\documentstyle[multicol,eqsecnum,aps,graphicx]{revtex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Evolution of the Yukawa coupling constants and seesaw operators \\ in the universal seesaw model} \author{\bf Yoshio Koide and Hideo Fusaoka$^{(a)}$} \address{ Department of Physics, University of Shizuoka 52-1 Yada, Shizuoka 422-8526, Japan \\ (a) Department of Physics, Aichi Medical University Nagakute, Aichi 480-1195, Japan} \date{\today} \maketitle \begin{abstract} The general features of the evolution of the Yukawa coupling constants and seesaw operators in the universal seesaw model with det$M_F=0$ are investigated. Especially, it is checked whether the model causes bursts of Yukawa coupling constants, because in the model not only the magnitude of the Yukawa coupling constant $(Y_L^u)_{33}$ in the up-quark sector but also that of $(Y_L^d)_{33}$ in the down-quark sector is of the order of one, i.e., $(Y_L^u)_{33} \sim (Y_L^d)_{33} \sim 1$. The requirement that the model should be calculable perturbatively puts some constraints on the values of the intermediate mass scales and $\tan\beta$ (in the SUSY model). \end{abstract} \pacs{12.15.Ff, 12.38.Bx, 12.60.-i} %\maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{multicols}{2} \narrowtext \section{Introduction} Recently, there has been considerable interest in the evolution (energy-scale dependency) of the Yukawa coupling constants of quarks and leptons. If we intend to build a model which gives a unified description of quark and lepton mass matrices, we cannot avoid investigating evolutions of the Yukawa coupling constants. The recent study on the quark masses and mixings has been given, for example, in Ref.~\cite{q-evol}. Especially, recently, the evolution of the neutrino seesaw mass matrix has been received considerable attention (for example, see Ref.~\cite{nu-evol}) in connection with the energy-scale dependence of the large mixing angle. %In general, if the mass spectrum in each fermion sector %is hierarchical, the energy-scale dependence of the mass %ratios (not the magnitudes of the masses) and mixings %in the fermion sector are not so large. As one of such unified models, there is a non-standard model, the so-called universal seesaw model" (USM) \cite{USM}. The model describes not only the neutrino mass matrix $M_\nu$ but also the quark mass matrices $M_u$ and $M_d$ and charged lepton mass matrix $M_e$ by the seesaw-type matrices universally: The model has hypothetical fermions $F_i$ ($F=U,D,N,E$; $i=1,2,3$) in addition to the conventional quarks and leptons $f_i$ ($f=u, d, \nu, e$; $i=1,2,3$), and these fermions are assigned to $f_L = (2,1)$, $f_R = (1,2)$, $F_L = (1,1)$ and $F_R = (1,1)$ of SU(2)$_L \times$ SU(2)$_R$. The 6 $\times$ 6 mass matrix which is sandwiched between the fields ($\overline{f}_L, \overline{F}_L$) and ($f_R, F_R$) is given by % M^{6 \times 6} = \left( \begin{array}{cc} 0 & m_L\\ m_R^\dagger & M_F \end{array} \right) \ , %\eqno(1.1) where $m_L$ and $m_R$ are universal for all fermion sectors ($f=u, d, \nu, e$) and only $M_F$ have structures dependent on the flavors $F$. For $\Lambda_L < \Lambda_R \ll \Lambda_S$, where $\Lambda_L = O(m_L)$, $\Lambda_R = O(m_R)$ and $\Lambda_S = O(M_F)$, the $3\times 3$ mass matrix $M_f$ for the fermions $f$ is given by the well-known seesaw expression $$M_f \simeq - m_L M^{-1}_F m_R^\dagger \ . %\eqno(1.2)$$ Thus, the model answers the question why the masses of quarks (except for top quark) and charged leptons are so small compared with the electroweak scale $\Lambda_L$ ($\sim$ 10$^2$ GeV). Recently, in order to understand the observed fact $m_t \sim \Lambda_L$ ($m_t$ is the top quark mass), the authors have proposed a universal seesaw mass matrix model with an ansatz \cite{KFzp,KFptp,Morozumi} ${\rm det} M_F = 0$ for the up-quark sector ($F=U$). In the model, one of the fermion masses $m(U_i)$ is zero [say, $m(U_3)=0$], so that the seesaw mechanism does not work for the third family, i.e., the fermions ($u_{3L}, U_{3R}$) and ($U_{3L}, u_{3R}$) acquire masses of $O(m_L)$ and $O(m_R)$, respectively. We identify $(u_{3L},U_{3_R})$ as the top quark $(t_L, t_R)$. Thus, we can understand the question why only the top quark has a mass of the order of $\Lambda_L$. Our interest is as follows: In the conventional model, the Yukawa coupling constants $y_f$ of the fermions $f$ are given by $y_f=m_f/\langle\phi_L^0\rangle$. Only the Yukawa coupling constants $y_t$ of the top quark $t$ takes a large value $y_t=m_t/\langle\phi_L^0\rangle\sim1$. The other Yukawa coupling constants $y_f$ are sufficiently smaller than one. On the contrast to the conventional model, in this USM, the matrices $m_L^f=Y_L^f\langle\phi_L^0\rangle$ are universal for all fermion sectors $f=u,d,e,\nu$, i.e., $Y_L^u=Y_L^d=Y_L^e=Y_L^{\nu}$. Therefore, when $(Y_L^u)_{33}$ is of the order of one, the other $(Y_L^f)_{33}$ will also be of the order of one. We are afraid that such a model causes bursts of the Yukawa coupling constants according as the energy scale increases. One of our interests is to investigate whether such a model can provide a set of reasonable parameter values which does not cause bursts of the Yukawa coupling constants (and also the seesaw operators $m_L M_F^{-1} m_R^\dagger$). We also take an interest in the democratic" USM \cite{KFzp,KFptp}, which is an extended version of USM and has successfully given the quark masses and the Cabibbo-Kobayashi-Maskawa (CKM) \cite{CKM} matrix parameters in terms of the charged lepton masses. However, the study is only phenomenology at the energy scale $\mu=m_Z$ ($m_Z$ is the neutral weak boson mass). Since the model is one of the promising models of the unified description of the quark and lepton mass matrices, it is important to investigate the evolutions of the mass matrices in the USM. The democratic USM is as follows: \noindent (i) The mass matrices $m_L$ and $m_R$ have the same structure except for their phase factors $$m_R^f = \kappa m_L^f \equiv \kappa m_0 Z^f \ , %\eqno(1.3)$$ where $\kappa$ is a constant and $Z^f$ are given by $$Z^f ={\rm diag}\left(z_1 \exp(i\delta_1^f),z_2 \exp(i\delta_2^f), z_3 \exp(i\delta_3^f)\right) \ , %\eqno(1.4)$$ with $z_1^2 + z_2^2 + z_3^2 = 1$. (The fermion masses $m_i^f$ are independent of the parameters $\delta_i^f$. Only the values of the CKM matrix parameters $|V_{ij}|$ depend on the parameters $\delta_i^f$). \noindent (ii) In the basis on which the matrices $m_L^f$ and $m_R^f$ are diagonal, the mass matrices $M_F$ are given by the form $$M_F = m_0 \lambda ({\bf 1} + 3b_f X), %\eqno(1.5)$$ $${\bf 1} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) ,\ \ \ X = \frac{1}{3} \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array} \right) \ . %\eqno(1.6)$$ \noindent (iii) The parameter $b_f$ for the charged lepton sector is given by $b_e$ = 0, so that in the limit of $\kappa/\lambda \ll 1$, the parameters $z_i$ are given by $$\frac{z_1}{\sqrt{m_e}} = \frac{z_2}{\sqrt{m_\mu}} = \frac{z_3}{\sqrt{m_\tau}} = \frac{1}{\sqrt{m_e + m_\mu + m_\tau}} \ . %\eqno(1.7)$$ Then, the up- and down-quark masses are successfully given by the choice of $b_u = -1/3$ and $b_d = -e^{i \beta_d}$ ($\beta_d = 18^{\circ}$), respectively. The CKM matrix is also successfully obtained by taking $$\delta_1^u -\delta_1^d=\delta_2^u-\delta_2^d=0 \ , \ \ \delta_3^u -\delta_3^d \simeq \pi \ ,$$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% However, when we take the evolution of the Yukawa coupling constants into the consideration, we should consider that the assumptions (i) and (ii) are required not at the electroweak energy scale $\mu=\Lambda_L$, but at a unification energy scale $\mu=\Lambda_X$, i.e., the assumptions (i) and (ii) should be replaced with $$Y_L^f(\Lambda_X) = Y_R^f(\Lambda_X) = \xi_{LR}^f Z^f \ , % \eqno(1.9)$$ $$\xi_{LR}^u=\xi_{LR}^d, %\eqno(1.10)$$ and $$Y_S^f(\Lambda_X) = \xi_S^f \left( {\bf 1} + 3b_f X\right) \ , %\eqno(1.11)$$ respectively, where mass matrices $m_L$, $m_R$ and $M_F$ are expressed by $$m_L^f= Y_L^f \langle\phi_L^0\rangle \ , \ \ m_R^f= Y_R^f \langle\phi_R^0\rangle \ , \ \ M_F= Y_S^f \langle\Phi^0\rangle \ , %\eqno(1.12)$$ respectively, and $$\langle\phi_L^0\rangle = \langle\phi_R^0\rangle/\kappa = \langle\Phi^0\rangle/\lambda %\eqno(1.13)$$ and $\phi_L$, $\phi_R$ and $\Phi$ are Higgs scalars whose vacuum expectation values (VEV) break SU(2)$_L$, SU(2)$_R$, and an additional U(1) symmetry U(1)$_X$, respectively. (For simplicity, we have assumed that the values of $\langle\phi_L^0\rangle$, $\langle\phi_R^0\rangle$ and $\langle\Phi^0\rangle$ are real.) Another interest in the present paper is to check whether or not the phenomenological study in the previous paper \cite{KFzp} is still approximately valid under the evolution of the Yukawa coupling constants. For example, the model with $b_e=0$ and $b_u=-1/3$ has led to the relation \cite{Koide-mpl,KFzp} $${\frac{m_u}{m_c}}\simeq{\frac{3}{4}}{\frac{m_e}{m_{\mu}}}, %\eqno(1.14)$$ almost independently of the value of the seesaw suppression factor $\kappa/\lambda$. One of the reasons to taking the value of $b_f$ in the up-quark sector as $b_u=-1/3$ exists in the successful relation (1.14). Therefore, we have interest whether the relation (1.14) still holds even when we take the evolution into consideration. Besides, even apart from such phenomenological interests, it is very important to investigate the general features of the evolution of the Yukawa coupling constants in the universal seesaw model with det$M_F=0$, because in the present model one of the fermions $F_i$ does not decouple from the theory at $\mu < \Lambda_S$, so that the evolution shows peculiar behavior in contrast with the conventional seesaw model. A similar study has been done in Ref.~\cite{evol-USM} by one of the authors (Y.K.). However, in Ref.~\cite{evol-USM}, instead of the seesaw operators $K^f$ which will be defined later in Eqs.~(3.8) corresponding to $m_L M_F^{-1} m_R^\dagger$, the evolution of the seesaw forms of the Yukawa coupling constants $Y_L^f (Y_S^f)^{-1} (Y_R^f)^\dagger$ were investigated by calculating the Yukawa coupling constants $Y_L^f$, $Y_R^f$ and $Y_S^f$ individually under the assumption that the heavy particles with the masses of the order of $\Lambda_S \equiv \langle\Phi^0\rangle$ do not contribute to the evolution of $Y_A^f$ ($A=L,R,S$) below $\mu=\Lambda_S$. In the present paper, we will calculate the evolution of the Yukawa coupling constants $Y_A^f$ above $\mu=\Lambda_S$ and that of the seesaw operators $K^f$ below $\mu=\Lambda_S$, except for $(Y_L^f)_{i3}$ as discussed in Sec.