be
 B \, \sim \frac{14\alpha_{h}m_{t}}{\pi} .
\label{mhc}
\ee

The assumption that the mass scale for the horizontal symmetry is of the order of the gut one is
plausible within our model, where all these symmetries will be broken at very high energies. The
dependence on the ratio of these masses in Eq.(\ref{mch1}) is only a logarithmic one.

\par The above result could be expected since in a similar model, but with fundamental scalars,
Barr and Zee \cite{ref5} obtained a quite similar result between the first and second generation
of leptons $m_{e} = \frac{N\alpha_{h}m_{\m}}{\pi}$. In their model the muon would obtain mass
at the 1-loop level coupling to heavier leptons $X$, and $N$ in their model would be a number
of $O(1)$. The difference between our models is that the top quark is the one obtaining mass
at 1-loop level, and the $c$ quark is obtaining mass at a 2-loop level, coupling to the top quark through
the horizontal symmetry. Considering the expected numerical values of all constants we obtain
$\overline{m}_{c}\sim 17 $ GeV, which is of the order of the previous determination.
In this section we obtained the entry of the mass matrix correspondent to the intermediate generation.
We have made several approximations and differences of factors of $O(2)$ or more would not be surprising in our
approach. Actually it would be too ambitious if we wanted to compute the vacuum alignment of
the two strongly interacting sectors (qcd and technicolor) with a better precision. In the next
section we proceed to the diagonalization of the mass matrix.

\section{The diagonalized mass matrix}

\par In this section we will proceed to the mass matrix diagonalization in order to obtain the fermionic
mass spectrum. To write the mass matrix we have to consider that for the heaviest generation, and in
particular for the top quark, there are several diagrams that contribute to the mass. The same happens
for the lighter generation. The top quark receive mass coupling to the techniquarks and technileptons
through all the possible interactions (the ones contained in the $SU(9)$ group as well as the ones due
to the horizontal symmetry). The different charges, couplings as well as condensates will also produce
one splitting in the masses of fermions of the same generation as indicated in Figs.(\ref{f12}) and (\ref{f13}).
For instance, when computing diagrams involving the technileptons and techniquarks condensates
we assume

\be
 <\bar{L} { L>} = \frac{ 1}{ 3}{ <\bar{Q} Q>}  ,
\ee
because the techniquarks carry also the three color degrees of freedom.

%\vspace{1.5cm}
\begin{figure}[ht]
\begin{center}
\epsfig{file=listfig2.eps}
%\includegraphics{listfig2.eps}
\vspace{0.5cm}
\caption{Diagrams contributing to the mass generation of 1/3 charged quarks.}
\label{f12}
\end{center}
\end{figure}

%\vspace{0.5cm}
\begin{figure}[ht]
\begin{center}
\epsfig{file=listfig3.eps}
%\includegraphics{listfig3.eps}
\vspace{0.5cm}
\caption{Diagrams contributing to lepton masses.}
\label{f13}
\end{center}
\end{figure}

\par Let us consider only the $\frac{2}{3}$ charged quarks and verify their different contributions
to the matrix in Eq.(\ref{e1}). These will come from the diagrams labeled (a), (b)  and (c)  in
Fig.(\ref{f11}) and are equal to

