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{\it University of Shizuoka}
\hspace*{9.5cm} {\bf US-97-02}\\[-.3in]
\hspace*{9.5cm} {\bf May 1997}\\[.3in]
\vspace*{.4in}
\begin{center}
{\large\bf Democratic Seesaw Mass Matrix Model}\\[.1in]
{\large\bf and New Physics
}\footnote{Talk at the Workshop on Masses and Mixings of
Quarks and Leptons, University of Shizuoka, March 19 -- 21, 1997.
To be appeared in the Proceedings, 1997.}
\vglue.3in
{\bf Yoshio Koide}\footnote{
E-mail: koide@u-shizuoka-ken.ac.jp} \\
Department of Physics, University of Shizuoka \\
395 Yada, Shizuoka 422, Japan \\[.1in]
\vspace{.3in}
{\large\bf Abstract}\\[.1in]
\end{center}
\begin{quotation}
A seesaw mass matrix model is reviewed as a unification model of quark
and lepton mass matrices.
The model can understand why top-quark mass $m_t$ is so singularly
enhanced compared with other quark masses, especially, why $m_t \gg m_b$
in contrast to $m_u\sim m_d$, and why only top-quark mass is of the order
of the electroweak scale $\Lambda_W$, i.e., $m_t\sim O(\Lambda_W)$.
The model predicts the fourth up-quark $t'$ with a mass
$m_{t'}\sim O(m_{W_R})$, and an abnormal structure of the right-handed
up-quark mixing matrix $U_R^u$.
Possible new physics is discussed.
\end{quotation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%
\noindent{\bf 1. Why seesaw mass matrix?}
\vglue.05in
The seesaw mechanism
$$
M_f \simeq m_L M_F^{-1} m_R \ .
\eqno(1.1)
$$
was first proposed [1] in order to answer the question
why neutrino masses are so invisibly small.
And then, in order to explain why quark masses are so small compared
with the electroweak scale $\Lambda_W$, the seesaw mechanism was
applied to the quarks [2].
However, the observation [3] of the top-quark with the large mass
$m_t\sim O(\Lambda_W)$ brought a new situation to the seesaw mass
matrix model:
Why is the top quark mass $m_t$ singularly large compared with $m_b$
in the third family in contrast to $m_u \sim m_d$ in the first family?
Why is the top-quark mass $m_t$ of the order of $\Lambda_W$?
It seems that the observation of the large top-quark mass rules out
the application of the seesaw mass matrix model to the quarks.
In the present talk, I would like to point out that the largeness
of $m_t$, especially, $m_t\sim O(\Lambda_W)$, is rather preferable
to the seesaw mass matrix model, and as an example, I will review
a specific model of a seesaw type mass matrix model,
``democratic seesaw mass matrix model" [4,5].
The most of the works were done in the collaboration with
H.~Fusaoka.
I would like to thank him for his energetic collaboration.
The basic idea is as follows.
We consider an $SU(2)_L \times SU(2)_R \times U(1)_Y$ gauge model.
We assume vector-like fermions $F_i$
in addition to the three-family quarks and leptons $f_i$
($f=u, d, \nu, e$; $i=1,2,3$).
These fermions and Higgs scalars belong to
$$
\begin{array}{lll}
f_L = (2,1) \ , \ \ & F_L = (1,1) \ , \ \ & \phi_L=(2,1) \ , \\
f_R = (1,2) \ , \ \ & F_R = (1,1) \ , \ \ & \phi_R=(1,2) \ , \\
\end{array} \eqno(1.2)
$$
of $SU(2)_L \times SU(2)_R$.
Then, the mass matrix for $(f, F)$ is given by
$$
M = \left(\begin{array}{cc}
0 & m_L \\
m_R & M_F \\
\end{array} \right) = m_0 \left(
\begin{array}{cc}
0 & Z \\
\kappa Z & \lambda O_f \\
\end{array} \right) \ \ .
\eqno(1.3)
$$
For simplicity, we have taken
$$
m_L = m_R/\kappa = m_0 Z \ .