~\ref{sec:3}. In Sec.~\ref{sec:2}, we will discuss an additional symmetry which is introduced for the purpose of preventing that the fermions $F$ acquire the masses $M_F$ at the energy scale $\mu=\Lambda_S$. In Sec.~\ref{sec:3}, we will give the general formulation of the evolution of the seesaw mass matrices with ${\rm det}M_U=0$. In Sec.~\ref{sec:4}, we give the explicit coefficients of the renormalization group equations. In Sec.~\ref{sec:5}, we discuss the evolution of an extended version of the USM, the democratic seesaw model" \cite{KFzp,KFptp}. The numerical results for a non-SUSY model and for a minimal SUSY model are given in Secs.~\ref{sec:6} and \ref{sec:7}, respectively. It will be emphasized that the energy scale dependencies in the SUSY model are quite different from those in the non-SUSY model. The evolution of the neutrino mass matrix is given in Sec.~\ref{sec:8}. It will be showed that, differently from the conventional seesaw model, the present neutrino mass matrix is form-invariant below $\mu=\Lambda_S$. Finally, Sec.~\ref{sec:9} will be devoted to the conclusions and remarks. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newpage %\narrowtext \section{U(1)$_X$ symmetry} \label{sec:2} %\vglue.1in In the present model, the gauge symmetries are broken as follows: $$\begin{array}{c} {\rm SU(2)}_L \times {\rm SU(2)}_R \times {\rm U(1)}_{LR} \times {\rm SU(3)}_c \times {\rm U(1)}_X \\ \downarrow \ \ \mu=\Lambda_S \\ {\rm SU(2)}_L \times {\rm SU(2)}_R \times {\rm U(1)}_{LR} \times {\rm SU(3)}_c \\ \downarrow \ \ \mu=\Lambda_R \\ {\rm SU(2)}_L \times {\rm U(1)}_{Y} \times {\rm SU(3)}_c \\ \downarrow \ \ \mu=\Lambda_L \\ {\rm U(1)}_{em} \times {\rm SU(3)}_c \ . \\ \end{array} %\eqno(2.1)$$ Here, the symmetry U(1)$_X$, which is spontaneously broken at the energy scale $\mu=\Lambda_S$, has been introduced for the purpose of preventing that the fermions $F$ acquire the masses $M_F$ at $\mu > \Lambda_S$. Hereafter, we call the ranges $\Lambda_L < \mu \leq \Lambda_R$, $\Lambda_R < \mu \leq \Lambda_S$, and $\Lambda_S < \mu \leq \Lambda_X$ as the ranges I, II, and III, respectively. In the present paper, the energy scale $\Lambda_X$ does not always mean a gauge unification energy scale. We assume that at the energy scale $\Lambda_X$ the mass matrices (Yukawa coupling constants) take simple forms discussed in the previous section. The Yukawa coupling constants $Y_L^f$, $Y_R^f$ and $Y_S^f$ are defined as follows: \end{multicols} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hspace{-0.5cm} \rule{8.7cm}{0.1mm}\rule{0.1mm}{2mm} \widetext \begin{eqnarray} H_{int} &=& Y^u_{Lij}\overline{q}_{Li}\widetilde{\phi}_L U_{Rj} + Y^d_{Lij}\overline{q}_{Li}{\phi}_L D_{Rj} + Y^\nu_{Lij}\overline{\ell}_{Li}\widetilde{\phi}_L N_{Rj} + Y^e_{Lij}\overline{\ell}_{Li}{\phi}_L E_{Rj} \nonumber\\ &+& Y^u_{Rij}\overline{q}_{Ri}\widetilde{\phi}_R U_{Lj} + Y^d_{Rij}\overline{q}_{Ri}{\phi}_R D_{Lj} + Y^\nu_{Rij}\overline{\ell}_{Ri}\widetilde{\phi}_R N_{Lj} + Y^e_{Rij}\overline{\ell}_{Ri}{\phi}_R E_{Lj} \\ &+& Y^u_{Sij}\overline{U}_{Li}\Phi U_{Rj} + Y^d_{Sij}\overline{D}_{Li}{\Phi}^\dagger D_{Rj} + Y^\nu_{Sij}\overline{N}_{Li}{\Phi} N_{Rj} + Y^e_{Sij}\overline{E}_{Li}{\Phi}^\dagger E_{Rj} + h.c. \ ,\nonumber %\eqno(2.2) \end{eqnarray} \hspace{9.1cm} \rule{-2mm}{0.1mm}\rule{8.7cm}{0.1mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{multicols}{2} \narrowtext \noindent where \begin{eqnarray} &&q_{L/R} =\left( \begin{array}{c} u \\ d \end{array} \right)_{L/R} \ , \ \ \ \ell_{L/R} =\left( \begin{array}{c} \nu \\ e^- \end{array} \right)_{L/R} \ ,\nonumber\\ &&\phi_{L/R} =\left( \begin{array}{c} \phi^+ \\ \phi^0 \end{array} \right)_{L/R} \ , \ \ \ \widetilde{\phi}_{L/R} =\left( \begin{array}{c} \overline{\phi}^0 \\ -\phi^- \end{array} \right)_{L/R} \ . %\eqno(2.3) \end{eqnarray} {}From Eq.~(2.2), the U(1)$_X$ charge assignment should satisfy the following relations %\end{multicols} % %\hspace{-0.5cm} %\rule{8.8cm}{0.1mm}\rule{0.1mm}{2mm} \begin{eqnarray} &&X(U_R)= X(U_L) -X(\Phi) \ , \ \ \nonumber\\ &&X(D_R)= X(D_L) +X(\Phi) \ , %\eqno(2.4) \end{eqnarray} \begin{eqnarray} &&X(q_L)= \frac{1}{2} \left[ X(U_R) +X(D_R) \right] \ , \ \nonumber\\ &&X(q_R)= \frac{1}{2} \left[ X(U_L) +X(D_L) \right] \ , %\eqno(2.5) \end{eqnarray} \begin{eqnarray} &&X(\phi_L)= \frac{1}{2} \left[ X(U_R) -X(D_R) \right] \ , \ \nonumber\\ &&X(\phi_R)= \frac{1}{2} \left[ X(U_L) -X(D_L) \right] \ , %\eqno(2.6) \end{eqnarray} for quark sectors, and equations similar to Eqs.~(2.4) - (2.6) for lepton sectors $f=\nu, e$. For simplicity, in the present paper, we choose $$X(q_{L/R}) = X(\ell_{L/R}) = 0 \ , \ \ X(\Phi)=+1 \ . %\eqno(2.7)$$ Then, the quantum numbers of the fermions $f$ and $F$ and Higgs scalars $\phi_L$, $\phi_R$ and $\Phi$ for ${\rm SU(2)}_L \times {\rm SU(2)}_R \times {\rm U(1)}_{LR} \times {\rm U(1)}_{X}$ are given in Table \ref{T-qn}. Note that the quantum number of the fermion $N_{L}$ is identical with that of the fermion $N_R^c$ [$\equiv (N_R)^c \equiv C \overline{N}_R^T$]. Therefore, the neutral fermions $N_L$ and $N_R$ can acquire the following Majorana mass terms at $\mu=\Lambda_S$: $$H_{Majorana} = \left( Y^L_{Sij} \overline{N}_{Li} N_{Lj}^c + Y^R_{Sij} \overline{N}_{Ri}^c N_{Rj} \right) \Phi + h.c. \ . %\eqno(2.8)$$ Then, the neutrino mass matrix is given as follows $$\left( \overline{\nu}_L\ \overline{\nu}_R^c\ \overline{N}_L\ \overline{N}_R^c \right) { \left( \begin{array}{cccc} 0 & 0 & 0 & m_L \\ 0 & 0 & m_R^{\dagger T} & 0 \\ 0 & m_R^\dagger & M_L & M_D \\ m_L^T & 0 & M_D^T & M_R \\ \end{array} \right) } \left( \begin{array}{c} \nu_L^c \\ \nu_R \\ N_L^c \\ N_R \end{array} \right) \ , %\eqno(2.9)$$ where $M_D= Y_S^\nu \langle \Phi \rangle$, $M_L= Y_S^L \langle \Phi \rangle$ and $M_R= Y_S^R \langle \Phi \rangle$. Since $O(M_D) \sim O(M_L) \sim O(M_R) \gg O(m_R) \gg O(m_L)$, we obtain the mass matrix $M_\nu$ for the active neutrinos $\nu_L$ $$M_\nu \simeq - m_L M_R^{-1} m_L^T \ . %\eqno(2.10)$$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{General features of the evolutions} \label{sec:3} In the present section, we give a general formulation of the evolution of the seesaw matrix with det$M_F=0$. The evolution of the neutrino seesaw mass matrix is well known. However, in such a model with det$M_F=0$ as the present model (the democratic seesaw model), a careful treatment is required. Without losing the generality, we can express the Yukawa coupling constants $Y_L^f$ and $Y_R^f$ ($f=u,d,\nu,e$) as $$Y_L^f(\mu)=\xi_L^f(\mu) Z_L^f(\mu) \ , \ \ Y_R^f(\mu)=\xi_R^f(\mu) Z_R^f(\mu) \ , %\eqno(3.1)$$ where $Z_A^f(\mu)$ ($A=L,R$) are defined by $$Z_A^f(\mu) = {\rm diag}(z_{A1}^f(\mu), z_{A2}^f(\mu), z_{A3}^f(\mu)) \ , %\eqno(3.2)$$ $$|z_{A1}^f(\mu)|^2+|z_{A2}^f(\mu)|^2+|z_{A3}^f(\mu)|^2=1 \ , %\eqno(3.3)$$ on the basis on which $Y_A^f(\mu)$ are diagonal. In the present model, the word universal" means the following initial conditions $$\xi_L^f(\Lambda_X) = \xi_R^f(\Lambda_X) \equiv \xi_{LR} \ , %\eqno(3.4)$$ $$|z_{Li}^f(\Lambda_X)|=|z_{Ri}^f(\Lambda_X)| \equiv z_i \ , %\eqno(3.5)$$ for all fermion sectors $f=u,d,\nu,e$ universally. In the range III ($\Lambda_S < \mu \leq \Lambda_X$), the evolutions of the Yukawa coupling constants $Y_L^f$, $Y_R^f$ and $Y_S^f$ are given by the one loop renomalization group equations (RGE) as follows: $$16\pi^2 \frac{d Y_A^f}{dt} = \left( T_A^f - G_A^f +H_A^f \right) Y_A^f \ , \ \ (A=L, R, S) \ , %\eqno(3.6)$$ where $t=\log \mu$, and $T_A^f$, $G_A^f$ and $H_A^f$ ($A=L, R, S$) denote contributions from fermion loop corrections, vertex corrections due to the gauge bosons, and vertex corrections due to the Higgs boson, respectively. Note that the matrices $T_A^f$ and $G_A^f$ are proportional to the unit matrix. As stated in the next section, the coefficients $H_A^f$ ($A=L, R, S$) take diagonal forms on the basis on which $Y_A^f$ are diagonal. Therefore, if we take a basis on which $Y_L^f$ (and $Y_R^f$) or $Y_S^f$ are diagonal at $\mu=\Lambda_X$, then the Yukawa coupling constants $Y_L^f$ (and $Y_R^f$) or $Y_S^f$ can keep the forms diagonal in the range III. Sometimes, the basis on which $Y_L^f$ (and $Y_R^f$) are diagonal is useful, but sometimes, another basis on which $Y_S^f$ are diagonal is useful, as we discuss later. In the present model, it is assumed that we can choose a flavor basis on which $Y_S^f(\Lambda_X)$ are simultaneously diagonal for all $f=u,d,\nu,e$. Then, on this basis, since the Yukawa coupling constants $Y_S^f(\mu)$ can keep the forms diagonal in the range III, we can find that all $Y_S^f$ are diagonal at $\mu=\Lambda_S$. We can denote those as $$Y_S^f (\Lambda_S) = {\rm diag}( y_{1S}^f, y_{2S}^f, y_{3S}^f)\ . %\eqno(3.7)$$ At the energy scale $\mu=\Lambda_S$, the fermions $F_i$ (except for $U_3$) acquire the heavy masses $(M_F)_{ii}= y_{iS}^f \langle\Phi^0\rangle$. In the conventional seesaw model with det$M_F\neq 0$, the energy scale behaviors of the fermion masses in $\mu<\Lambda_S$ are described by evolutions of the following operators $$(K^f)_{ij} = \left[ Y_L^f (Y_S^f)^{-1} (Y_R^f)^{\dagger} \right]_{ij} = \sum_{k=1}^3 \frac{1}{y_{kS}^f} (Y_L^f)_{ik} (Y_R^f)_{jk}^{\ast} \ , %\eqno(3.8)$$ and $$(K^\nu)_{ij} = \left[ Y_L^\nu (Y_S^\nu)^{-1} (Y_L^\nu)^T \right]_{ij} =\sum_{k=1}^3 \frac{1}{y_{kS}^\nu} (Y_L^\nu)_{ik} (Y_L^\nu)_{jk} \ . %\eqno(3.9)$$ (Hereafter, for convenience, we will denote the Yukawa coupling constants $Y_S^R$ for the Majorana mass $M_R=Y_S^R \langle\Phi^0 \rangle$ in Eq.~(2.10) as $Y_S^\nu$.) The quark and lepton mass matrices $M_f$ are given by $$M_f = K^f \langle\phi_L^0\rangle \langle\phi_R^0\rangle /\langle\Phi^0\rangle \ , \ \ (f=u, d, e) \ , %\eqno(3.10)$$ $$M_\nu = K^\nu \langle\phi_L^0\rangle^2 /\langle\Phi^0\rangle \ . %\eqno(3.11)$$ As explicitly shown in Sec.~\ref{sec:4}, the evolutions of the operators $K^f$ are described by the one-loop RGE's with the following forms \begin{eqnarray} 16\pi^2 \frac{d K^f}{dt} = \left( T_K^f -G_K^f \right) K^f + H_{KL}^f K^f +K^f H_{KR}^{f\dagger} \ ,\nonumber\\ (f=u,d,e), %\eqno(3.12) \end{eqnarray} $$16\pi^2 \frac{d K^\nu}{dt} = \left( T_K^\nu -G_K^\nu \right) K^\nu + H_{KL}^\nu K^\nu +K^\nu H_{KL}^{\nu T} \ , %\eqno(3.13)$$ where $T_K^f$, $G_K^f$ and ($H_{KL}^f$, $H_{KR}^f$) denote contributions from fermion loop corrections, vertex corrections due to the gauge bosons, and vertex corrections due to the Higgs bosons $\phi_L$ and $\phi_R$, respectively. However, in the seesaw mass matrix with det$M_F=0$, since one of the eigenvalues of $Y_S^f$ ($f=u$) is zero (say, $y_{3S}^u=0$), we must calculate the following operator $$(K^u)_{ij} = \left[ Y_L^u (Y_S^u)^{-1} Y_R^{u\dagger} \right]_{ij} =\sum_{k=1}^2 \frac{1}{y_{kS}^u} (Y_L^u)_{ik} (Y_R^u)_{jk}^{\ast} \ , %\eqno(3.14)$$ where $Y_S^u= {\rm diag}(y_{1S}^u, y_{2S}^u)$. Note that the matrices $Y_U$ and $Y_L^u$ ($Y_R^u$) in Eq.