\br A\,\,=\,\,&&\frac{1\mu_{qcd}}{10 c\alpha_{qcd}}\left[ 1 + bg^2_{qcd}ln\frac{M^2_{5}}{\m^2_{qcd}}\right]^{-\gamma_{qcd} + 1}\!\!\!\!\!\!\!\!\!\!\!\!+\,\,\, \frac{4\mu_{qcd}}{135 c\alpha_{qcd}}\left[ 1 + bg^2_{qcd}ln\frac{M^2_{h}}{\m^2_{qcd}}\right]^{-\gamma_{qcd} + 1}\nonumber\hspace{-1.2cm},\\
B\,\,=\,\,&& \frac{28\mu_{tc}}{675\pi c\alpha_{tc}}\left[ 1 + bg^2_{tc}ln\frac{M^2_{9}}{\m^2_{tc}}\right]^{-\gamma_{tc} + 1}\nonumber\hspace{-1.0cm},\\
C\,\,=\,\,&&\,\,\,\,\frac{2\mu_{tc}}{15 c\alpha_{tc}}\left[ 1 + bg^2_{tc}ln\frac{M^2_{9}}{\m^2_{tc}}\right]^{-\gamma_{tc} + 1}\hspace{-1cm}.
\label{abc}
\er
Where the contributions for $A$, $B$ and $C$ come respectively from the diagrams (a), (b)  and (c) displayed in
Fig.(\ref{f11})  and  again we assumed $\alpha_{k}\sim \frac{1}{45}$.

As the mass matrix is the same obtained in Ref.\cite{fritzsch} we can
use the same diagonalization procedure to obtain the $t$, $c$ and $u$ quark masses.
Therefore, as in Ref.\cite{fritzsch}, we assume the following rotation matrix to diagonalize
Eq.(\ref{e1})

\br
R(\theta_1,\theta_2) = \left(\begin{array}{ccc} \cos\theta_1 & \sin\theta_1 & 0\\ -\sin\theta_1 & \cos\theta_1 & 0 \\
0 & 0 & 1
\end{array}\right)\left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & \cos\theta_2 & \sin\theta_2 \\
0 & -\sin\theta_2  & \cos\theta_2
\end{array}\right) , \nonumber
\er

\noindent which is useful in the limit $m_u \ll m_c \ll m_t$.
\par The  diagonalized matrix is giving by

\be
M^{\frac{2}{3}}_{f_{Diag}} = R^{-1}(\theta_1,\theta_2)M_{f}R(\theta_1,\theta_2),
\ee
\noindent  and after diagonalization we obtain

\be m_{u}\, \sim\, \frac{{\mid A\mid}^2}{{\mid B\mid}^2}\mid C\mid\,\,,\,\, m_{c}\, \sim\, \frac{{\mid B\mid}^2}{\mid C\mid}\,\,\,\,{\rm and}\,\,\,\,m_{t}\,\sim\, \,\,\mid C\mid,
\ee

\noindent where  in Eq.(\ref{abc}) we defined the entries $A$, $B$ and $C$.  We will
also assume the gut mass scale as $M_{gut} \sim 10^{16}$ GeV and the horizontal mass scale
equal to  $M_{h}\sim 10^{13}$ GeV as discussed in Section V. The several constants contained in
Eq.(\ref{abc})  are  $b_{tc}= \frac{1}{16\pi^2}\frac{26}{3}$, $b_{qcd}= \frac{7}{16\pi^2}$,
$\gamma_{tc}=\frac{15}{23}$ and $ \gamma_{qcd}=\frac{4}{7}$, below  we  plot the behavior  of
the  fermion masses  as a function of the  parameter $c\alpha_{[tc\,,\,qcd]}$, in the calculation we also
assumed $\alpha_{h} \sim \alpha_{k} \sim \frac{1}{45}$,  $\m_{tc} = 250 GeV$ and $\m_{qcd} = 250 MeV$.

\begin{figure}[hbt]
\begin{center}
\epsfig{file=graficomassauctJHEP.EPS,width=0.74\textwidth}
\caption{Plot showing the behavior of the $u$, $c$  and $t$ quark  masses  as a function of the  parameter $c\alpha$}
\end{center}
\label{plottop}
\end{figure}

In Fig.(10) it is shown the behavior of the $u$, $c$  and $t$ quark masses as a function of
the parameter  $c\alpha = c\alpha _{tc} = c\alpha_{qcd}$.  In Fig.(\ref{fbottom}) it is shown the behavior of the $d$, $s$ and $b$ quark masses.

