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\begin{flushright} UCRHEP-T352\\January 2003\
\end{flushright}
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{\LARGE \bf Truly Minimal Left-Right Model of\\ Quark and Lepton Masses\\}
\vspace{1.0in}
{\bf Biswajoy Brahmachari$^{1,2}$, Ernest Ma$^3$, and Utpal Sarkar$^4$
\\}

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{$^1$ \sl Theoretical Physics Division, Saha Institute of Nuclear Physics, \\
AF/1 Bidhannagar, Kolkata - 700 064, India} \\
\vspace{0.1in}
{$^2$ \sl Physics Department, Vidyasagar Evening College,\\
39, Sankar Ghosh Lane, Kolkata 700 006, India\\}
\vspace{0.1in}
{$^3$ \sl Physics Department, University of California, Riverside, California
92521, USA\\}
\vspace{0.1in}
{$^4$ \sl Theory Group, Physical Research Laboratory, Ahmedabad 380 009, India\\}
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\begin{abstract}
We propose a left-right model of quarks and leptons based on the gauge group 
$SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$, where the scalar 
sector consists of only two doublets: (1,2,1,1) and (1,1,2,1).  As a result, 
any fermion mass, whether it be Majorana or Dirac, must come from 
dimension-five operators.  This allows us to have a common view of quark 
and lepton masses, including the smallness of Majorana neutrino masses as 
the consequence of a double seesaw mechanism.
\end{abstract}

\vspace{1cm}
\newpage
\baselineskip 24pt

In the standard model of electroweak interactions, neutrinos are massless.  
On the other hand, recent experimental data on atmospheric \cite{atm} and 
solar \cite{sol} neutrinos indicate strongly that they are massive and mix 
with one another \cite{others}. To allow neutrinos to be massive 
theoretically, the starting point is the observation of Weinberg 
\cite{wein79} over 20 years ago that a unique dimension-five operator 
exists in the standard model, i.e. 
\begin{equation}
{\cal L}_\Lambda = {f_{ij} \over 2 \Lambda} (\nu_i \phi^0 - e_i \phi^+) (\nu_j 
\phi^0 - e_j \phi^+) + H.c. \label{eq1}
\end{equation}
which generates a Majorana neutrino mass matrix given by
\begin{equation}
({\cal M}_\nu)_{ij} = {f_{ij} v^2 \over \Lambda}, \label{eq2}
\end{equation}
where $v$ is the vacuum expectation value of $\phi^0$.  This also shows 
that whatever the underlying mechanism for the Majorana neutrino mass, it 
has to be ``seesaw'' in character, i.e. $v^2$ divided by a large mass 
\cite{ma98}.

If the particle content of the standard model is extended to include 
left-right symmetry \cite{lr}, then the gauge group becomes ${\cal G}_{LR}
\equiv SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$, whose 
diagonal generators satisfy the charge relationship
\begin{equation}
Q = T_{3L} + T_{3R} + \frac{(B-L)}{2} = T_{3L} + \frac{Y}{2}. \label{eq3}
\end{equation}
Quarks and leptons transform as
\begin{eqnarray}
q_L = (u,d)_L &\sim& (3,2,1,1/3), \label{eq04} \\ 
q_R = (u,d)_R &\sim& (3,1,2,1/3), \label{eq4} \\ 
l_L = (\nu,e)_L &\sim& (1,2,1,-1), \label{eq05} \\ 
l_R = (N,e)_R &\sim& (1,1,2,-1), \label{eq5}
\end{eqnarray}
where a new fermion, i.e.~$N_R$, has been added in order that the left-right 
symmetry be maintained.

In all previous left-right models, a scalar bidoublet transforming as 
$(1,2,2,0)$ is then included for the obvious reason that we want masses 
for the quarks and leptons.  Suppose however that we are only interested in 
the spontaneous breaking of $SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ to 
$U(1)_{em}$ with $v_R >> v_L$, then the simplest way is to introduce two 
Higgs doublets transforming as
\begin{eqnarray}
\Phi_L = (\phi^+_L,\phi^0_L) &\sim& (1,2,1,1), \label{eq06} \\ 
\Phi_R = (\phi^+_R,\phi^0_R) &\sim& (1,1,2,1). \label{eq6}
\end{eqnarray}
Suppose we now do not admit any other scalar multiplet.  This is analogous 
to the situation in the standard model, where $SU(2)_L \times U(1)_Y$ is 
spontaneously broken down to $U(1)_{em}$ by a Higgs doublet and we do not 
admit any other scalar multiplet.  In that case, we find that quark and 
charged-lepton masses are automatically generated by the existing Higgs 
doublet, but neutrinos obtain Majorana masses only through the dimension-five 
operator of Eq.~(\ref{eq1}).  
In our case, in the absence of the bidoublet, all 
fermion masses, be they Majorana or Dirac, must now have their origin in 
dimension-five operators, as shown below.