~(3.14) are $2\times 2$ and $3\times 2$ matrices, respectively. Note that in (3.14) we have taken the sum over $k=1$ and $2$ only. In the range II, the evolutions of the Yukawa coupling constants $Y_{Li3}^u$ and $Y_{R i3}^u$ ($i=1,2,3$) are still described by the equation (3.6). At the energy scale $\mu=\Lambda_R$, we obtain a new mass term $$H_{mass}=\sum_i (Y_R^u)_{i3}^{\ast} \overline{U}_{L3} u_{Ri} \langle\phi_R^0\rangle \ . %\eqno(3.15)$$ By defining a mixing state $$u'_{R3}=\frac{(Y_R^u)_{13}^{\ast} u_{R1} +(Y_R^u)_{23}^{\ast} u_{R2} +(Y_R^u)_{33}^{\ast} u_{R3}}{ \sqrt{ |(Y_R^u)_{31}|^2 +|(Y_R^u)_{32}|^2 +|(Y_R^u)_{33}|^2 }} \ , %\eqno(3.16)$$ we obtain a mass $m_{t'}$ of the fourth up-quark $t'=(t'_L, t'_R)=(U_{L3}, u'_{R3})$, $$m_{t'}= \langle\phi_R^0\rangle \sqrt{ |(Y_R^u)_{13}|^2 +|(Y_R^u)_{23}|^2 +|(Y_R^u)_{33}|^2 } \ . %\eqno(3.17)$$ Similarly, in the approximation in which the terms suppressed by $y_{1S}^u$ and $y_{2S}^u$ are neglected, the mass $m_t$ of the third up-quark (i.e., top quark) $t=(t_L, t_R)=(u'_{L3}, U_{R3})$ is given by $$m_t \simeq \langle\phi_L^0\rangle \sqrt{ |(Y_L^u)_{13}|^2 +|(Y_L^u)_{23}|^2 +|(Y_L^u)_{33}|^2 } \ , %\eqno(3.18)$$ where $$u'_{L3}\simeq \frac{(Y_L^u)_{13}^{\ast} u_{L1} +(Y_L^u)_{23}^{\ast} u_{L2} +(Y_L^u)_{33}^{\ast} u_{L3}}{ \sqrt{ |(Y_L^u)_{13}|^2 +|(Y_L^u)_{23}|^2 +|(Y_L^u)_{33}|^2 }} \ . %\eqno(3.19)$$ More precisely speaking, the masses $(m_u, m_c, m_t,m_{t'})$ are obtained by diagonalizing the following mass matrix \end{multicols} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hspace{-0.5cm} \rule{8.7cm}{0.1mm}\rule{0.1mm}{2mm} \widetext $$M^u = \langle\phi_L^0\rangle \left( \begin{array}{cccc} -(\kappa/\lambda)K_{11}^u & -(\kappa/\lambda)K_{12}^u & -(\kappa/\lambda)K_{13}^u & Y_{L13}^u \\ -(\kappa/\lambda)K_{21}^u & -(\kappa/\lambda)K_{22}^u & -(\kappa/\lambda)K_{23}^u & Y_{L23}^u \\ -(\kappa/\lambda)K_{31}^u & -(\kappa/\lambda)K_{32}^u & -(\kappa/\lambda)K_{33}^u & Y_{L33}^u \\ \kappa Y_{R13}^{u\ast} & \kappa Y_{R23}^{u\ast} & \kappa Y_{R33}^{u\ast} & 0 \\ \end{array} \right) \ , %\eqno(3.20)$$ \hspace{9.1cm} \rule{-2mm}{0.1mm}\rule{8.7cm}{0.1mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{multicols}{2} \narrowtext \noindent which is sandwiched by the fields $(\overline{u}_{L1}, \overline{u}_{L2}, \overline{u}_{L3}, \overline{U}_{L3})$ and $({u}_{R1},{u}_{R2}, {u}_{R3},{U}_{R3})$, where $\kappa= \langle\phi_R^0\rangle/\langle\phi_L^0\rangle$ and $\lambda=\langle\Phi^0\rangle/\langle\phi_L^0\rangle$ as defined in Eq.~(1.13). Of the Yukawa coupling constants $(Y_L^u)_{ij}$ and $(Y_R^u)_{ij}$, the twelve components $(Y_L^u)_{ik}$ and $(Y_R^u)_{ik}$ ($i=1,2,3; j=1,2$) are absorbed into the operator $K^u$ defined by (3.14), while the rest $(Y_L^u)_{i3}$ and $(Y_R^u)_{i3}$ are still described by the equation (3.6). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newpage %\narrowtext \section{Coefficients of the RGE} \label{sec:4} %\vglue.1in In the present section, we give the coefficients of the renomalization group equations (RGE) (3.6), (3.12) and (3.13). \subsection{Evolution in the range III} In the non-SUSY model, the terms $T_A^f$, $G_A^f$ and $H_A^f$ ($A=L,R,S$) are given as follows: %\end{multicols} %\hspace{-0.5cm} %\rule{8.7cm}{0.1mm}\rule{0.1mm}{2mm} \begin{eqnarray} T_A^u&=&T_A^d=T_A^\nu=T_A^e \nonumber\\ &=&3 {\rm Tr}\left( Y_A^u Y_A^{u \dagger} + Y_A^d Y_A^{d \dagger}\right) + {\rm Tr}\left( Y_A^\nu Y_A^{\nu \dagger} + Y_A^e Y_A^{e \dagger} \right) \ , \nonumber \\ %\eqno(4.1) \end{eqnarray} \begin{eqnarray} &&G_A^u= \frac{17}{8}g_1^2 +\frac{9}{4}g_{2A}^2 +8 g_3^2 +\frac{3}{4} g_X^2 \ , \ \ \ \nonumber\\ &&G_A^d= \frac{5}{8}g_1^2 +\frac{9}{4}g_{2A}^2 +8 g_3^2 +\frac{3}{4} g_X^2 \ , \nonumber \\ &&G_A^\nu= \frac{9}{8}g_1^2 +\frac{9}{4}g_{2A}^2 +\frac{3}{4} g_X^2 \ , \ \ \nonumber\\ &&G_A^e= \frac{45}{8}g_1^2 +\frac{9}{4}g_{2A}^2 +\frac{3}{4} g_X^2 \ , %\eqno(4.2) \end{eqnarray} \begin{eqnarray} &&H_A^u=-H_A^d =\frac{3}{2}\left( Y_A^u Y_A^{u \dagger} - Y_A^d Y_A^{d \dagger} \right) \ ,\nonumber\\ &&H_A^\nu=-H_A^e =\frac{3}{2}\left( Y_A^\nu Y_A^{\nu \dagger} - Y_A^e Y_A^{e \dagger} \right) \ , %\eqno(4.3) \end{eqnarray} where $A=L,R$, and \begin{eqnarray} T_S^u&=&T_S^d=T_S^\nu=T_S^e \nonumber\\ &=&3 {\rm Tr}\left( Y_S^u Y_S^{u \dagger} + Y_S^d Y_S^{d \dagger}\right) + {\rm Tr}\left( Y_S^\nu Y_S^{\nu \dagger} + Y_S^e Y_S^{e \dagger} \right) \ , \nonumber\\ %\eqno(4.4) \end{eqnarray} \begin{eqnarray} &&G_S^u= 4 g_1^2 +8 g_3^2 +\frac{3}{2} g_X^2 \ , \ \ \ \nonumber\\ &&G_S^d= g_1^2 +8 g_3^2 +\frac{3}{2} g_X^2 \ , \nonumber \\ &&G_S^\nu= +\frac{3}{2} g_X^2 \ , \ \ \nonumber\\ &&G_S^e= 9 g_1^2 +\frac{3}{2} g_X^2 \ , %\ eqno(4.5) \end{eqnarray} $$H_S^f= Y_S^f Y_S^{f \dagger} \ , \ \ \ \ \ (f=u,d,\nu,e) \ . %\eqno(4.6)$$ The coefficients $T_A^f$, $G_A^f$ and $H_A^f$ in the minimal SUSY model are given in the Appendix. As seen from Eq.~(4.6), since the matrix $H_A^f$ is diagonal on the diagonal basis of $M_F(\Lambda_X)$, the Yukawa coupling constants $Y_S^f(\mu)$ can keep the forms diagonal. Similarly, when we choose the diagonal basis of $M_L(\Lambda_X)$ [$M_R(\Lambda_X)$], the matrices $Y_L^f(\mu)$ [$Y_R^f(\mu)$] keep their forms diagonal. For a model with $g_{2L}(\mu)=g_{2R}(\mu)$ and $Y_L^f (\mu)=Y_L^f (\mu)$ at $\mu=\Lambda_X$, we can assert that $$Y_L^f(\mu) = Y_R^f(\mu) \ , %\eqno(4.7)$$ in the range III ($\Lambda_S < \mu \leq \Lambda_X$), because on the diagonal basis of $Y_L$ we obtain $$16 \pi^2 \frac{d}{dt} \ln \frac{(Y_L^f)_{ii}}{(Y_R^f)_{ii}} =(T_L^f-G_L^f+H_L^f)_{ii} -(T_R^f-G_R^f+H_R^f)_{ii} \ . %\eqno(4.8)$$ The case $g_{2L}=g_{2R}$ is likely in the L-R symmetric model. For convenience, in the numerical evaluation in the present paper, we will take $g_{2L}(\mu)=g_{2R}(\mu)$ in the range III ($\Lambda_S < \mu \leq \Lambda_X$). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Evolution in the ranges I and II} %\vglue.1in In the ranges I and II, all the fermions $F_i$ except for $U_3$ are decoupled from the equation (3.6). In the present section, we will take the diagonal basis of $M_F$. Therefore, it is convenient that we define a spurion $$S=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) \ . %\eqno(4.9)$$ Then, the surviving Yukawa coupling constants $(Y^u_A)_{i3}$ are expressed as $(Y^u_{A} S)_{ij} =(Y^u_A)_{i3}\delta_{3j}$. The evolution of $Y^u_AS$ is still described by the RGE (3.6) by substituting $Y_A^u S$ for $Y_A^f$. Here, the terms $T_A^u$, $G_A^u$ and $H_A^u$ ($A=L,R$) are expressed as follows [$Y_A^f$ ($f=d,e,\nu$) are already absorbed into the operators $K^f$]: $$T_A^u= 3 {\rm Tr}\left( Y_A^u S Y_A^{u\dagger} \right) \ , %\eqno(4.10)$$ $$G_A^u = \frac{17}{8}g_1^2 +\frac{9}{4}g_{2A}^2 +8g_3^2 \ , %\eqno(4.11)$$ $$H_A^u = \frac{3}{2}Y_A^u S Y_A^{u\dagger} \ , %\eqno(4.12)$$ ($A=L,R$) in the range II, and $$T_L^u= 3 {\rm Tr}\left( Y_L^u S Y_L^{u \dagger} \right) \ , %\eqno(4.13)$$ $$G_L^u = \frac{17}{20}g_1^2 +\frac{9}{4}g_{2L}^2 +8g_3^2 \ , %\eqno(4.14)$$ $$H_L^u = \frac{3}{2}Y_L^u S Y_L^{u\dagger} \ , %\eqno(4.15)$$ in the range I. Here, the coupling constant $g_1\equiv g_{1LR}$ in the range II is that for the U(1) operator $(1/2)Y_{LR}$ which is defined by the relation $$Q=I_3^L + I_3^R +\frac{1}{2} Y_{LR} \ , %\eqno(4.16)$$ for the symmetry ${\rm SU(2)}_L \times {\rm SU(2)}_R \times {\rm U(1)}_{LR}$, while the coupling constant $g_1\equiv g_{1Y}$ in the range I is that for the U(1) operator $(1/2)Y$ which is defined by the relation $$Q=I_3^L +\frac{1}{2} Y \ , %\eqno(4.17)$$ for the symmetry ${\rm SU(2)}_L \times {\rm U(1)}_{Y}$, and they are connected by $$\alpha_{em}^{-1}(\Lambda_L)= \alpha_{2L}^{-1}(\Lambda_L) +\frac{5}{3} \alpha_{1LR}^{-1}(\Lambda_L) \ , %\eqno(4.18)$$ $$\frac{5}{3} \alpha_{1Y}^{-1}(\Lambda_R) = \alpha_{2R}^{-1}(\Lambda_R) +\frac{2}{3} \alpha_{1LR}^{-1}(\Lambda_R) \ , %\eqno(4.19)$$ where $\alpha_i=g_i^2/4\pi$. Similarly, the terms $T_K^f$, $G_K^f$ $H_{KL}^f$ and $H_{KR}^f$ ($f=u,d,e$) are given by $$T_K^u=T_K^d=T_K^e=3{\rm Tr}\left( Y_L^u S Y_L^{u\dagger} +Y_R^u S Y_R^{u\dagger} \right) \ , %\eqno(4.20)$$ \begin{eqnarray} G_K^u&=&G_K^d=\frac{5}{2}g_1^2 +\frac{9}{4}g_{2L}^2 +\frac{9}{4}g_{2R}^2 + 8 g_3^2 \ , \nonumber \\ G_K^e&=&\frac{9}{2}g_1^2 +\frac{9}{4}g_{2L}^2 +\frac{9}{4}g_{2R}^2 \ , %\eqno(4.21) \end{eqnarray} $$H_{KA}^u=H_{KA}^d=\frac{3}{2} Y_A^{u} S Y_A^{u\dagger}\ , \ \ H_{KA}^e =0\ , \ \ (A=L,R) \ , %\eqno(4.22)$$ in the range II, and $$T_K^u=T_K^d=T_K^e=3{\rm Tr}\left( Y_L^u S Y_L^{u\dagger} \right) \ , %\eqno(4.23)$$ \begin{eqnarray} G_K^u&=&\frac{17}{20}g_1^2 +\frac{9}{4}g_{2L}^2 + 8 g_3^2 \ , \nonumber \\ G_K^d&=&\frac{5}{20}g_1^2 +\frac{9}{4}g_{2L}^2 + 8 g_3^2 \ ,\\ G_K^e&=&\frac{45}{20}g_1^2 +\frac{9}{4}g_{2L}^2 \ , \nonumber %\eqno(4.24) \end{eqnarray} $$H_{KL}^u= H_{KL}^d= \frac{3}{2} Y_L^{u} S Y_L^{u\dagger}\ , \ \ H_{KR}^f=0 \ , \ \ f=u,d \ ,$$ $$H_{KL}^e= H_{KR}^e=0 \ , %\eqno(4.25)$$ in the range I. The terms $T_K^\nu$, $G_K^\nu$ and $H_{KL}^\nu$ have rather simple forms in contrast with those in the conventional neutrino seesaw model, because the partners of the fermions $f_L$ which couple to the Higgs scalar $\phi_L$ are not $f_R$, but $F_R$ which are already decoupled at $\mu=\Lambda_R$: $$T_K^\nu=6{\rm Tr}\left( Y_L^u S Y_L^{u\dagger} \right) \ , %\eqno(4.26)$$ $$G_K^\nu= 3 g_{2L}^2 \ , %\eqno(4.27)$$ $$H_{KL}^\nu= \lambda_{HL} \ , %\eqno(4.28)$$ in the ranges I and II, where $\lambda_{HL}$ is the coupling constant of the Higgs scalar $\phi_L$ defined by $$H_\phi = \frac{1}{2} \lambda_{HL} (\phi_L^\dagger \phi_L)^2 \ , %\eqno(4.29)$$ and the mass of the physical Higgs scalar $H_L^0$ is given by $$m_{HL}^2 = 2 \lambda_{HL} \langle \phi_L^0 \rangle^2 \ . %\eqno(4.30)$$ The similar coefficients in the minimal SUSY model are given in the Appendix. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newpage \section{Case of the democratic seesaw model} \label{sec:5} %\vglue.1in In the democratic seesaw model, on the diagonal basis of $Y_L^f(\Lambda_X)$ and $Y_R^f(\Lambda_X)$, the Yukawa coupling constants of heavy fermions $Y_S^f(\Lambda_X)$ are given by the democratic form (1.11). Since on this basis the Yukawa coupling constants $Y_S^f$ keep the forms democratic: $$Y_S^f(\mu) = \xi_S^f(\mu) \left( {\bf 1} + 3b_f(\mu) {X}\right) \ , %\eqno(5.1)$$ we will call this basis the democratic basis of $M_F$" hereafter. On the other hand, if we take a basis on which $Y_S^f$ are diagonal, i.e., the matrix forms are given by $$\widetilde{Y}_S^f(\mu) = \xi_S^f(\mu) \left( {\bf 1} + 3b_f(\mu) \widetilde{X} \right) \ , %\eqno(5.2)$$ $$\widetilde{X} = A X A^T ={\rm diag}(0, 0, 1) \ , %\eqno(5.3)$$ $$A= \left( \begin{array}{ccc} \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & - \frac{2}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \end{array} \right) \ . %\eqno(5.4)$$ Especially, on this basis, the Yukawa coupling constants $(\widetilde{Y}_S^e)_{ii}$ and $(\widetilde{Y}_S^u)_{ii}$ of the fermions $E_i$ and $U_i$ satisfy the relations $$[\widetilde{Y}_S^e(\mu)]_{11}= [\widetilde{Y}_S^e(\mu)]_{22} =[\widetilde{Y}_S^e(\mu)]_{33} = \xi_S^e(\mu) \ , %\eqno(5.5)$$ $$[\widetilde{Y}_S^u(\mu)]_{11}= [\widetilde{Y}_S^u(\mu)]_{22}= \xi_S^u(\mu) \ , \ \ \ [\widetilde{Y}_S^u(\mu)]_{33}=0 \ , %\eqno(5.6)$$ in the range III ($\Lambda_S < \mu \leq \Lambda_X$), i.e., $$b_e(\mu)=0 \ , \ \ \ \ b_u(\mu)=-1/3 \ . %\eqno(5.7)$$ On the other hand, on this basis, the Yukawa coupling constants $\widetilde{Y}_L^f(\mu)$ and $\widetilde{Y}_R^f(\mu)$ are not diagonal. However, we can easily obtain their diagonal forms by $A^T \widetilde{Y}_L^f(\mu) A$ and $A^T \widetilde{Y}_R^f(\mu) A$. At the energy scale $\mu=\Lambda_S$, the fermions $F_i$ (except for $U_3$) acquire the heavy masses $(M_F)_{ii}$. Therefore, for $\mu < \Lambda_S$, the operators $K^f$ are given as follows: \end{multicols} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hspace{-0.5cm} \rule{8.7cm}{0.1mm}\rule{0.1mm}{2mm} \widetext $$(K^u)_{ij} = \left[ \widetilde{Y}_L^u (\widetilde{Y}_S^u)^{-1} \widetilde{Y}_R^{u\dagger} \right]_{ij} =\frac{1}{\xi^u_S(\Lambda_S)} \sum_{k=1,2} (\widetilde{Y}_L^u)_{ik} (\widetilde{Y}_R^u)_{jk}^{\ast} \ , %\eqno(5.8)$$ $$(K^d)_{ij} = \left[ \widetilde{Y}_L^d (\widetilde{Y}_S^d)^{-1} \widetilde{Y}_R^{d\dagger} \right]_{ij} =\frac{1}{\xi^d_S(\Lambda_S)}\left( \sum_{k=1,2} (\widetilde{Y}_L^d)_{ik} (\widetilde{Y}_R^d)_{jk}^{\ast} +\frac{1}{1+3b_d(\Lambda_S)} (\widetilde{Y}_L^d)_{i3} (\widetilde{Y}_R^d)_{j3}^{\ast} \right) \ , %\eqno(5.9)$$ $$(K^e)_{ij} = \left[ \widetilde{Y}_L^e (\widetilde{Y}_S^e)^{-1} \widetilde{Y}_R^{e\dagger} \right]_{ij} =\frac{1}{\xi_S^e(\Lambda_S)} \sum_{k=1,2,3} (\widetilde{Y}_L^e)_{ik} (\widetilde{Y}_R^e)_{jk}^{\ast} \ , %\eqno(5.10)$$ $$(K^\nu)_{ij} = \left( \widetilde{Y}_L^\nu (\widetilde{Y}_S^\nu)^{ -1} \widetilde{Y}_L^{\nu T} \right)_{ij} =\frac{1}{\xi_S^\nu(\Lambda_S)} \left( \sum_{k=1,2} (\widetilde{Y}_L^\nu)_{ik} (\widetilde{Y}_L^\nu)_{jk} +\frac{1}{1+3b_\nu(\Lambda_S)} (\widetilde{Y}_L^\nu)_{i3} (\widetilde{Y}_L^\nu)_{j3} \right) \ . %\eqno(5.11)$$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hspace{9.1cm} \rule{-2mm}{0.1mm}\rule{8.7cm}{0.1mm} \begin{multicols}{2} \narrowtext \noindent In Eq.~(5.11), we have assumed that the structure of the Majorana mass term $M_R(\Lambda_S)= Y_S^R(\Lambda_S) \langle\Phi^0\rangle$ for the neutral fermions $N_R$ has a structure similar to the Dirac mass matrices $M_F$ which is given by Eq.~(5.1). Since the Yukawa coupling constants $Y_A^f(\mu)$ ($A=L,R$) in the range III keep their forms diagonal on the democratic basis of $M_F$, it is convenient to express $Y_A^f(\mu)$ as follows, $$Y_A^f(\mu) = \xi_A^f (\mu) Z_A^f (\mu) \ , %\eqno(5.12)$$ where the diagonal matrix $Z_A^f(\mu)$ is given by Eq.~(3.2). Then, the matrix $\widetilde{Y}_A^f$ on the diagonal basis of $M_F$ is given by $$\widetilde{Y}_A^f(\mu) = \xi_A^f(\mu) \widetilde{Z}^f (\mu) \ , %\eqno(5.13)$$ where \end{multicols} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hspace{-0.5cm} \rule{8.7cm}{0.1mm}\rule{0.1mm}{2mm} \widetext $$\widetilde{Z}^f = A Z^f A^T = \frac{1}{6} \left( \begin{array}{ccc} 3(z_2+z_1) & -\sqrt{3} (z_2-z_1) & -\sqrt{6} (z_2-z_1) \\ -\sqrt{3} (z_2-z_1) & 4z_3 +z_2+z_1 & -\sqrt{2} (2z_3-z_2-z_1) \\ -\sqrt{6} (z_2-z_1) & -\sqrt{2} (2z_3-z_2-z_1) & 2(z_3+z_2+z_1) \end{array} \right) \ , %\eqno(5.14)$$ \hspace{9.1cm} \rule{-2mm}{0.1mm}\rule{8.7cm}{0.1mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{multicols}{2} \narrowtext \noindent (we have dropped the indices $A$ and $f$, and for simplicity, we have taken $\delta_i^f=0$). Although the Yukawa coupling constants $\widetilde{Y}_L^u$ and $\widetilde{Y}_R^u$ in the range II and $\widetilde{Y}_L^f$ in the range I have the physical meaning only for the one column matrix components $(\widetilde{Y}_A^f)_{i3}$ ($i=1,2,3$), we still use the expressions (5.12) and (5.13), because the matrix $K^e(\mu)$ ($\Lambda_L <\mu \leq \Lambda_S$) which is proportional to $Y_L^e(\mu) Y_R^{e\dagger}(\mu)$ is still diagonal on the democratic basis of $M_F$ as discussed in Sec.~\ref{sec:4}, so that we regard that $Y_A^u(\mu)$ is also diagonal". Then, the top quark mass $m_t(\mu)$ is approximately expressed as \end{multicols} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hspace{-0.5cm} \rule{8.7cm}{0.1mm}\rule{0.1mm}{2mm} \widetext $$m_t(\mu) \simeq \langle\phi_L^0\rangle \sqrt{ \sum _i |(\widetilde{Y}_L^u(\mu))_{i3}|^2} = \langle\phi_L^0\rangle \xi_L^u(\mu) \sqrt{ \frac{1}{3} \sum_i |z_{Li}^u|^2 } = \frac{1}{\sqrt{3}} \xi_L^u(\mu) \langle\phi_L^0\rangle \ . %\eqno(5.15)$$ \hspace{9.1cm} \rule{-2mm}{0.1mm}\rule{8.7cm}{0.1mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{multicols}{2} \narrowtext \noindent The expression (5.15) is valid in the whole ranges $\Lambda_L < \mu \leq \Lambda_X$. Since $$\sum_{i=1}^3 \sum_{k=1}^2 \left(\widetilde{Z}_{ik}\right)^2 =\frac{2}{3} (z_1^2+z_2^2+z_3^2) = \frac{2}{3} \ , %\eqno(5.16)$$ we obtain $$m_c(\mu)+m_u(\mu) \simeq \frac{2}{3} \frac{\xi^u_L(\mu) \xi^u_R(\mu)} {\xi^u_S(\mu)} \frac{\langle\phi^0_L\rangle \langle\phi^0_R\rangle} {\langle\Phi^0\rangle} \ , %\eqno(5.17)$$ from Eq.~(5.8). Note that the expression (5.17) is valid only in the range III. In the ranges I and II, the ratio $\xi^u_L \xi^u_R/\xi^u_S$ behaves as a operator $K^u(\mu)$ which obeys Eq.~(3.12). {}From Eq.~(5.17), the ratio $m_c/m_t$ is given by $$\frac{m_c(\mu)}{m_t(\mu)} \simeq \frac{2}{\sqrt{3}} \frac{\xi^u_R(\mu)}{\xi^u_S(\mu)} \frac{\langle\phi^0_R\rangle}{\langle\Phi^0\rangle} \ . %\eqno(5.18)$$ Since $H^e_{KL}=H^e_{KR}=0$ in the ranges I and II, the form of $K^e(\mu)$ is invariant in the ranges, i.e., $$Z^e_L(\Lambda_L) Z^{e\dagger}_R(\Lambda_L) = Z^e_L(\Lambda_S) Z^{e\dagger}_R(\Lambda_S) \ , %\eqno(5.19)$$ especially, since $$Z^f_L(\mu) = Z_R^f(\mu) \equiv Z^f(\mu) \ , %\eqno(5.20)$$ for a model with $g_{2R}(\Lambda_R)=g_{2L}(\Lambda_R)$, we obtain $$Z^e(\Lambda_L) = Z^e(\Lambda_S) \ . %\eqno(5.21)$$ Therefore, in preliminary evaluations prior to fixing the final values of the parameters, we will sometimes use the values of $z_i(m_Z)$ which are obtained from the observed charge lepton masses $m^e_i(m_Z)$ by using Eq.~(1.7) instead of the values of $z_i(\Lambda_X)$ which are defined in Eq.~(1.9) as the initial condition at $\mu=\Lambda_X$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newpage %\narrowtext %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Numerical results in the non-SUSY model} \label{sec:6} %\vglue.1in We define % $$\Lambda_L =\langle\phi_L^0\rangle \ , \ \ \Lambda_R =\langle\phi_R^0\rangle \ , \ \ \Lambda_S =\langle\Phi^0\rangle \ . %\eqno(6.1)$$ However, for convenience, in the numerical evaluations, instead of physical quantities at $\mu=\Lambda_L$, we will use those at $\mu=m_Z$ ($m_Z$ is the neutral weak boson mass). First, in order to overlook the behavior of the Yukawa coupling constant $Y_L^f(\mu)$, we illustrate the behavior of $\xi_L^u(\mu)$ in the non-SUSY model in Fig.~\ref{xi-nonSUSY}. Here, we have used the approximate relation (5.15) and the input values $m_t(m_Z)=181$ GeV and $\langle \phi_L^0 \rangle = 174$ GeV: $$\xi_L^u(m_Z)={\sqrt{3}} {\frac{m_t(m_Z)}{\langle\phi_L^0\rangle}}=1.80. %\eqno(6.2)$$ In other words, the behavior of $\xi^u_L(\mu)$ corresponds to that of $m_t(\mu)$ because of $\xi^u_L(\mu)=(m_t(\mu)/m_t(m_Z))\xi^u_L(m_Z)$. In the ranges I and II, since the terms $T_L^u$ and $H_L^u$ are expressed only in terms of $\widetilde{Y}_L^u S\widetilde{Y}_L^{u\dagger}$, the evolution of the factor $|\xi_L^u(\mu)|^2= 3 {\rm Tr}[ \widetilde{Y}_L^u S \widetilde{Y}_L^{u\dagger} ]$ is described by the equation $$16\pi^2 \frac{d}{dt} |\xi_L^u|^2 =2 \left[ \left( \frac{1}{3} |\xi_L^u|^2 -G_L^u\right) |\xi_L^u|^2 + \frac{1}{2} |\xi_L^u|^4 \right] \ . %\eqno(6.3)$$ However, in the range III, the terms $T_L^u$ and $H_L^u$ contain other factors $Y_L^f Y_L^{f\dagger}$ in addition to $Y_L^u Y_L^{u\dagger}$, so that the evolution of $\xi_L^u$ cannot be expressed so simply such as (6.3). For the evaluation of $\xi_L^u$ in the range III, we have tentatively substituted the values $z_i(m_Z)$ given by (1.7) for the initial values $z_i(\Lambda_X)$. For simplicity, as we discussed in (4.7), we have taken as $g_{2L}(\Lambda_R) =g_{2R}(\Lambda_R)$. In Fig.~\ref{xi-nonSUSY}, the ratio $\Lambda_S/\Lambda_R$ has been taken as $\Lambda_S/\Lambda_R=107$, which has determined from the fitting of the observed ratio $m_t/m_c$ as we discuss later. The behavior of $\xi^u_L(\mu)$ is insensitive to the ratio $\Lambda_S/\Lambda_R$. As seen in Fig.~\ref{xi-nonSUSY}, a case with a lower $\Lambda_S$ ($\Lambda_S < 10^5$ GeV) causes a burst of $\xi^u_L(\mu)$, so that the case is ruled out. On the other hand, a case with a higher $\Lambda_S$ ($\Lambda_S > 10^{19}$ GeV) causes $\alpha_1(\mu)\rightarrow \infty$ at $\mu\rightarrow \Lambda_S$, so that the case is also ruled out. Taking account of the behavior of $\xi^u_L(\mu)$ shown in Fig.