\eqno(1.4)
$$
We assume that the matrix $Z$ is universal for $f = u, d, \nu, e$.
Further, we assume that the heavy fermion mass matrix $M_F$ has
a form [(unit matrix) + (rank-one matrix)]:
$$
M_F = \lambda m_0 O_f = \lambda m_0 ({\bf 1} + 3 b_f X ) \ ,
\eqno(1.5)
$$
where $b_f$ is an $f$-dependent complex parameter,
{\bf 1} is the $3\times 3$ unit matrix, and $X$ is a rank-one matrix
normalized by Tr$M_F=0$ at $b_f=-1/3$.
Then, for $b_f=-1/3$, we will find [4,5,6] the following mass spectrum,
$$
\begin{array}{l}
m_1, m_2 \sim \frac{\kappa}{\lambda} m_0 \ , \\
m_3 \simeq \frac{1}{\sqrt{3}} m_0 \sim O(m_L) \ , \\
m_4 \simeq \frac{1}{\sqrt{3}}\kappa m_0 \sim O(m_R) \ , \\
m_5, m_6 \sim \lambda m_0 \sim O(M_F) \ ,
\end{array} \eqno(1.6)
$$
independently of the datails of the matrix $Z$ ($\sim O(1)$).
(Also see Fig.~1 later.)
Therefore, if we assume that Tr$M_F=0$ for up-quark sector,
we can naturally understand why only the top quark has a mass of
the order of the electroweak scale $\Lambda_W\sim O(m_L)$.
This point will also be emphasized by T.~Satou [7] in this session
from more general study of the seesaw quark mass matrix.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vglue.2in
\noindent{\bf 2. Why democratic $M_F$?}
\vglue.05in
So far, we have never assumed that
the rank-one matrix $X$ is a democratic type.
Next, I would like to talk about why our model is called
``democratic" [8].
We know that we can always take the rank-one matrix $X$
as a democratic type
$$
X = \frac{1}{3} \left(\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1 \\
\end{array} \right) \ \ ,
\eqno(2.1)
$$
without losing generality.
The naming ``democratic" for the model is motivated by the following
phenomenological success [4] of taking $M_F$ ``democratic":
if we assume that the matrix $Z$ is given
by a diagonal matrix
$$
Z = \left(
\begin{array}{ccc}
z_1 & 0 & 0 \\
0 & z_2 & 0 \\
0 & 0 & z_3
\end{array} \right) \propto
\left(
\begin{array}{ccc}
\sqrt{m_e} & 0 & \\
0 &\sqrt{m_\mu} & 0 \\
0 & 0 & \sqrt{m_\tau}
\end{array} \right) \ .
\eqno(2.2)
$$
we can obtain reasonable values of the quark masses $m_i^q$ and
Cabibbo-Kobayashi-Maskawa (CKM) [9] matrix $V$.
For example, we can obtain the successful relation [10]
$$
\frac{m_u}{m_c}\simeq \frac{3}{4}\frac{m_e}{m_\mu} \ , \eqno(2.3)
$$
for $b_u\simeq -1/3$.
So, hereafter, we call the seesaw mass matrix model
with (2.1) and (2.2) the ``democratic seesaw mass matrix model".
Such a structure of the matrix $Z$ was suggested from
the following phenomenology:
Experimentally well-satisfied charged lepton mass formula [11]
$$
m_\tau + m_\mu + m_e = \frac{2}{3} \left(\sqrt{m_\tau} + \sqrt{m_\mu}
+ \sqrt{m_e} \right)^2
\eqno(2.4)
$$
can be derived from the bi-liner form
$$
M_e \propto Z \cdot {\bf 1} \cdot Z \ ,
\eqno(2.5)
$$
where
$$
Z = \left(
\begin{array}{ccc}
z_1 & 0 & 0 \\
0 & z_2 & 0 \\
0 & 0 & z_3
\end{array} \right) \ , \ \ \ \
\begin{array}{l}
z_i\equiv x_i +x_0 \ , \\
x_1+x_2+x_3=0 \ , \\
x_0^2=(x_1^2+x_2^2+x_3^2)/3 \ .