\begin{figure}[hbt]
\begin{center}
\epsfig{file=graficomassadsbJHEP.EPS,width=0.75\textwidth}
\caption{Behavior of the $d$, $s$ and $b$  quark  masses as a function of the  parameter $c\alpha$}
\label{fbottom}
\end{center}
\end{figure}


\par The behavior of the lepton masses as a function of the  parameter  $c\alpha$ are show in the Fig.(12).

\begin{figure}[htb]
\begin{center}
\epsfig{file=graficomassaemtJHEP.EPS,width=0.75\textwidth}
\caption{Lepton  masses  as function of the parameter $c\alpha$.}
\end{center}
\end{figure}

\par We display in Table 1 the fermionic mass spectrum obtained in this model. Considering   the figures shown above, we select the values for the masses fixing the parameter $c\alpha = 0.5$ for simplicity. The two columns were obtained using the two different determinations of the coefficient $B$ described in Section V.

In Table 1  the values indicated by $(*)$ are the ones with larger disagreement in comparison to the
experimental values,  the  other values show a quite reasonable agreement if we consider all the
approximations that we have performed and the fact that we have a totally dynamical scheme. The values of the $u$ and $e$ masses can be easily lowered with
a smaller value of $\mu_{qcd}$. Of course, we are also assuming a very particular form for
the mass matrix based in one particular family symmetry. Better knowledgement of the symmetry
behind the mass matrix, and a better understanding of the strong interaction group
alignment will certainly improve the comparisom between data and theory. The high value for
the masses obtained for some of the second generation fermions also come out from the
overestimation of the $b$ and $\tau$ masses. The mass splitting between the $t$ and
$b$ quarks, which is far from the desirable result, is a problem that has not been satisfactorily solved
in most of the dynamical models of mass generation up to now. It is possible that an extra symmetry,
preventing these fermions to obtain masses at the leading order as suggested by Raby \cite{raby} can be
easily implemented in this model.  We will discuss these points again in the conclusions.
Finally considering that we do not have any fcnc problems, assume only the existence
of quite expected symmetries as the gut and family symmetries, the model does quite
well in comparison with many other models.

\begin{center}
\TABLE[ht]{\begin{tabular}{|c|c|c|c|c|c|}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
   Masses &
   $ B=\frac{14\alpha_{h}m_t}{\pi}$
   &$B=\left(\frac{M^4_{W}G^2_{F}\m^4_{tc}}{18\pi^3c\alpha_{tc}}\right)^{\frac{1}{4}}$ & Masses & $B=\frac{14\alpha_{h}m_t}{\pi}$
   & $B=\left(\frac{M^4_{W}G^2_{F}\m^4_{tc}}{18\pi^3c\alpha_{tc}}\right)^{\frac{1}{4}}$  \\
  \hline
   $m_{t}$  &  160.3 GeV     &   160.3 GeV  &   $m_{d}$   &  15.6 Mev  &  6.86 Mev  \\
   $m_{c}$  & 1.57 GeV  &  1.76 GeV  &   $m_{e}$   &   5.5 Mev  &  3.32 Mev \\   $m_{u}$  &  29.6 Mev    & 26.4 Mev  & $m_{\mu}^{(*)}$ &  1.30 GeV &  2.15 GeV \\
   $m_{b}^{(*)}$  &   113 GeV  & 113 GeV  & $m_{\tau}^{(*)}$ & 131.2 GeV & 131.2 GeV \\
   $m_{s}^{(*)}$  &  1.10 GeV &  2.50 GeV  &     &  & \\
   \hline
\end{tabular}
\caption{Approximate values for  quarks and leptons  masses according
to the possible values of couplings and strongly interacting mass scales.}}
\end{center}
%\newpage

\section{Pseudo-Goldstone boson masses}

\par  Another problem in technicolor models is the proliferation of pseudo-Goldstone bosons \cite{weisus,hillsi,ref6}.
After the chiral symmetry breaking of the strongly interacting sector a large number of Goldstone bosons
are formed, and only few of these degrees of freedom are absorbed by the weak interaction gauge bosons. The others
may acquire small masses resulting in light  pseudo-Goldstone bosons that have not been observed experimentally.
We will discuss how in our model these bosons obtain masses that are large enough to have escaped detection
at the present accelerator energies, but will show up at the next generation of accelerators (for instance, LHC).