~\ref{xi-nonSUSY}, as a trial, we take $$\Lambda_X = 2\times 10^{16} \ {\rm GeV} \ , %\eqno(6.4)$$ which is known as the unification energy scale in the minimal SUSY model. (However, in the present paper, we do not consider the gauge unification.) As a value of $\Lambda_S$, we tentatively take $$\Lambda_S = 3\times 10^{13} \ {\rm GeV} \ , %\eqno(6.5)$$ which leads to the mass-squared difference $\Delta{m}_{32}^2\equiv{m}_{{\nu}_3}^2-m_{{\nu}_2}^2\sim (10^{-3}-10^{-2}){\rm eV}^2$ as we demonstrate later. For the values (6.4) and (6.5), we obtain $\xi^u_L(\Lambda_X)=1.2$. Next, we determine the values of $\xi^u_S(\Lambda)$ and $\Lambda_S/\Lambda_R$. Since we have already obtained the value $\xi^u_L(\Lambda_X)=1.2$, it seems that we can fix the value of $\xi^u_S(\Lambda) \Lambda_S/\Lambda_R$ from the observed value of $m_t(m_Z)/m_c(m_Z)$ because of the relation (5.18). However, the value of $\xi^u_S(\Lambda_X)$ [also $\xi^f_S(\Lambda_X)$] is sensitive to the value of $\xi^u_S(\Lambda_S)$ [$\xi^f_S(\Lambda_S)$] [in other words, a small deviation of $\xi^f_S(\Lambda_S)$ causes a large deviation of $\xi^f_S(\Lambda_X)$]. Therefore, we cannot fix the values $\xi^u_S(\Lambda_X)$ unless we put a tentative model for $\xi^f_{LR}$ and $\xi^f_S$. The basic assumption in the universal seesaw model is to consider that the mass matrices $m_L$ and $m_R$ in Eq.~(1.1) are universal" (common) for all fermion sectors (quarks and leptons). Therefore, we put the following initial condition $$\xi^u_{LR}(\Lambda_X)=\xi^d_{LR}(\Lambda_X)=\xi^e_{LR}(\Lambda_X)= \xi^\nu_{LR}(\Lambda_X)\equiv \xi_{LR}(\Lambda_X) \ . %\eqno(6.6)$$ Then, a model with $\xi^u_S(\Lambda_X)=\xi^d_S(\Lambda_X)= \xi^e_S(\Lambda_X)$ is obviously ruled out because we cannot give the observed values of quark and charged lepton masses simultaneously. We must consider $$\xi^u_S(\Lambda_X)=\xi^d_S(\Lambda_X) \equiv \xi^q_S(\Lambda_X) \neq \xi^e_S(\Lambda_X) \ . %\eqno(6.7)$$ We tentatively put $\xi^e_S(\Lambda_X)=\xi_{LR}(\Lambda_X)$. The numerical results are as follows: $$\xi_{LR}(\Lambda_X)=\xi^e_S(\Lambda_X)=1.20 \ , \ \ \ \xi^q_S(\Lambda_X)=0.80 \ , %\eqno(6.8)$$ $$\Lambda_S/\Lambda_R = 107 \ , %\eqno(6.9)$$ $$z_1=0.01617\ , \ \ z_2=0.2349\ , \ \ z_3=0.9719\ . %\eqno(6.10)$$ In the quark and charged lepton mass expressions (3.19) the factors $\xi^e_S$ and $\xi^q_S$ appear only in terms of the combinations $\xi^q_S \Lambda_S$ and $\xi_S^e \Lambda_S$, respectively, so that the absolute values of $\xi^e_S$ and $\xi^q_S$ depend on the choice of the input value of $\Lambda_S$. Only the ratio $\xi^e_S/\xi^q_S$ is substantial for the fitting of the quark and charged lepton mass. (However, as we state in the Sec.~\ref{sec:8}, the neutrino mass difference between $m_{\nu 3}$ and $m_{\nu 2}$ rapidly varies in the range III. Therefore, in the neutrino mass matrix, the choice of the input value $\Lambda_S$ is important.) We can obtain $$\xi^e_S(\Lambda_X)/\xi^q_S(\Lambda_X) \simeq 1.5 \ , %\eqno(6.11)$$ for any initial values of $\xi^e_S(\Lambda_X)$ with $O(1)$. The values (6.10) are nearly in agreement with the values $z_1=0.01622$, $z_2=0.2357$, and $z_3=0.9717$ which are obtained from Eq.~(1.7) at $\mu=m_Z$. We can see that the effect of the evolution is not so large for $Z^e$. The value of the parameter $b_d(\Lambda_X)$ is determined from the fitting of the observed down-quark mass ratios $m_d/m_s$ and $m_s/m_b$ and the CKM matrix parameter $|V_{us}(m_Z)|=0.22$. In Fig.~\ref{bd-nonSUSY}, we illustrate the mass ratios $m_d(\mu)/m_s(\mu)$ and $m_s(\mu)/m_b(\mu)$ and the CKM parameter $|V_{us}(\mu)|$ at $\mu=m_Z$ versus the parameters $b_d$ and $\beta_d$, where we have re-defined the complex parameter $b_d$ by $b_d e^{i\beta_d}$ with two real parameters. For convenience, in Fig.~\ref{bd-nonSUSY}, the quantities are expressed in the unit of the corresponding observed values at $\mu=m_Z$ (for example, in Fig.~\ref{bd-nonSUSY}, the curve $m_d/m_s$ denotes $[m_d(\mu)/m_s(\mu)]_{\mu=m_Z} /[m_d/m_s]_{observed}$). We obtain $$b_d(\Lambda_X)=-1.20 \ , \ \ \ \beta_d(\Lambda_X)=19.2^\circ \ , %\eqno(6.12)$$ which give the following predictions at $\mu=m_Z$: \begin{eqnarray} &&m_u(m_Z)=2.60 \times 10^{-3} \ {\rm GeV} \ , \ \ \nonumber\\ &&m_c(m_Z)=6.92 \times 10^{-1} \ {\rm GeV}\ , \nonumber\\ &&m_t(m_Z)=182 \ {\rm GeV} \ , \nonumber \\ &&m_d(m_Z)=4.38 \times 10^{-3} \ {\rm GeV} \ , \nonumber\\ &&m_s(m_Z)=9.84 \times 10^{-2} \ {\rm GeV}\ , \\ &&m_b(m_Z)=3.02 \ {\rm GeV} \ , \nonumber \\ &&m_e(m_Z)=4.90 \times 10^{-4} \ {\rm GeV} \ , \nonumber\\ &&m_\mu(m_Z)=1.03 \times 10^{-1} \ {\rm GeV}\ , \nonumber\\ &&m_\tau(m_Z)=1.76 \ {\rm GeV} \ . \nonumber %\eqno(6.13) \end{eqnarray} The experimental values corresponding to the results (6.13) are as follows \cite{qmass}: \begin{eqnarray} && m_u(m_Z)=(2.33^{+0.42}_{-0.45}) \times 10^{-3} \ {\rm GeV} \ , \nonumber\\ && m_c(m_Z)=(6.85^{+0.56}_{-0.61}) \times 10^{-1} \ {\rm GeV}\ , \nonumber\\ && m_t(m_Z)=(181\pm 13) \ {\rm GeV} \ , \nonumber\\ && m_d(m_Z)=(4.69^{+0.60}_{-0.66}) \times 10^{-3} \ {\rm GeV} \ , \nonumber\\ && m_s(m_Z)=(0.934^{+0.118}_{-0.130}) \times 10^{-1} \ {\rm GeV}\ , \\ && m_b(m_Z)=(3.00 \pm 0.11) \ {\rm GeV} \ , \nonumber\\ && m_e(m_Z)=(4.8684727\pm 0.00000014) \times 10^{-4} \ {\rm GeV} \ , \nonumber\\ && m_\mu(m_Z)=(1.0275138 \pm 0.0000033) \times 10^{-1} \ {\rm GeV}\ , \nonumber\\ && m_\tau(m_Z)=(1.7467 \pm 0.0003) \ {\rm GeV} \ .\nonumber %\eqno(6.14) \end{eqnarray} The results (6.13) is in agreement with the observed values (6.14) within the experimental errors. The predicted values of $|V_{ij}|$ depends on the phase parameters $\delta_i^f$ given by Eq.~(1.4). Only when we take those as (1.8) (at $\mu=\Lambda_X$), we can obtain reasonable values of $|V_{ij}|$. For example, for $\delta_3^u -\delta_3^d=\pi$, we obtain the predictions at $\mu=m_Z$ \begin{eqnarray} && |V_{us}|=0.220 \ , \ \ |V_{cb}|=0.0668 \ , \nonumber\\ && |V_{ub}/V_{cb}|=0.0558 \ , \nonumber\\ && |V_{td}|=0.0177 \ , \\ && J=3.25 \times 10^{-5} \ . \nonumber %\eqno(6.15) \end{eqnarray} The observed values \cite{PDG00} are \begin{eqnarray} &&|V_{us}|=0.2196\pm 0.0023 \ , \ \ \nonumber\\ &&|V_{cb}|=0.0402\pm 0.0019 \ , \ \ \\ &&|V_{ub}/V_{cb}|=0.090 \pm 0.025 \ . \nonumber %\eqno(6.16) \end{eqnarray} Although the results (6.15) are roughly consistent with experiments, the value $|V_{cb}|=0.066$ is somewhat large compared with the observed value $|V_{cb}|=0.040$. This discrepancy can be adjusted by considering a small deviation from $\pi$ of the relative phase $\delta_3^u -\delta_3^d$ as demonstrated in Ref.~\cite{KFptp}. Related to the phenomenological requirement (1.8), it is interesting to consider that $Y_L^u$ which is the coefficient of the Higgs scalar $\widetilde{\phi}_L$ is related to $Y_L^d$ which is the coefficient of the scalar $\phi_L$ as $$Y_L^u(\Lambda_X) =[Y_L^d(\Lambda_X)]^\dagger \ . %\eqno(6.17)$$ Then, the relations (1.8) mean that $(Y_L^f)_{11}$ and $(Y_L^f)_{22}$ are real, while $(Y_L^f)_{33}$ is almost pure imaginary. We take $$(Z^u)^\dagger = Z^d ={\rm diag}(z_1, z_2, z_3 e^{i\delta_3}) \ . %\eqno(6.18)$$ The parameter $\delta_3$ ($=\delta_3^d =-\delta_3^u$) does not affect the masses, but only the CKM mixings. It is interesting to consider that the parameter $\delta_3(\Lambda_X)$ takes its value such as the CKM mixings become minimum, i.e., such as the value $\sum_{i\neq j}|V_{ij}(\Lambda_X)|^2$ takes the minimum. This requirement gives the initial value $\delta_3(\Lambda_3)=84^\circ$ (see Fig.~\ref{d3-vinjx}). Then, we obtain the predictions of $|V_{ij}|$ at $\mu=m_Z$ \begin{eqnarray} && |V_{us}| = 0.220 \ , \ \ |V_{cb}| =0.0418 \ , \nonumber \\ && |V_{ub}/V_{cb}|=0.0726 \ ,\nonumber\\ && |V_{td}|=0.0109 \ , \\ && J=2.38 \times 10^{-5} \ . \nonumber % %\eqno(6.19) \end{eqnarray} which is in excellent agreement with the experimental values (6.16). In Fig.~\ref{d3-vij}, we illustrate the predicted values $|V_{ij}(m_Z)|$ versus $\delta_3(\Lambda_X)$. As seen in Fig.~\ref{d3-vij}, the value of $\delta_3(\Lambda_X)$ at which $\sum_{i\neq j}|V_{ij}(\Lambda_X)|^2$ takes the minimum also gives the minimum of the CKM mixings at $\mu=m_Z$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Numerical results in the SUSY model} \label{sec:7} %\vglue.1in The behavior of $\xi_L^u(\mu)$ in the SUSY model is somewhat different from that in the non-SUSY model. Since in the SUSY model, the top quark mass $m_t(\mu)$ is given by $$m_t(\mu) = \frac{1}{\sqrt{3}} \xi^u_L(\mu) \frac{v_L}{\sqrt{2}} \sin\beta \ , %\eqno(7.1)$$ where $v_L/\sqrt{2}= 174$ GeV and $\tan\beta\equiv \tan\beta_L =v_L^u/v_L^d$, the initial value of $\xi^u_L(m_Z)$ in the SUSY model corresponds to $$\left[\xi^u_L(m_Z)\right]_{SUSY}=\left[\xi^u_L(m_Z)\right]_{nonSUSY} \frac{1}{\sin\beta} \ . %\eqno(7.2)$$ However, this does not mean $[\xi^u_L(\Lambda_X)]_{SUSY}= [\xi^u_L(\Lambda_X)]_{nonSUSY}/{\sin\beta}$, because the behavior of $[\xi^u_L(\mu)]_{SUSY}$ is considerably different from that of $[\xi^u_L(\mu)]_{nonSUSY}$. In Fig.~\ref{xi-SUSY}, we illustrate the behavior of $\xi_L^u(\mu)$ in the SUSY model for the case of $\tan\beta=3.5$. If we take $\tan\beta < 2.5$, the initial value of $\xi_L^u(m_Z)$ becomes $\xi_L^u(m_Z,\tan\beta>2.5) > \xi_L^u(m_Z,\tan\beta=2.5)$ from Eq.~(7.2), so that the curve of $\xi_L^u(\mu)$ will be illustrated in the upper side of the curve given in Fig.~\ref{xi-SUSY}. Therefore, a case with a small value of $\tan\beta$ yields a burst of $\xi^u_L(\mu)$ at a relatively lower energy scale. We consider that the model should be calculable perturbatively, so that a case with such a large value of $\xi_L^u$ should be ruled out. As seen in Fig.~\ref{xi-SUSY}, since the model gives, in general, $\xi_L^u(\Lambda_X)>\xi_L^u(\mu)$ ($m_Z < \mu < \Lambda_X$), the value $\xi_L^u(\Lambda_X)$ should, at least, be $[\xi_L^u(\Lambda_X)]^2/4\pi <1$, i.e., $\xi_L^u(\Lambda_X) < \sqrt{4\pi} = 3.54$. However, when we take contributions from the higher order corrections into consideration, even the value $\xi_L^u(\Lambda_X)=3.0$ is still dangerous. Therefore, we put the constraint $\xi_L^u(\Lambda_X)=2.0$ for the results of the present one loop calculation. In Fig.~\ref{mt-tanb}, we illustrate the predicted value of $m_t(m_Z)$ for the initial values $\xi_L^u(\Lambda_X)=\sqrt{4\pi}=3.54$ and $\xi_L^u(\Lambda_X) \alt 2.0$, where we have used the input values $$\Lambda_X=2\times 10^{16} \ {\rm GeV} \ , \ \ \Lambda_S=6\times 10^{13} \ {\rm GeV} \ . %\eqno(7.3)$$ The value of $\Lambda_S$ has been chosen as the neutrino mass-squared difference $\Delta m_{32}^2$ is of the order of $(10^{-3}-10^{-2})$ eV$^2$. {}From Fig.