\end{array} \eqno(2.6)
$$
The form (2.5) suggests a seesaw mass matrix model
with a U(3)-family nonet Higgs boson [12].
However, in the present talk, I will skip this topic because
I have no time sufficient to discuss it.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vglue.2in
\noindent{\bf 3. Phenomenology of $m_i^q$ ($q=u, d$) and $V$}
\vglue.05in
We take the rank-one matrix $X$ as the democratic form (2.1).
Then, the successful results for $m_i^q$ and $V$ are obtained
from the following assumptions and inputs.
[Assumption I]: The matrix $Z$ takes a diagonal form
$Z={\rm diag}(z_1, z_2, z_3)$, when $X$ is in a democratic basis (2.1).
[Assumption II]: The parameter $b_f$ takes $ b_e =0$,
in the charged lepton sector.
The assumption II was put in order to fix the parameters $z_i$ as a trial.
Then, the parameters $z_i$ are given by
$$
\frac{z_1}{\sqrt{m_e}} = \frac{z_2}{\sqrt{m_\mu}}
= \frac{z_3}{\sqrt{m_\tau}}
= \frac{1}{\sqrt{m_e + m_\mu + m_\tau}} \ .
\eqno(3.1)
$$
In Fig.~1, we show the behavior of mass spectra $m_i^f$
($i=1,2,\cdots , 6$) versus the parameter $b_f$.
As seen in Fig.~1, the third fermion mass $m_3^f$ is sharply enhanced at
$b_f=-1/3$.
Also, note that the masses $m_2^f$ and $m_3^f$ (masses $m_1^f$ and $m_2^f$)
degenerate at $b_f=-1/2$ and $b_f=-1$, and the degeneration disappears
for the case of arg$b_f=0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Fig.1
\begin{figure}[htbp]
\begin{minipage}[tl]{10cm}
\epsfile{file=fig1.eps,scale=0.45}
\end{minipage}
\begin{minipage}[tr]{4cm}
{\small Fig.~1. Masses $m_i$ ($i=1,2,\cdots,6$) versus $b_f$ for the
case $\kappa=10$ and $\kappa/\lambda=0.02$.
The solid and broken lines represent the cases arg$b_f=0$ and
arg$b_f=18^\circ$, respectively. The figure was quoted from Ref.~[5].
}
\end{minipage}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In addition to (2.3), we can obtain many interesting relations [4,5]:
$$
\frac{m_c}{m_b}\simeq 4\frac{m_\mu}{m_\tau} \ , \ \ \
\frac{m_d m_s }{m_b^2}\simeq 4\frac{m_e m_\mu}{m_\tau^2} \ , \ \ \
\frac{m_u}{m_d}\simeq 3\frac{m_s}{m_c} \ , \eqno(3.2)
$$
around $b_u \sim - 1/3$, and $b_d \sim -e^{i\beta_d}$
$(1\ll \beta_d^2 \neq 0)$.
Therefore, we put the following assumption.
[Assumption III]: We fix the values of $|b_f|$ for the quark-sector as
$$
b_u=-\frac{1}{3} \ , \ \ \
b_d=-e^{i\beta_d} \ (1\ll \beta_d^2 \neq 0) .
\eqno(3.3)
$$
The former means the ansatz of
``the maximal top-quark-mass enhancement",
but, at present, there is not good naming for the latter.
For phenomenological fitting, we have used the following inputs:
$\kappa/\lambda=0.02$ from the observed ratio $m_c/m_t$;
$\beta_d=18^\circ $ from the observed ratio $m_d/m_s$.
Then we obtain reasonable quark mass ratios
and CKM matrix parameters [4]:
$$
\begin{array}{ll}
|V_{us}|=0.220 \ , \ \ \ & |V_{cb}|=0.0598 \ , \\
|V_{ub}|=0.00330 \ , \ \ \ & |V_{td}|=0.0155 \ .