We can list the possible pseudo Goldstone bosons according to their different quantum numbers:

\par \textit{ Colored pseudos}: They carry color degrees of freedom and can be divided into the
{\bf{3}} or {\bf{8}} color representations. We can indicate them by $$\Pi^a \sim \bar{Q}\gamma_5\lambda^aQ. $$
\par \textit{ Charged pseudos}: These ones carry electric charge and we can take as one example the following
current $$\Pi^{+}\sim \bar{L}\gamma_5Q.$$
\par \textit{ Neutral pseudos}: They do not carry color or charge and one example is
$$\Pi^0 \sim \bar{N}\gamma_5N.$$
\par Following closely Ref.\cite{ref6} the standard procedure to determine the  $SU(3)_{qcd}$
contribution to the mass ($M_c$) of a colored pseudo Goldstone boson gives

\be
M_{c}\sim \left(\frac{C_2(R)\alpha_{c}(\mu)}{\alpha_{el}}\right)^{\frac{1}{2}}\frac{F_\Pi}{f_\pi}35.5 MeV \sim 170\sqrt{C_2(R)} GeV \sim O(300)GeV.
\ee
\par While the electromagnetic contribution to the mass of the charged pseudos Goldstone bosons is
estimated to be \cite{ref6}

\be
M_{em}\sim Q_{ps}\frac{F_{\Pi}}{f_{\pi}}35.5 MeV \sim Q_{ps}47 GeV \sim O(50 GeV),
\ee
 \noindent in the equations above we assumed that the technipion and pion decay constants
 are given by $F_{\Pi}\approx 125 GeV$ and  $f_{\pi}\approx 95 MeV$, $Q_{ps}$ is the electric
 charge of the pseudo-Goldstone boson, and  $C_2(R)$ is the quadratic Casimir operator in the representation
 $R$ of the pseudo-Goldstone boson under the tc group. There is not much to change in these standard
 calculations, except that due to the particular form of the technifermion self energy the technifermion will
 acquire large masses, and subsequently the  pseudos-Goldstone bosons formed with these ones.

We know that any chiral current
 $\Pi^f$ can be written as a vacuum term $m_f  <\bar{\psi}_f\psi_f> $ plus electroweak (color, ...)
 corrections \cite{adriano}, where $m_f$ is the current mass of the fermion $\psi_f$ participating
 in the composition of the current  $\Pi^f$, neglecting the electroweak corrections and
 using PCAC in the case of qcd we obtain the Dashen relation

\be
m^2_{\pi} \approx \frac{m_{q}<\bar{q}q>}{f^2_{\pi}},
\ee

\noindent where $<\bar{q}q>$ is the quark condensate. Of course this relation is valid for any
chiral current and in particular for the technifermions we can write

\be
M^2_{\Pi} \approx \frac{M_{T_{f}}<\bar{T_{f}}T_{f}>}{F^2_{\Pi}},
\label{MPG}
\ee

\noindent where $M_{T_{f}}$ is the technifermion current mass. In the usual models (with $\alpha =1$
in Eq.(\ref{sa})) the technifermions are massless or acquire very tiny masses  leading to negligible
values for  $M_{\Pi}$. In our model this is not true. All technifermions acquire masses due to the
self-interaction with their own condensates through the interchange of grand unified bosons.