~\ref{mt-tanb}, we conclude that the value of $\tan\beta$ must be $$\tan\beta \agt 3 \ . %\eqno(7.4)$$ Prior to the numerical investigation of the evolutions in the SUSY model, in order to see the difference between the parameter structures in the non-SUSY and SUSY models, let us give a rough sketch for the parameters in the case of the SUSY model by neglecting the evolution effects. The quark mass matrices $M_u$ and $M_d$ are given by $$\left( M_u \right)_{ij} =\sum_{k=1}^2 \left(\widetilde{Z}\right)_{ik} \left(\widetilde{Z}\right)_{jk} \left(O^u\right)_{kk} \frac{\xi^u_L \xi^u_R}{\xi^u_S} \frac{\Lambda_L \Lambda_R}{\Lambda_S} \sin\beta \ , %\eqno(7.5)$$ $$\left( M_d \right)_{ij} =\sum_{k=1}^3 \left(\widetilde{Z}\right)_{ik} \left(\widetilde{Z}\right)_{jk} \left(O^d\right)_{kk} \frac{\xi^d_L \xi^d_R}{\xi^d_S} \frac{\Lambda_L \Lambda_R}{\Lambda_S} \cos\beta \ , %\eqno(7.6)$$ where $O^u={\rm diag}(1,1)$, $O^d={\rm diag}(1,1, 1/(1+3b_d))$, and $\widetilde{Z}$ is given by Eq.~(5.14). Here, for simplicity, we have assumed $\beta_L =\beta_R =\beta_S \equiv \beta$. For $\tan\beta > 3$, the factors $\sin\beta$ and $\cos\beta$ are approximated as $\sin\beta\simeq 1$ and $\cos\beta \simeq 1/\tan\beta$, respectively. Obviously, the model with $\xi_S^u =\xi_S^d$ in addition to the constraint $$\xi_{LR}^u (\Lambda_X) = \xi_{LR}^d (\Lambda_X) \equiv \xi_{LR}^q (\Lambda_X) \ , %\eqno(7.7)$$ is ruled out, because we cannot fit the up- and down-quark masses simultaneously due to the existence of the factor $\cos\beta$. Therefore, we must consider a model with $\xi_S^u \neq \xi_S^d$ differently from the constraint (6.6) in the non-SUSY model. If we consider $$\xi_S^u \simeq \xi_S^q \sin\beta \ , \ \ \ \xi_S^d \simeq \xi_S^q \cos\beta \ , %\eqno(7.8)$$ then the model becomes similar to the case of the non-SUSY model, because $$\frac{\xi^u_L \xi^u_R}{\xi^u_S} \frac{\Lambda_L \Lambda_R}{\Lambda_S} \sin\beta \simeq \frac{\xi^d_L \xi^d_R}{\xi^d_S} \frac{\Lambda_L \Lambda_R}{\Lambda_S} \cos\beta \ , %\eqno(7.9)$$ and we will obtain reasonable fittings for the quark masses and CKM matrix parameters as well as in the non-SUSY model. Note that for a large value of $\tan\beta$, the value of $K^d\equiv \xi_L^d \xi_R^d/\xi_S^d$ becomes large because $K^d \simeq K^u \tan\beta$ from the relation (7.9), so that we cannot evaluate the RGE (3.12) perturbatively. We must take the value of $\tan\beta$ near to the lower bound given by Eq.~(7.4). When we take the evolution effects into consideration, the situation is further complicated. The evolutions of $\xi_L^f(\mu)$, $\xi_R^f(\mu)$ and $\xi_S^f(\mu)$ in the SUSY model are quite different from those in the non-SUSY model. We illustrate the behaviors of $m_i^f(\mu)/m_i^f(\Lambda_X)$ which correspond to the behaviors of $[\xi_L^f(\mu)\xi_R^f(\mu) /\xi_S^f(\mu)]/[\xi_L^f(\Lambda_X)\xi_R^f(\Lambda_X) /\xi_S^f(\Lambda_X)]$ in the non-SUSY model and those in the SUSY model in Figs.~\ref{m-nonSUSY} and \ref{m-SUSY}, respectively. In Fig.~\ref{m-SUSY}, we see that the values $m_u(\mu)$ and $m_c(\mu)$ cause rapid changes in the range III. In the non-SUSY model, the charged lepton mass ratios are almost invariant, i.e., $m_e(\mu)/m_\mu(\mu) \simeq {\rm constant}$ and $m_\mu(\mu)/m_\tau(\mu)\simeq {\rm constant}$, while, in the SUSY model, the mass ratio $m_\mu(\mu)/m_\tau(\mu)$ shows a considerable change (although $m_e(\mu)\simeq m_\mu(\mu)$ still holds). The situation is critical for the input values. If we adhere to the input value $m_t(m_Z)=181$ GeV, then it is hard to obtain reasonable values of the other quark mass values $m_c$, $m_u$, $m_b$, $m_s$ and $m_d$ for any parameter values of $\Lambda_S/\Lambda_R$ and $b_d$. However, if we take a slightly lower value of $m_t(m_Z)$, for example, $m_t(m_Z)=168$ GeV [cf. $[m_t(m_Z)]_{observed} =181 \pm 13$ GeV], we can find the following parameter values $$\tan\beta = 3.5 \ , \ \ \ \ \Lambda_S/\Lambda_R =38 \ , %\eqno(7.10)$$ $$z_1=0.01449 \ , \ \ z_2=0.2117 \ , \ \ z_3=0.9772 \ , %\eqno(7.11)$$ $$\xi_{LR}^u (\Lambda_X) = \xi_{LR}^d (\Lambda_X) \equiv \xi_{LR}^q (\Lambda_X) = 1.3 \ , \ \ \xi_{LR}^e (\Lambda_X) = 1.0 \ , %\eqno(7.12)$$ $$\xi_{S}^u (\Lambda_X) = 1.7 \ , \ \ \xi_{S}^d (\Lambda_X) = 0.50 \ , \ \ \xi_{S}^e (\Lambda_X) = 1.0 \ , %\eqno(7.13)$$ $$b_d = -1.2 \ , \ \ \ \beta_d = 19.4^\circ \ , %\eqno(7.14)$$ which leads to the following quark and charged lepton masses and CKM matrix parameters: \begin{eqnarray} &&m_u(m_Z)=2.47 \times 10^{-3} \ {\rm GeV} \ , \nonumber \\ &&m_c(m_Z)=6.46 \times 10^{-1} \ {\rm GeV}\ , \nonumber \\ &&m_t(m_Z)=167 \ {\rm GeV} \ , \nonumber \\ &&m_d(m_Z)=4.49 \times 10^{-3} \ {\rm GeV} \ , \nonumber \\ &&m_s(m_Z)=1.00 \times 10^{-1} \ {\rm GeV}\ , \\ &&m_b(m_Z)=2.83 \ {\rm GeV} \ , \nonumber \\ &&m_e(m_Z)=4.87 \times 10^{-4} \ {\rm GeV} \ , \nonumber \\ &&m_\mu(m_Z)=1.03 \times 10^{-1} \ {\rm GeV}\ , \nonumber \\ &&m_\tau(m_Z)=1.75 \ {\rm GeV} \ ,\nonumber \end{eqnarray} %$$%\eqno(7.15)$$ \begin{eqnarray} &&|V_{us}|=0.220 \ , \ \ |V_{cb}|=0.0665 \ , \ \ \nonumber\\ &&|V_{ub}/V_{cb}|=0.0603 \ , \nonumber\\ &&|V_{td}|=0.0179 \ , \\ &&J=3.38 \times 10^{-5} \ .\nonumber \end{eqnarray} %$$\eqno(7.16)$$ The values $|V_{ij}|^2$ are again desirably adjustable by the phase parameter $\delta_3$ defined by (6.18). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Evolution of the Neutrino Mass Matrix} \label{sec:8} %\vglue.1in The evolution of the neutrino mass matrix $M_\nu=K^\nu \langle \phi_L^0\rangle^2/\langle\Phi^0\rangle$ is described by the RGE (3.13). Since the coefficient $H_{KL}^\nu$ in the ranges I and II is given by $H_{KL}^\nu=\lambda_{HL}$, (4.28), for the non-SUSY model, and by $H_{KL}^\nu=0$, (A.15) and (A.24), for the SUSY model, the form of the matrix $K^\nu$ at $\mu=\Lambda_L$ does not vary from that at $\mu=\Lambda_S$, so that the mass ratios and mixing matrix $U_\nu$ are also invariant. Since the coefficients $H_{KL}^e$ and $H_{KR}^e$ in the charged lepton sector are given by $H_{KL}^e=H_{KR}^e=0$ in the ranges I and II for the non-SUSY and SUSY models, the form of the charged lepton mass matrix $M_e$ is also invariant below $\mu=\Lambda_S$. Therefore, the Maki-Nakagawa-Sakata (MNS) \cite{MNS} matrix $U=U_{eL}^\dagger U_\nu$ is invariant in the ranges I and II. Note that in the conventional model, the neutrino seesaw mass matrix can vary the from. The neutrino mass matrix in the present model can vary the form only in the range III ($\Lambda_S < \mu \leq \Lambda_X$). The reason is that in the conventional model the scalar $\phi_L^+$ couples to $\overline{\nu}_L e_R$, while that in the present model couples to $\overline{\nu}_L E_R$, so that the contribution of $\phi_L$ to $H_{KL}^\nu$ in the latter case is decoupled below $\mu=\Lambda_S$. For the numerical study, the case with $b_\nu=-1/2$ is most interesting, because the inverse matrix of $Y_S^\nu(\Lambda_X)=\xi_S^\nu(\Lambda_X) [{\bf 1} + 3b_\nu(\Lambda_X) X]$ with $b_\nu(\Lambda_X) =-1/2$ has the form $$\left[ Y_S^\nu(\Lambda_X) \right]^{-1} = -\frac{1}{\xi_S^\nu(\Lambda_X) } \left( \begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} \right) \ , %\eqno(8.1)$$ so that $$Y_L^\nu (Y_S^\nu)^{-1}(Y_L^\nu)^T = - \frac{\xi_L^\nu(\Lambda_X)]^2}{\xi_S^\nu(\Lambda_X)} \left( \begin{array}{ccc} 0 & z_1 z_2 & z_1 z_3 \\ z_1 z_2 & 0 & z_2 z_3 \\ z_1 z_3 & z_2 z_3 & 0 \\ \end{array} \right) \ . %\eqno(8.2)$$ The form (8.2) is well known as the Zee-type \cite{Zee} mass matrix, which can lead to a large mixing \cite{mixing-Zee}. The eigenstates $m_{\nu i}$ and mixing matrix $U$ at $\mu=\Lambda_X$ are given by \cite{nu-Koide} \begin{eqnarray} &&m_{\nu 1}\simeq -2 z_1 m_0^\nu \ , \nonumber \\ &&m_{\nu 2}\simeq - z_2\left( 1-\frac{1}{2} z_2^2\right) + z_1^2 \ , \\ &&m_{\nu 3} \simeq z_2\left( 1-\frac{1}{2} z_2^2\right) + z_1^2 \ , \nonumber \end{eqnarray} %$$%\eqno(8.3)$$ $$m_0^\nu= \frac{(\xi_L^\nu)^2}{\xi_X^\nu} \frac{\Lambda_L^2}{\Lambda_S} \ , %\eqno(8.4)$$ $$U=\left( \begin{array}{ccc} 1 & \frac{1}{\sqrt{2}} \frac{z_1}{z_2} (1-z_2) & \frac{1}{\sqrt{2}} \frac{z_1}{z_2} (1+z_2) \\ -\frac{z_1}{z_2} & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ -z_1 & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array} \right) \ . %\eqno(8.5)$$ The model with $b_d=-1/2$ gives highly degenerate mass-squared levels $m_{\nu 2}^2 \simeq m_{\nu 3}^2$ and a large mixing between $\nu_\mu$ and $\nu_\tau$ at $\mu=\Lambda_X$. Therefore, the model has a possibility that it can give a reasonable explanation for the atmospheric neutrino data \cite{nu-atm}. In Figs.~\ref{dm-nonSUSY} and \ref{dm-SUSY}, we illustrate the behaviors of the mass-squared differences $\Delta m_{ij}^2=m_i^2 -m_j^2$ in the non-SUSY and SUSY models, respectively. As seen in Figs.~\ref{dm-nonSUSY} and \ref{dm-SUSY}, the mass-squared difference $\Delta m_{32}^2$ rapidly increase according as the energy scale decreases in the range III. The numerical results are given in Table \ref{T-dm}. We can see that the neutrino mass ratios are invariant in the ranges I and II. In general, since the mixing angle $\theta_{23}$ is given by $$\sin 2\theta_{23} \simeq \frac{2 (M_\nu)_{23}}{ m_{\nu 3}-m_{\nu 2}} \ , %\eqno(8.6)$$ the mixing angle $\theta_{23}$ in the conventional democratic type neutrino mass matrix model is sensitive to the energy scale \cite{dem-nu-evol}, because $\Delta m_{32}^2(\mu)$ has a large energy scale dependency. In contrast to the conventional model, the mixing angle $\theta_{23}$ in the present model does not so drastically vary. The reason is as follows: the neutrino mass matrix $M_\nu$ in the present democratic" seesaw model is not democratic, i.e., the form of $M_\nu$ is given by Eq.~(8.2). In fact, the present model gives not $m_{\nu 2}\simeq m_{\nu 3}$, but $m_{\nu 2}\simeq -m_{\nu 3}$, so that the evolution effect on $U_{23}$ is not so sensitive as seen in Eq.~(8.6). As seen in Table \ref{T-dm}, the model can fit the value $\Delta m^2_{32}$ to the atmospheric neutrino data \cite{nu-atm} $(\Delta m_{32}^2)_{observ}=3.2 \times 10^{-3}$ eV$^2$ by adjusting the value of $\Lambda_S$, but it cannot give any explanation of the solar neutrino data \cite{nu-solar}, because of $\Delta m_{21}^2 \gg \Delta m_{32}^2 \equiv \Delta m_{atm}^2$. We must introduce a further mechanism for the explanation of the solar neutrino data, for example, as discussed in Ref.~\cite{nu-DUSM}. %The present model gives $\sin^2 2\theta_{12} \simeq 0.02$. %If we want to explain the LSND data \cite{LSND} by the %$\nu_e$-$\nu_\mu$ mixing, the value of $\Delta m_{21}^2 %\simeq 0.3$ is required corresponding to the value %$\sin^2 2\theta_{12} \simeq 0.02$. %The ratio $\Delta m_{32}^2/\Delta m_{21}^2$ rapidly %decreases according as the energy scale decreases %at the range III. %If we take a value of $\Lambda_S$ larger than the %tentative value $\Lambda_S=10^{13}$ GeV, we will can %reduce the value of $\Delta m_{32}^2/\Delta m_{21}^2$ %from the value of $\Delta m_{32}^2/\Delta m_{21}^2$ %for $\Lambda_S=10^{13}$ GeV. However, since the purpose of the present model is not to propose a plausible neutrino mass matrix model in the framework of the USM, but to see the characteristic features of the neutrino mass matrix evolution in contrast to the conventional seesaw model. Therefore, we do not touch the numerical fitting furthermore. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusions} \label{sec:9} %\vglue.1in In conclusion, we have investigated the evolutions of the quark and lepton mass matrices $M_f$ ($f=u,d,\nu,e$) in the universal seesaw mode with ${\rm det}M_F=0$ in the up-quark sector $F=U$. The quark and charged lepton mass matrices $M_f$ ($f=u,d,e$) and neutrino mass matrix $M_\nu$ are given by $$M_f=Y_L^f (Y_S^f)^{-1}(Y_R^f)^\dagger (\Lambda_L\Lambda_R/\Lambda_S) \ , %\eqno(9.1)$$ $$M_\nu = Y_L^\nu (Y_S^\nu)^{-1}(Y_L^\nu)^T (\Lambda_L^2/\Lambda_S) \ , %\eqno(9.2)$$ except for the top quark, where $\Lambda_L=\langle\phi^0_L\rangle$, $\Lambda_R=\langle\phi^0_R\rangle$ and $\Lambda_S=\langle\Phi^0\rangle$. The evolutions below $\mu=\Lambda_S$ are described by the RGE (3.12) and (3.13) for the seesaw operators. On the other hand, the top quark mass $m_t(\mu)$ given by the expression (3.18) is still described by RGE (3.6) for the Yukawa coupling constants below $\mu=\Lambda_S$. Although the heavy fermions $F$ do not contribute to the evolutions below $\mu=\Lambda_S$, the third family would-be" heavy up-quark $U_3$ can contribute to the RGE even below $\mu=\Lambda_S$. However, as far as the $H_{KL}^f$ ($F=\nu, e$) and $H_{KR}^e$ terms in the lepton sectors are concerned, the would-be heavy quark $U_3$ cannot contribute to those, so that the forms of the mass matrices $M_\nu(\mu)$ and $M_e(\mu)$ are invariant below $\mu=\Lambda_S$. For a model with $g_{2L}(\mu)=g_{2R}(\mu)$ and $Y_L^f (\mu)=Y_R^f (\mu)$ at $\mu=\Lambda_X$, we can assert $Y_L^f(\mu) = Y_R^f(\mu)$ in the range III ($\Lambda_S < \mu \leq \Lambda_X$) as we have shown in Eq.~(4.8). Therefore, for simplicity, for the numerical evaluations, we adopt the left-right symmetric model with $g_{2L}(\Lambda_X)=g_{2R}(\Lambda_X)$. As a more concrete model of the USM, we interest in the democratic USM \cite{KFzp,KFptp}. In the model, the mass matrices $M_F$ are given by a simple form (5.1), i.e., by the form  the unit matrix pulse a democratic matrix", whose forms are invariant under the evolution in the range III. In this model, the top quark mass $m_t(\mu)$ is given by (5.15). The behavior of $m_t(\mu)$, i.e., $\xi_L^u(\mu)$, is given in Figs.~\ref{xi-nonSUSY} and \ref{xi-SUSY} for the non-SUSY and SUSY models, respectively. We can obtain the constraint on the values of the intermediate energy scales $\Lambda_R$ and $\Lambda_S$ by considering that the model should be calculable perturbatively. In the non-SUSY model, since $\Lambda_S/\Lambda_R \sim 10^2$ from the ratio $m_t/m_c$, we find the constraint $$10^{10}\ {\rm GeV} < \Lambda_S < 10^{19}\ {\rm GeV} \ , %\eqno(9.3)$$ for $\Lambda_X \sim 10^{16}$ GeV. In the SUSY model, the results highly depend on the input parameter $\tan\beta$. {}From the numerical study, we obtain the constraints $$3 < \tan\beta < 4 \ , %\eqno(9.4)$$ $$10^{10}\ {\rm GeV} < \Lambda_S < 10^{19}\ {\rm GeV} \ , %\eqno(9.5)$$ for $\Lambda_X \sim 10^{16}$ GeV. In the non-SUSY model, we have been able to choose $$\xi_S^u(\Lambda_X)=\xi_S^d(\Lambda_X) \ , %\eqno(9.6)$$ in addition to the ansatz of universality" $$\xi_{LR}^u(\Lambda_X)=\xi_{LR}^d(\Lambda_X) =\xi_{LR}^\nu(\Lambda_X)=\xi_{LR}^e(\Lambda_X) \ , %\eqno(9.7)$$ while in the SUSY model we have not been able to choose such a condition (9.6) in order to fit the numerical results to the observed quark masses and CKM parameters, because the up$\leftrightarrow$down symmetry is broken due to the factor $\tan\beta \neq 1$ in the SUSY model. As seen in Figs.~\ref{m-nonSUSY} and \ref{m-SUSY}, both in the non-SUSY and SUSY models, the mass ratios $m_e/m_\mu$ and $m_u/m_c$ are almost constant (although the ratio $m_u/m_c$ is slightly changed in the SUSY model), so that the phenomenologically well-satisfied relation (1.14) still holds under the evolutions. For the neutrino mass matrix $M_\nu$, we have investigate the case with $b_\nu (\Lambda_X)=-1/2$, which leads to a large mixing $\sin^2 \theta_{23}\simeq 1$. Although the mass-squared difference $\Delta m_{32}^2(\mu)$ is highly sensitive to the energy scale $\mu$ in the range III ($\Lambda_S <\mu \leq \Lambda_X$), the mixing angle $\theta_{23}$ is not sensitive to the energy scale. In contrast to the conventional seesaw neutrino mass matrix, note that the present neutrino mass matrix $M_\nu$ is form-invariant below $\mu=\Lambda_S$, so that the neutrino mass ratios and mixings are invariant below $\mu=\Lambda_S$. In the present paper, we have assumed ${\rm SU(3)}_c\times {\rm SU(2)}_L \times {\rm SU(2)}_R \times {\rm U(1)}_{LR} \times {\rm U(1)}_X$ symmetries above $\mu=\Lambda_S$. As seen in Figs.~\ref{xi-nonSUSY} and \ref{xi-SUSY}, in general, the burst of the Yukawa coupling constant $Y_L^u(\mu)$ causes above $\mu=\Lambda_S$, although we have been able to find a set of the reasonable parameter values without the burst. The burst is mainly due to the burst of the gauge coupling constant $g_1$ above $\mu=\Lambda_S$. If we want to build a unification model with a unified gauge symmetry G, we may consider that the U(1) symmetry is embedded into the unified symmetry G. (For example, see an ${\rm SO(10)}_L \times {\rm SO(10)}_R$ model \cite{Koide-so10}, where ${\rm SO(10)}_L \times {\rm SO(10)}_R$ is broken into $[{\rm SU(2)}\times {\rm SU(2)}' \times {\rm SU(4)}]_L \times [{\rm SU(2)}\times {\rm SU(2)}' \times {\rm SU(4)}]_R$.) Then, the gauge structure above $\mu=\Lambda_S$ is different from the present model, so that the evolutions will be also different from the present results. (Of course, the evolutions below $\mu=\Lambda_S$ are still the same as those in the present paper.) It is likely that the gauge structure above $\mu=\Lambda_S$ is different from the present model. Our next task is to investigate what gauge structure above $\mu=\Lambda_S$ is promising for a unified description of the quark and lepton masses and mixings. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \acknowledgements{ We thank N.~Okamura and A.~Ghosal for their helpful comments on the SUSY version of the universal seesaw model. } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newpage \vspace{0.2in} \appendix {\large\bf Appendix}\\ \renewcommand{\theequation}{A.\arabic{equation}} \setcounter{equation}{0} In Secs. 4 and 5, the coefficients of RGE (3.6), (3.12) and (3.13) have been given only for the case of non-SUSY scenario with one SU(2)-doublet Higgs scalar. In the present Appendix, we give the coefficients of RGE in the minimal SUSY scenario. \vspace{2mm} {\bf [Range III]} %$$\begin{eqnarray} T_A^u&=&T_A^{\nu}=3{\rm Tr}(Y_A^uY_A^{u\dagger}) +{\rm Tr}(Y_A^{\nu}Y_A^{\nu\dagger}), \nonumber\\ %$$ %$$T_A^d&=&T_A^e=3{\rm Tr}(Y_A^dY_A^{d\dagger}) +{\rm Tr}(Y_A^eY_A^{e\dagger}), %\eqno(A.1) %$$ \end{eqnarray} %$$\begin{eqnarray} &&G_A^u={\frac{13}{6}}g_1^2+3g_2^2+{\frac{16}{3}}g_3^2+g_X^2, \nonumber\\ %$$ %$$&&G_A^d={\frac{7}{6}}g_1^2+3g_2^2+{\frac{16}{3}}g_3^2+g_X^2, \nonumber\\ %$$ %$$&&G_A^{\nu}={\frac{9}{6}}g_1^2+3g_2^2+g_X^2, \\ %$$ %$$&&G_A^e={\frac{27}{6}}g_1^2+3g_2^2+g_X^2, \nonumber %\eqno(A.2) %$$ \end{eqnarray} %$$\begin{eqnarray} &&H_A^u=3Y_A^uY_A^{u\dagger}+Y_A^dY_A^{d\dagger},\nonumber\\ %$$ %$$&&H_A^d=3Y_A^dY_A^{d\dagger}+Y_A^uY_A^{u\dagger},\nonumber\\ %$$ %$$&&H_A^{\nu}=3Y_A^{\nu}Y_A^{\nu\dagger}+Y_A^eY_A^{e\dagger},\\ %$$ %$$&&H_A^e=3Y_A^eY_A^{e\dagger}+Y_A^{\nu}Y_A^{\nu\dagger},\nonumber %\eqno(A.3) \end{eqnarray} %$$ %$$\begin{eqnarray} &&T_S^u=T_S^{\nu}=3{\rm Tr}(Y_S^uY_S^{u\dagger})+{\rm Tr} (Y_S^{\nu}Y_S^{\nu\dagger}), \nonumber\\ %$$ %$$&&T_S^d=T_S^e =3{\rm Tr}(Y_S^dY_S^{d\dagger})+ {\rm Tr}(Y_S^eY_S^{e\dagger}), %\eqno(A.4) \end{eqnarray} %$$ %$$\begin{eqnarray} &&G_S^u={\frac{8}{3}}g_1^2+{\frac{16}{3}}g_3^2+3g_X^2, \nonumber\\ %$$ %$$&&G_S^d={\frac{2}{3}}g_1^2+{\frac{16}{3}}g_3^2+3_X^2,\nonumber\\ %$$ %$$&&G_S^{\nu}=3g_X^2, \\ %$$ %$$&&G_S^e={\frac{18}{3}}g_1^2+3g_X^2,\nonumber %\eqno(A.5) \end{eqnarray} %$$ $$H_S^f=2Y_S^f Y_S^{f\dagger}, %\eqno(A.6)$$ where $A=L,R$ and $f=u,d,\nu,e$. \vspace{2mm} {\bf [Range II]} $$T_A^u= 3{\rm Tr}(Y_A^uSY_A^{u\dagger}), %\eqno(A.7)$$ $$G_A^u={\frac{13}{6}}g_1^2+3g_{2A}^2+{\frac{16}{3}}g_3^2, %\eqno(A.8)$$ $$H_A^u=3Y_A^uSY_A^{u\dagger}, %\eqno(A.9)$$ %$$\begin{eqnarray} &&T_K^u=3{\rm Tr}(Y_L^uSY_L^{u\dagger}+Y_R^uSY_R^{u\dagger}),\nonumber\\ %\eqno(A.10) %$$ %$$&&T_K^d=T_K^e=0, %\eqno(A.11) \end{eqnarray} %$$ $$G_K^u=G_K^d=G_K^e={\frac{9}{2}}g_1^2+{\frac{9}{2}} (g_{2L}^2+g_{2R}^2), %\eqno(A.12)$$ %$$\begin{eqnarray} &&H_{KA}^u=H_{KA}^d={\frac{2}{3}}Y_ASY_A^{u\dagger},\nonumber\\ %\eqno(A.13) %$$ %$$&&H_{KA}^e=0, %\eqno(A.14) %$$ \end{eqnarray} $$T_K^{\nu}=6{\rm Tr}(Y_L^uSY_L^{u\dagger}), %\eqno(A.15)$$ $$G_K^{\nu}={\frac{9}{2}}g_1^2+9g_{2L}^2, %\eqno(A.16)$$ $$H_{KL}^{\nu}=0, %\eqno(A.17)$$ where $A=L,R$. \vspace{2mm} {\bf [Range I]} $$T_L^u=3{\rm Tr}(Y_L^uSY_L^{u\dagger}), %\eqno(A.18)$$ $$G_L^u=\frac{13}{15}g_1^2 +3g_{2A}^2+{\frac{16}{3}}g_3^2 , %\eqno(A.19)$$ $$H_L^u=3Y_uSY_u^{\dagger}, %\eqno(A.20)$$ %$$\begin{eqnarray} &&T_K^u=3{\rm Tr}(Y_L^uSY_L^{u\dagger}),\nonumber\\ %\eqno(A.21) %$$ %$$&&T_K^d=T_K^e=0, %\eqno(A.22) %$$ \end{eqnarray} $$G_K^u=G_K^d=G_K^e={\frac{9}{10}}g_1^2+{\frac{9}{2}}g_{2L}^2, %\eqno(A.23)$$ %$$\begin{eqnarray} &&H_{KL}^u=H_{KL}^d={\frac{2}{3}}Y_L^uSY_L^{u\dagger},\nonumber\\ %\eqno(A.24) %$$ %$$&&H_{KR}^u=H_{KR}^d=0,\\ %\eqno(A.25) %$$ %$$&&H_{KL}^e=H_{KR}^e=0,\nonumber %\eqno(A.26) %$$ \end{eqnarray} $$T_K^{\nu}=6{\rm Tr}(Y_L^uSY_L^{u\dagger}), %\eqno(A.27)$$ $$G_K^{\nu}={\frac{9}{10}}g_1^2 + 9g_{2L}^2, %\eqno(A.28)$$ $$H_{KL}^{\nu}=0. %\eqno(A.29)$$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{references} %%%%%%%%%%%%%%%% \vspace{.3in} \begin{thebibliography}{99} % \bibitem{q-evol} S.~R.~Ju\'{a}rex W., S.F.~Herrera H., P.~Kielanowski and G.~Mora H., talk presented at the {\it IX Mexican School on Particles and Fields}, August 9-19, 2000, Pue., Mexico. hep-ph/0009148 (2000); C.~R.~Das and M.~K.~Parida, NEHU/PHYS-MP-03/2000, hep-ph/0010004 (2000). % \bibitem{nu-evol} P.~H.~Chankowski and Z.