\end{array} \eqno(3.4)
$$
(The value of $|V_{cb}|$ is somewhat larger than the observed value [13]
$ |V_{cb}|_{exp}=0.041\pm 0.003$.
For the improvement of the numerical value,
see Ref.~[5].)
%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4.
\vglue.2in
\noindent{\bf 4. Application to neutrino mass matrix}
\vglue.05in
As seen in Fig.~1, the choice of
$b_f\simeq -1/2$ gives
$$
m_1 \ll m_2 \simeq m_3 \ , \eqno(4.1)
$$
$$
U_{L} \simeq \left(
\begin{array}{ccc}
1 & \frac{1}{\sqrt{2}}\left(\sqrt{\frac{m_e}{m_\mu}}-
\sqrt{\frac{m_e}{m_\tau}}\right)
& \frac{1}{\sqrt{2}}\left(\sqrt{\frac{m_e}{m_\mu}}+
\sqrt{\frac{m_e}{m_\tau}}\right) \\
-\sqrt{m_e/m_\mu} & \frac{1}{\sqrt{2}} &
-\frac{1}{\sqrt{2}} \\
-\sqrt{m_e/m_\tau} & \frac{1}{\sqrt{2}} &
\frac{1}{\sqrt{2}} \\
\end{array} \right) \ .\eqno(4.2)
$$
On the other hand, the atmospheric neutrino data (Kamiokande) [14]
have suggested a large neutrino mixing
$\sin^22\theta_{\mu\tau}\simeq 1$ with
$\Delta m^2_{\tau\mu} \simeq 1.6 \times 10^{-2}$ ${\rm eV}^2$, and
the solar neutrino data (with MSW effects) [15] have suggested
a neutrino mixing $\sin^22\theta_{e x}\simeq 0.007$ with
$\Delta m^2_{x e} \simeq 6 \times 10^{-6}$ eV$^2$.
The results (4.1) and (4.2) are preferable to these data.
In order to make the model more explicit, we put the following assumption:
We assume that $\nu_R$ has a Majorana mass of the order of $\xi m_0$
($\xi\gg\lambda\gg\kappa\gg 1$)
in addition to the heavy neutrino masses $M_N \sim O(\lambda m_0)$.
Then, for example, for $b_\nu = -0.41$, we obtain
$$
\begin{array}{ll}
\sin^22\theta_{\mu\tau}\simeq 0.58 \ , \ \ \ \ \ &
\Delta m^2_{\tau\mu} \simeq 1.0 \times 10^{-2} \ {\rm eV}^2 \ , \\
\sin^22\theta_{e\mu}\simeq 0.0061 \ , \ \ \ \ \ &
\Delta m^2_{\mu e} \simeq 6 \times 10^{-6} \ {\rm eV}^2 \ ,
\end{array} \eqno(4.3)
$$
with $\xi m_0 = 1.9 \times 10^9$ GeV.
More details have been given in Ref.~[16].
%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5.
\vglue.2in
\noindent{\bf 5. Abnormal Structure of $U_R^u$}
\vglue.05in
In the down-quark sector, where the seesaw expression
$M_f \simeq m_L M_F^{-1} m_R$ is valid,
the mixing matrices $U_L^d$ and $U_R^d$ are given by
$$
U_L^d = \left(
\begin{array}{cc}
A_d & \frac{1}{\lambda} C_d \\
\frac{1}{\lambda} C'_d & B_d
\end{array} \right) \ , \ \
U_R^d \simeq \left(
\begin{array}{cc}
A_d^* & \frac{\kappa}{\lambda} C_d^* \\
\frac{\kappa}{\lambda} C^{\prime \ast}_d & B_d
\end{array} \right) \ .