\par There are several bosons in the $SU(9)$ (and also in the $SU(3)_H$) theory connecting to technifermions
and generating a current mass as is shown in Fig.(\ref{fx}).

\begin{figure}[ht]
\begin{center}
\epsfig{file=figurmasstc.eps}
%\includegraphics{figurmasstc.eps}
\caption{Diagram responsible for the technifermion mass generation.}
\label{fx}
\end{center}
\end{figure}

\par A simple estimative, based on  Eq.(\ref{mftc}), of the contribution of Fig.(\ref{fx}) to the technifermion
masses  gives

\be
M_{T_{f_{SU(9)}}}\sim O(80 - 130)GeV .
\ee

\noindent  If we also include the contribution of the same diagram where the exchanged boson
is a horizontal  $SU(3)_{H}$ boson coupling technifermions of different generations,
we must add to the above value the following one

\be
M_{T_{f_{SU(3)_{H}}}} \sim O(10 - 40) GeV .
\ee

\par Therefore, we expect that the technifermion masses are at least of the order of
$M_{T_{f}}\approx O(100)GeV$. Now, according to Eq.(\ref{MPG}) and assuming
$<\bar{T_{f}}T_{f}>\sim F^3_{\Pi}$ we have the following estimative for the
pseudo-Goldstone boson masses

\be
M_{\Pi}\sim O(100) GeV .
\ee

\par Note that in this calculation we have not considered the qcd or electroweak corrections
discussed previously. Therefore, even if the pseudo-Goldstones bosons do not acquire masses
due to qcd or electroweak corrections they will at least have masses of order of  $100$ GeV
because of the ``current" technifermion masses obtained at the $SU(9)$ (or $SU(3)_H$) level.

\section{Conclusions}

We have presented a grand unified technicolor theory based on the group structure
 $SU(9) \times SU(3)_H$. The model is based on a particular ansatz
 for the tc and qcd self energy. We justify this choice in the appendix, but even if
 it is considered as an ``ad-hoc" choice for the self energy the main point of the
 model is that we can build a consistent model for fermion masses. This is the
 only new ingredient in the model, all the others (gut and horizontal symmetry)
 are naturally expected in the current scenario of particle physics.

 One of the characteristics of the model is that the first fermionic generation basically obtain
 masses due to the interaction with the qcd condensate, whereas the third generation
 obtain masses due to its coupling with the tc condensate. The reason for this particular
 coupling and for the alignment of the strong theory sectors generating intermediate masses is
 provided by the $SU(3)_H$  horizontal symmetry. Of course, our model is not successful in predicting
  all the fermion masses although it has a series of advantages. It does not need
  the presence of many etc boson masses to generate the different fermionic mass
  scales. The etc theory is replaced by the gut and horizontal symmetries. It has no
  flavor changing neutral currents or unwanted light pseudo-Goldstone bosons.

  There are many points that still need some work in this line of model. The breaking
  of the gut and horizontal symmetries is not discussed, and just assumed to happen
  near the Planck scale and possibly could be promoted by fundamental scalar bosons.
  The mass splitting in the third generation could be produced with the introduction of
  a new symmetry. For instance, if in the $SU(9)$ breaking besides the standard model
  interactions and the technicolor one we leave an extra $U(1)$, maybe we could have quantum numbers
  such that only the top quark would be allowed to couple to the tc condensate at leading order.
  This possibility should be further studied because it also may introduce large quantum corrections
  in the model.

  In this class of models we can choose different gut theories, different horizontal symmetries with
  different textures for the mass matrix which will certainly modify the values of the fermion
  masses that we have obtained. The alignment of the strongly interacting sectors can be
  studied only with many approximations, but it is quite possible that it generates more entries
  to the mass matrix than only the term $B$. With the evidences for neutrino masses that appeared
  recently it would be also interesting to study models containing right handed components for the
  light neutrinos.