~Pluciennik, Phys.~Lett. {\bf B316}, 312 (1993); K.~S.~Babu, C.~N.~Leung and J.~Pantaleone, Phys.~Lett. {\bf B319}, 191 (1993). 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Phys.~Lett. {\bf 11}, 2849 (1996); Phys.~Rev. {\bf D57}, 5836 (1998). % \bibitem{nu-atm} Y.~Fukuda {\it et al.}, Phys.~Lett. {\bf B335}, 237 (1994); Super-Kamiokande collaboration, Y.~Fukuda, {\it et. al.}, Phys.~Rev.~Lett. {\bf 81}, 1562 (1998); H.~Sobel, Talk presented at {\it Neutrino 2000}, Sudbury, Canada, June 2000 (http://nu2000.sno.laurentian.ca/). % \bibitem{dem-nu-evol} N.~Haba, Y.~Matsui, N.~Okamura and T.~Suzuki, Phys.~Lett. {\bf B489}, 184 (2000). % %Solar \bibitem{nu-solar} Y.~Suzuki, Talk presented at {\it Neutrino 2000}, Sudbury, Canada, June 2000 (http://nu2000.sno.laurentian.ca/). Also see, M.~Gonzalez-Garcia, Talk presented at {\it Neutrino 2000}, Sudbury, Canada, June 2000 (http://nu2000.sno.laurentian.ca/). %{nu-DUSM} \bibitem{nu-DUSM} Y.~Koide and H.~Fusaoka, Phys.~Rev. {\bf D59}, 053004 (1999); Y.~Koide and A.~Ghosal, Phys.~Lett. {\bf B488}, 344 (2000). % % LSND %\bibitem{LSND} C.~Athanassopoulos {\it et al.}, Phys.~Rev.~Lett. %{\bf 75}, 2650 (1995); %Phys.~Rev.~Lett. {\bf 77}, 3082 (1996); nucl-ex/9706006 (1997); %G.~Mills, Talk presented at {\it Neutrino 2000}, %Sudbury, Canada, June 2000 (http://nu2000.sno.laurentian.ca/). % \bibitem{Koide-so10} Y.~Koide, Phys.~Rev. {\bf D61}, 035008 (2000). \end{thebibliography} \end{multicols} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \vspace{5mm} \mediumtext \begin{table} \caption{Quantum numbers of the fermions $f$ and $F$ and Higgs scalars $\phi_L$, $\phi_R$ and $\Phi$ for ${\rm SU(2)}_L \times {\rm SU(2)}_R \times {\rm U(1)}_{LR} \times {\rm U(1)}_{X}$. } \label{T-qn} \vglue.1in \begin{tabular}{|c|cccc|c|cccc|} \hline & $I_3^L$ & $I_3^R$ & $\frac{1}{2} Y_{LR}$ & $X$ & & $I_3^L$ & $I_3^R$ & $\frac{1}{2} Y_{LR}$ & $X$ \\ \hline $u_L$ & $+\frac{1}{2}$ & 0 & $+\frac{1}{6}$ & 0 & $u_R$ & 0 & $+\frac{1}{2}$ & $+\frac{1}{6}$ & 0 \\ $d_L$ & $-\frac{1}{2}$ & 0 & $+\frac{1}{6}$ & 0 & $d_R$ & 0 & $-\frac{1}{2}$ & $+\frac{1}{6}$ & 0 \\ \hline $\nu_L$ & $+\frac{1}{2}$ & 0 & $-\frac{1}{2}$ & 0 & $\nu_R$ & 0 & $+\frac{1}{2}$ & $-\frac{1}{2}$ & 0 \\ $e_L$ & $-\frac{1}{2}$ & 0 & $-\frac{1}{2}$ & 0 & $e_R$ & 0 & $-\frac{1}{2}$ & $-\frac{1}{2}$ & 0 \\ \hline $U_L$ & 0 & 0 & $+\frac{2}{3}$ & $+\frac{1}{2}$ & $U_R$ & 0 & 0 & $+\frac{2}{3}$ & $-\frac{1}{2}$ \\ $D_L$ & 0 & 0 & $-\frac{1}{3}$ & $-\frac{1}{2}$ & $D_R$ & 0 & 0 & $-\frac{1}{3}$ & $+\frac{1}{2}$ \\ \hline $N_L$ & 0 & 0 & 0 & $+\frac{1}{2}$ & $N_R$ & 0 & 0 & 0 & $-\frac{1}{2}$ \\ $E_L$ & 0 & 0 & $-1$ & $-\frac{1}{2}$ & $E_R$ & 0 & 0 & $-1$ & $+\frac{1}{2}$ \\ \hline $\phi_L^+$ & $+\frac{1}{2}$ & 0 & $+\frac{1}{2}$ & $-\frac{1}{2}$ & $\phi_R^+$ & 0 & $+\frac{1}{2}$ & $+\frac{1}{2}$ & $+\frac{1}{2}$ \\ $\phi_L^0$ & $-\frac{1}{2}$ & 0 & $+\frac{1}{2}$ & $-\frac{1}{2}$ & $\phi_R^0$ & 0 & $-\frac{1}{2}$ & $+\frac{1}{2}$ & $+\frac{1}{2}$ \\ \hline $\Phi^0$ & 0 & 0 & 0 & $+1$ & & & & & \\ \hline \end{tabular} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \widetext \begin{table} \caption{The squared mass difference $\Delta m_{ij}^2 = m_{\nu i}^2-m_{\nu j}^2$. The values of the input parameters are the same as in Figs.~9 and 10. %Figs.~\ref{dm-nonSUSY} and \ref{dm-SUSY}. The absolute values of $\Delta m^2_{ij}$ should not be taken rigidly, because we can adjust those by the value of $\Lambda_S$. } \label{T-dm} \vglue.1in \begin{tabular}{|c|ccc|ccc|} \hline & \multicolumn{3}{c|}{non-SUSY model} & \multicolumn{3}{c|}{SUSY model} \\ \hline & at $\mu=\Lambda_L$ & at $\mu=\Lambda_S$ & at $\mu=\Lambda_X$ & at $\mu=\Lambda_L$ & at $\mu=\Lambda_S$ & at $\mu=\Lambda_X$ \\ \hline $\Delta m_{32}^2$ [eV$^2$] & $2.39\times 10^{-3}$ & $9.32\times 10^{-3}$ & $3.49\times 10^{-4}$ & $2.72\times 10^{-3}$ & $2.51\times 10^{-3}$ & $4.08\times 10^{-4}$ \\ \hline $\Delta m_{21}^2$ [eV$^2$] & $1.83\times 10^{-2}$ & $7.15\times 10^{-1}$ & $7.67\times 10^{-2}$ & $1.35\times 10^{-2}$ & $1.25\times 10^{-2}$ & $1.01\times 10^{-2}$ \\ \hline $\Delta m_{32}^2/\Delta m_{21}^2$ & $1.30\times 10^{-1}$ & $1.30\times 10^{-1}$ & $4.56\times 10^{-3}$ & $2.02\times 10^{-1}$ & $2.02\times 10^{-1}$ & $4.04\times 10^{-2}$ \\ \hline $|V_{23}|^2$ & $0.485$ & $0.485$ & $0.500$ & $0.478$ & $0.478$ & $0.500$ \\ \hline $|V_{12}|^2$ & $0.00484$ & $0.00484$ & $0.00471$ & $0.00492$ & $0.00492$ & $0.00466$ \\ \hline \end{tabular} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \begin{multicols}{2} \narrowtext \begin{figure} \begin{center} %\epsfile{file=xi-nonSUSY.eps,scale=0.65} \includegraphics[width=8.6cm]{xi-nonSUSY.eps} %\includegraphics[width=14cm]{xi-nonSUSY.eps} \end{center} \caption{ Behaviors of $\xi^u(\mu)$ in a non-SUSY model for the cases (a) $\Lambda_S=10^6$ GeV, (b) $\Lambda_S=10^9$ GeV, (c) $\Lambda_S=10^{12}$ GeV, and (d) $\Lambda_S=10^{15}$ GeV. The input values are $m_t(m_Z)=181$ GeV and $\Lambda_S/\Lambda_R=107$. } \label{xi-nonSUSY} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig.2 \narrowtext \begin{figure} \begin{center} %\epsfile{file=bd-nonSUSY.eps,scale=0.65} \includegraphics[width=8.6cm]{bd-nonSUSY.eps} %\includegraphics[width=14cm]{bd-nonSUSY.eps} \end{center} \caption{ Predictions of $m_d/m_s$, $m_s/m_b$ and $|V_{us}|$, and their dependency on the parameters $b_d$ and $\beta_d$. The dashed, solid and dotted lines denote $b_d=-1.1$, $-1.2$ and $-1.3$, respectively. } \label{bd-nonSUSY} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig.3 \narrowtext \begin{figure} \begin{center} \includegraphics[width=8.6cm]{d3-vinjx.eps} %\includegraphics[width=14cm]{d3-vinjx.eps} \end{center} \caption{ $\sum_{i\neq j} |V_{ij}(\Lambda_X)|^2$ versus $\delta_3^d(\Lambda_X)$. } \label{d3-vinjx} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig.4 \narrowtext \begin{figure} \begin{center} \includegraphics[width=8.6cm]{d3-vij.eps} %\includegraphics[width=14cm]{d3-vij.eps} \end{center} \caption{ Predicted values of the CKM matrix parameters $|V_{ij}(m_Z)|$ versus the parameter $\delta_3^d(\Lambda_X)$. Other input values of the parameters are $\Lambda_S=3\times 10^{13}$ GeV, $\Lambda_S/\Lambda_R=107$, $b_d= -1.2$ and $\beta_d=19.2^\circ$. } \label{d3-vij} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig.5 \narrowtext \begin{figure} \begin{center} \includegraphics[width=8.6cm]{xi-SUSY.eps} %\includegraphics[width=14cm]{xi-SUSY.eps} \end{center} \caption{ Behaviors of $\xi^u(\mu)$ in a SUSY model for the cases (a) $\Lambda_S=10^6$ GeV, (b) $\Lambda_S=10^9$ GeV, (c) $\Lambda_S=10^{12}$ GeV, and (d) $\Lambda_S=10^{15}$ GeV. The input values are $m_t(m_Z)=181$ GeV, $\Lambda_S/\Lambda_R=38$ and $\tan\beta=3.5$. } \label{xi-SUSY} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig.6 \begin{figure} \begin{center} %\epsfile{file=mt-tanb.eps,scale=0.65} \includegraphics[width=8.6cm]{mt-tanb.eps} %\includegraphics[width=14cm]{mt-tanb.eps} \end{center} \caption{ The top-quark mass $m_t(m_Z)$ versus $\tan\beta$ in a SUSY model. The solid and broken lines denote the cases with the initial conditions (a) $\xi_L^u(\Lambda_X)=2.0$ and (b) $\xi_L^u(\Lambda_X)=\sqrt{4\pi}=3.54$, respectively. The other input values are $\Lambda_S=6 \times 10^{13}$ GeV and $\Lambda_S/\Lambda_R=38$. The horizontal solid and broken lines denote the center and lower values of the observed top quark mass at $\mu=m_Z$, respectively. } \label{mt-tanb} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig.7 \begin{figure} \begin{center} %\epsfile{file=m-nonSUSY.eps,scale=0.65} \includegraphics[width=8.6cm]{m-nonSUSY.eps} %\includegraphics[width=14cm]{m-nonSUSY.eps} \end{center} \caption{ Behaviors of $m^f_i(\mu)/m^f_i(\Lambda_X)$ ($f=u,d,\nu,e$; $i=1,2,3$) in the non-SUSY model. The dotted, broken and solid lines denote the first, second and third fermion masses, respectively. The input parameter values are $\Lambda_S=3\times 10^{13}$ GeV, $\Lambda_S/\Lambda_R=107$ and $b_d(\Lambda_X)=-1.2 e^{i 19.2^\circ}$. } \label{m-nonSUSY} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig.8 \begin{figure} \begin{center} %\epsfile{file=m-SUSY.eps,scale=0.65} \includegraphics[width=8.6cm]{m-SUSY.eps} %\includegraphics[width=14cm]{m-SUSY.eps} \end{center} \caption{ Behaviors of $m^f_i(\mu)/m^f_i(\Lambda_X)$ ($f=u,d,\nu,e$; $i=1,2,3$) in the non-SUSY model. The dotted, broken and solid lines denote the first, second and third fermion masses, respectively. The input parameter values are $\Lambda_S=6\times 10^{13}$ GeV, $\Lambda_S/\Lambda_R=38$, $\tan\beta=3.5$ and $b_d(\Lambda_X)=-1.2 e^{i 19.4^\circ}$. } \label{m-SUSY} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure} \begin{center} %\epsfile{file=dm-nonSUSY.eps,scale=0.65} \includegraphics[width=8.6cm]{dm-nonSUSY.eps} %\includegraphics[width=14cm]{dm-nonSUSY.eps} \end{center} \caption{ Behavior of $\Delta m_{ij}^2(\mu)$ in the non-SUSY model. The input parameter values are the same as in Fig.~7 %Fig.~\ref{m-nonSUSY} with $\xi_A^\nu=\xi_A^e$ ($A=L, R, S$). } \label{dm-nonSUSY} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \narrowtext \begin{figure} \begin{center} %\epsfile{file=dm-SUSY.eps,scale=0.65} \includegraphics[width=8.6cm]{dm-SUSY.eps} %\includegraphics[width=14cm]{dm-SUSY.eps} \end{center} \caption{ Behavior of $\Delta m_{ij}^2(\mu)$ in the SUSY model. The input parameter values are the same as in %Fig.~\ref{m-SUSY} Fig.~8 with $\xi_A^\nu=\xi_A^e$ ($A=L, R, S$). } \label{dm-SUSY} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{figure} %\begin{center} %\epsfile{file=xif-non.eps,scale=0.65} %\end{center} %\caption{ %Behaviors of $\xi^f_A(\mu)$ ($f=u,d,\nu,e$; $A=L,R,S$) %in a non-SUSY model %for the cases (a) $\Lambda_S=10^6$ GeV, %(b) $\Lambda_S=10^9$ GeV, (c) $\Lambda_S=10^{12}$ GeV, %and (d) $\Lambda_S=10^{15}$ GeV. %The input values are $m_t(m_Z)=181$ GeV, %$\Lambda_S/\Lambda_R=35$ and $\tan\beta=2.5$. %}\label{xif-nonSUSY} % %\end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{figure} %\begin{center} %\epsfile{file=xif-SUSY.eps,scale=0.65} %\end{center} %\caption{ %Behaviors of $\xi^f_A(\mu)$ ($f=u,d,\nu,e$; $A=L,R,S$) %in a SUSY model %for the cases (a) $\Lambda_S=10^6$ GeV, %(b) $\Lambda_S=10^9$ GeV, (c) $\Lambda_S=10^{12}$ GeV, %and (d) $\Lambda_S=10^{15}$ GeV. %The input values are $m_t(m_Z)=181$ GeV, %$\Lambda_S/\Lambda_R=35$ and $\tan\beta=2.5$. %}\label{xif-SUSY} % %\end{figure} %%%%%%%%%%%%%%%%%%%%%%%5 \end{multicols} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document} 