\eqno(5.1)
$$
On the contrary, in the up-quark sector,
where the seesaw expression is not valid any longer,
the mixing matrices $U_L^u$ and $U_R^u$ are given by
$$
U_L^u = \left(
\begin{array}{lll|lll}
+0.9994 & -0.0349 & -0.0084 & -0.0247 \frac{1}{\lambda} &
+6\times 10^{-5}\frac{1}{\lambda} & +4\times 10^{-6}\frac{1}{\lambda} \\
+0.0319 & +0.9709 & -0.2373 & -0.2051\frac{1}{\lambda}
& -0.4346\frac{1}{\lambda} & +0.0259\frac{1}{\lambda} \\
+0.0165 & +0.2369 & +0.9714 & +0.8990\frac{1}{\lambda} &
+0.8431\frac{1}{\lambda} & -0.0444\frac{1}{\lambda} \\
\hline
+0.0934\frac{1}{\lambda} & +0.1114\frac{1}{\lambda}
& -1.0365\frac{1}{\lambda} & +0.5774 & +0.5774 & +0.5772 \\
-0.0118\frac{1}{\lambda} & +0.1649\frac{1}{\lambda}
& +0.0209\frac{1}{\lambda} & -0.7176 & +0.6961 & +0.0215 \\
-0.0064\frac{1}{\lambda} & -0.1011\frac{1}{\lambda}
& +0.7927\frac{1}{\lambda} & -0.3894 & -0.4267 & +0.8163
\end{array} \right) \ ,
\eqno(5.2)
$$
$$
U_R^u = \left(
\begin{array}{lll|lll}
+0.9994 & -0.0349 & -0.0084 & -0.0247\frac{\kappa}{\lambda} &
+6\times 10^{-5}\frac{\kappa}{\lambda}
& +4\times 10^{-6}\frac{\kappa}{\lambda} \\
+0.0319 & +0.9709 & -0.2373 & -0.2051\frac{\kappa}{\lambda}
& -0.4346\frac{\kappa}{\lambda} & +0.0259\frac{\kappa}{\lambda} \\
+0.0256\frac{\kappa}{\lambda} & +0.3459\frac{\kappa}{\lambda}
& -0.0747\frac{\kappa}{\lambda} & +0.5773 & +0.5773 & +0.5774 \\
\hline
+0.0165 & +0.2369 & +0.9713 & +0.3274\frac{\kappa}{\lambda}
& +0.2716\frac{\kappa}{\lambda} & -0.6160\frac{\kappa}{\lambda} \\
-0.0118\frac{\kappa}{\lambda} & +0.1649\frac{\kappa}{\lambda}
& +0.0209\frac{\kappa}{\lambda} & -0.7176 & +0.6961 & +0.0215 \\
-0.0064\frac{\kappa}{\lambda} & -0.1011\frac{\kappa}{\lambda}
& +0.7929\frac{\kappa}{\lambda} & -0.3894 & -0.4267 & +0.8161
\end{array} \right) \ .
\eqno(5.3)
$$
Note that the right-handed up-quark mixing
matrix $U_R^u$ has a peculiar structure
as if third and fourth rows of $U_R^u$ are exchanged
in contrast to $U_L^u$.
Why does such an abnormal structure appear in $U_R^u$?
In order to see this, let us change the heavy fermion basis
from the ``democratic basis" to the ``diagonal basis":
$$
M_F\ \ \Longrightarrow \ \ \widetilde{M}_F\equiv
A M_F A^{-1} = m_0 \lambda \left(
\begin{array}{ccc}
1+3b_f & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) \ .