\section*{Acknowledgments}


This research was supported by the Conselho Nacional de Desenvolvimento
Cient\'{\i}fico e Tecnol\'ogico (CNPq) (AAN) and by Fundac\~ao de Amparo \`a
Pesquisa do Estado de S\~ao Paulo (FAPESP) (AD).


\newpage

\section*{The self energy ansatz}

The expression for the quarks or techniquarks self energy have been obtained
mainly with two techniques: solution of the Schwinger-Dyson equations (SDE) for the
fermionic propagators and the operator product expansion (OPE).

We will not discuss too long the SDE solutions. First because it became clear in the last
years that  the high energy behavior of these chiral symmetry breaking solutions can be
dominated by higher order interactions (like four fermion interactions), that are relevant at or above
the tc or qcd scale of chiral symmetry breaking \cite{hillsi}. It is also worth mentioning that
in the numerical techniques it is extremely difficult to isolate a very slowly decreasing
of the self energy, if it happens much above the confinement scale. In this way the SDE solutions may
come out with the coefficient  $\a$ of Eq.(\ref{sa}) varying from $0$ to $1$, depending on
how large is the effect of these higher order interactions. Secondly, recently it has been
pointed out that the gauge boson propagator (gluon or technigluon) could be infrared
finite containing a new mass scale \cite{alkofer}. Considering that chiral symmetry breaking has
not been fully understood with this new infrared ingredient and within the SDE approach, we do
not believe that we can use SDE solutions to definitively settle the question of the self energy behavior
of the strongly interacting fermions at high momenta.

The determination of the quark self energy through OPE indicates that $\a = 1$ in Eq.(\ref{sa})
and not  $\a = 0$ as we assume here \cite{politzer}. Of course, this result could be modified by
the higher order operators as assumed in Ref.\cite{carpenter,soni}. But this is not the strongest
argument against this result, and we will argue that exactly for  $\a = 0$ the OPE result may
not be correct. Let us recall some of the results of Ref.\cite{aa}, where the vacuum energy for
composite operators was computed for the general expression of Eq.(\ref{sa}). There are two
important points: a) It does not matter what is the theory (or the coefficients $\a$ and $\beta$) the minimum
of energy always happens for $\a = 0$, b) For $\a = 0$ the value of the vacuum energy is proportional
to $1/g^2$, i.e. in Ref.\cite{aa} the deepest minimum of the vacuum energy $\< \Omega \>$
computed with Eq.(\ref{sa}) is given by
$$\< \Omega  (g,\mu) \> \propto \frac{1}{g^2} \mu^4\nonumber  . $$
Exactly for this case it was shown by Gupta and Quinn \cite{sq} that standard OPE
may fail! Because for such vacuum expectation value OPE is correct only at the leading-twist
contribution \cite{sq}. However, there is no leading-twist contribution for the mass operator in
the chiral limit! For this reason we believe that it is doubtfull that OPE can also be decisive to select
the asymptotic form of the self energy.

Considering what was discussed above, we cannot be totally sure that the  SDE and OPE techniques
have undoubtfully decided which is the specific asymptotic behavior of the
quark or techniquark self energy. On the other hand in favor of  the solution with $\a = 0$ we
have the following reasonings: 1) It is the only one that leads to an absolute minimum of energy \cite{aa}.
2) It is the only one that can reproduce Regge-pole solutions \cite{langacker}. 3) It is the only one that
has been obtained in  two other nonperturbative techniques \cite{chang,stam}. Where one of
these methods makes use of the Nambu-Jona-Lasinio procedure to treat the nonperturbative vacuum, which
brought so many information about the physics of chiral symmetry breaking \cite{chang}.  Therefore, we believe
that there are enough reasons to consider it as a viable solution. Finally, in the most
pessimistic point of view that chiral symmetry breaking in qcd (or tc) is explained by confinement,
as numerous authors in the eighties have considered in detail, we could just consider Eq.(\ref{sa})
with  $\a = 0$ as a reasonable ansatz.


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\end{document}
  
 