\eqno(5.4)
$$
Then, the Hermitian matrices $H_L\equiv \widetilde{M}\widetilde{M}^\dagger$
and $H_R \equiv \widetilde{M}^\dagger\widetilde{M}$ take the following forms:
$$
\begin{array}{ll}
H_L = \widetilde{M}\widetilde{M}^\dagger &
H_R = \widetilde{M}^\dagger\widetilde{M} \\
=
m_0^2 \left(\begin{array}{cc}
\widetilde{Z}^T \widetilde{Z} & \lambda\widetilde{Z}^T\widetilde{O}_u \\
\lambda\widetilde{O}_u\widetilde{Z} &
\lambda^2\widetilde{O}_u^2 + \kappa^2 \widetilde{Z}\widetilde{Z}^T \\
\end{array} \right) \ &
= m_0^2
\left(\begin{array}{cc}
\kappa^2\widetilde{Z}^T\widetilde{Z}
& \kappa \lambda \widetilde{Z}^T \widetilde{O}_u \\
\kappa\lambda \widetilde{O}_u\widetilde{Z} &
\lambda^2 \widetilde{O}_u^2 + \widetilde{Z}\widetilde{Z}^T \\
\end{array} \right) \ \\
\end{array}$$
$$
\begin{array}{ll}
=\left(
\begin{array}{ccc|ccc}
\ast & * & * & 0 & * \lambda & * \lambda \\
\ast & * & * & 0 & * \lambda & * \lambda \\
\ast & * & * & 0 & * \lambda & * \lambda \\
\hline
0 & 0 & 0 & * \kappa^2 & * \kappa^2 & * \kappa^2 \\
\ast \lambda & * \lambda & * \lambda &
* \kappa^2 & * \lambda^2 & * \kappa^2 \\
\ast \lambda & * \lambda & * \lambda &
* \kappa^2 & * \kappa & * \lambda^2 \\
\end{array} \right)
&
=\left(
\begin{array}{ccc|ccc}
\ast \kappa^2 & * \kappa^2 & * \kappa^2 &
0 & * \kappa\lambda & * \kappa\lambda \\
\ast \kappa^2 & * \kappa^2 & * \kappa^2 &
0 & * \kappa\lambda & * \kappa\lambda \\
\ast \kappa^2 & * \kappa^2 & * \kappa^2 &
0 & * \kappa\lambda & * \kappa\lambda \\
\hline
0 & 0 & 0 & * & * & * \\
\ast \kappa \lambda & * \kappa\lambda & * \kappa\lambda &
* & * \lambda^2 & * \\
\ast \kappa \lambda & * \kappa\lambda & * \kappa\lambda &
* & * & * \lambda^2 \\
\end{array} \right)
\end{array}
\eqno(5.5)
$$
where $\widetilde{Z}=AZ$ and $\ast \sim O(1)$.
The result(5.5) means that the top quark $t\equiv u_3$ and
the fourth up-quark $t'\equiv u_4$ consist of
the following components
$$
\begin{array}{l}
t \equiv u_3 \simeq (u_{3L}, U_{1R}) \ , \\
t'\equiv u_4 \simeq (U_{1L}, u_{3R}) \ ,
\end{array} \eqno(5.6)
$$
although $u\simeq (u_{1L}, u_{1R})$, $c\simeq (u_{2L}, u_{2R})$,
$u_5\simeq (U_{2L}, U_{2R})$ and $u_6\simeq (U_{3L}, U_{3R})$.
Therefore, we can expect a single $t'$ production
through the exchange of the right-handed weak boson $W_R$
as we state later.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6.
\vglue.2in
\noindent{\bf 6. New physics from DSMM}
\vglue.05in
Since we want to observe new effects from the present model,
we take $\kappa=10$ tentatively.
Then, we can expect $m_{t'} \simeq \kappa m_t \sim $ a few TeV.
The single $t'$ production may be observed
through the exchange of $W_R$ as $ d+u \rightarrow t' +d$,
with $|V_{t'd}^R|= 0.0206$ and $|V_{ud}^R|=0.976$.
For example, we will observe the production
$ p +p \rightarrow t' +X$ at LHC.
On the other hand, in the present model, FCNC effects appear
proportionally to the factor [17]
$$
\xi^f = U_{fF} U_{fF}^\dagger \ ,
\ \
{\rm where}
\ \
U=\left(
\begin{array}{cc}
U_{ff} & U_{fF} \\
U_{Ff} & U_{FF}
\end{array} \right) \ . \eqno(6.1)
$$
Note that the FCNC effects appear visibly
in the modes related to top-quark, because the large elements are
only $(\xi_R^u)_{tc}=-0.00709$ and $(\xi_R^u)_{tu}=-0.000284$, and
the other elements are harmlessly small, e.g.,
$(\xi_R^u)_{cu}=\kappa^2 (\xi_L^u)_{cu}=2.01 \times 10^{-6}$,
$|(\xi_R^d)_{ds}|=\kappa^2 |(\xi_L^d)_{ds}|=4.03 \times 10^{-8}$,
and so on.
For example, we may expect the single top-quark production
$ e^- + p\rightarrow e^- +t +X $ at HERA.
Unfortunately, the values $(\xi_L^u)_{tu}=-8.85 \times 10^{-8}$ and
$(\xi_R^u)_{tu}=-2.84 \times 10^{-4}$ lead to an invisibly small value
of the cross section $\sigma (e^-+ p\rightarrow e^- + t+ X) \sim 10^{-8}$ pb.
Only possibility of the observation will be at a future TeV collider:
for example, $e^- +e^+ \rightarrow t +\overline{c}$ at JLC:
$$
\begin{array}{ll}
\sigma =6.0\times 10^{-7}\ {\rm pb} &
{\rm at}\ \sqrt{s}=0.2 \ {\rm TeV} \ , \\
\sigma =3.1\times 10^{-5}\ {\rm pb} &
{\rm at}\ \sqrt{s}=2m_t=0.36 \ {\rm TeV} \ , \\
\sigma =1.1\times 10^{-4}\ {\rm pb} &
{\rm at}\ \sqrt{s}=0.5 \ {\rm TeV} \ , \\
\sigma =7.5\times 10^{-4}\ {\rm pb} &
{\rm at}\ \sqrt{s}=0.7 \ {\rm TeV} \ , \\
%\sigma =0.085\ {\rm pb} &
%{\rm at}\ \sqrt{s}=m_{Z_R}=0.9 \ {\rm TeV} \ , \\
\end{array}
\eqno(6.2)
$$
where $\sigma=\sigma(t\overline{c})+\sigma(c\overline{t})$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7.
\vglue.2in
\noindent{\bf 7. Summary}
\vglue.05in
(i) Seesaw Mass Matrix with
$M_F$=[(unit matrix)+(rank-one matrix)]
can naturally understand the observed facts $m_t \gg m_b$ in
contrast to $m_u \sim m_d$, and $m_t \sim \Lambda_W$.
(ii) Democratic seesaw mass matrix model with the input $ b_e=0 $
can give reasonable quark mass ratios and CKM matrix
by taking $b_u=-1/3$ and $b_d= - e^{i18^\circ}$,
and a large neutrino mixing $\nu_\mu$-$\nu_\tau$
by taking $b_\nu \simeq -1/2$.
However, at present, we must take ad hoc parameter values
$b_u = -1/3$, $b_\nu \simeq -1/2$, $b_d \simeq -1$, $b_e = 0$.
I do not know whether there is some regularity among the values of $b_f$
or not, and what is the meaning of the parameter $b_f$.
(iii) The model will provide new physics in abundance:
\noindent
(a) $m_{t'}\sim$ a few TeV: we may expect a fourth up-quark production.
\noindent
(b) Abnormal structure of $U_R^u$: we may expect a single top-quark
production.
However, whether these effects are visible or not in the near future
depends on the value of $\kappa$ although we tentatively take $\kappa=10$
at the present study.
If $\kappa\simeq 10$, these effects cannot observe until starting of JLC.
Rather, there is a possibility that the effects due to the abnormal
structure of $U_R^u$ are sensitive to the $K^0$-$\overline{K}^0$
mixing which was pointed by T.~Kurimoto [18].
However, since our right-handed current structure
is different from the conventional $SU(2)_L\times SU(2)_R$ models,
more careful study will be required.
(iv) Present model is still a semi-phenomenological model,
so that an embedding of the present model into a
field-theoretical unification scenario is hoped.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vglue.2in
\centerline{\large\bf Acknowledgments}
A series of works based on the democratic seesaw mass matrix model
was first started in collaboration with H.~Fusaoka.
The author would like to thank H.~Fusaoka for his enjoyable
collaboration.
This work was supported by the Grant-in-Aid for Scientific Research, the
Ministry of Education, Science and Culture, Japan (No.08640386).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%
\vglue.2in
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\end{document}