Using Eqs.~(\ref{eq04}) to (\ref{eq6}), it is clear that
\begin{equation}
(l_L \Phi_L) = \nu_L \phi_L^0 - e_L \phi_L^+ \label{eq7}
\end{equation}
and
\begin{equation}
(l_R \Phi_R) = N_R \phi_R^0 - e_R \phi_R^+  \label{eq8}
\end{equation}
are invariants under ${\cal G}_{LR}$.  Hence we have the dimension-five 
operators given by
\begin{equation}
{\cal L}_M = {f^L_{ij} \over 2 \Lambda_M} (l_{iL} \Phi_L)(l_{jL} \Phi_L) 
+ {f^R_{ij} \over 2 \Lambda_M} (l_{iR} \Phi_R)(l_{jR} \Phi_R) + H.c.,
\label{eq9}
\end{equation}
which will generate Majorana neutrino masses proportional to $v_L^2/\Lambda_M$ 
for $\nu_L$ and $v_R^2/\Lambda_M$ for $N_R$.  In addition, we have
\begin{equation}
{\cal L}_D = {f^D_{ij} \over \Lambda_D} (\bar l_{iL} \Phi_L^*)
(l_{jR} \Phi_R) + H.c.
\label{eq10}
\end{equation}
and the corresponding dimension-five operators which will generate Dirac 
masses for all the quarks and charged leptons.

From Eq.~(\ref{eq10}), it is clear that
\begin{equation}
(m_D)_{ij} = {f^D_{ij} v_L v_R \over \Lambda_D}, \label{eq11}
\end{equation}
hence $\nu_L$ gets a double seesaw \cite{dss} mass of order
\begin{equation}
{m_D^2 \over m_N} \sim {v_L^2 v_R^2 \over \Lambda_D^2} {\Lambda_M \over v_R^2} 
= {v_L^2 \Lambda_M \over \Lambda_D^2}, \label{eq12}
\end{equation}
which is much larger than $v_L^2/\Lambda_M$ if $\Lambda_D << \Lambda_M$.  
Take for example $\Lambda_M$ to be the Planck scale of $10^{19}$ GeV and 
$\Lambda_D$ to be the grand-unification scale of $10^{16}$ GeV, then the 
neutrino mass scale is 1 eV (for $v_L$ of order 100 GeV).  The difference 
between $\Lambda_M$ and $\Lambda_D$ may be due to the fact that if we assign 
a global fermion number $F$ to $l_L$ and $l_R$, then ${\cal L}_M$ has $F = 
\pm 2$ but ${\cal L}_D$ has $F=0$.

Since the Dirac masses of quarks and charged leptons are also given by 
Eq.~(\ref{eq11}), $v_R$ cannot be much below $\Lambda_D$.  This means 
that $SU(2)_R 
\times U(1)_{B-L}$ is broken at a very high scale to $U(1)_Y$, and our model 
at low energy is just the standard model.  We do however have the extra 
singlet neutrinos $N_R$ with masses of order $v_R^2/\Lambda_M$, i.e. below 
$10^{13}$ GeV, which are useful for leptogenesis, as is well-known \cite{lg}.

For $m_t = 174.3 \pm 5.1$ GeV, we need $v_R/\Lambda_D$ to be of order unity 
in Eq.~(\ref{eq11}).  One may wonder in that case whether we can still write 
Eq.~(\ref{eq10}) as an effective operator.  The answer is yes, as can be 
seen with the following specific example \cite{dwr}.  Consider the singlets
\begin{equation}
U_L, U_R \sim (3,1,1,4/3), \label{eq13}
\end{equation}
with invariant mass $M_U$ of order $\Lambda_D$, then the $2 \times 2$ 
mass matrix linking $(\bar t_L, \bar U_L)$ to $(t_R,U_R)$ is given by
\begin{equation}
{\cal M}_{tU} = \pmatrix{0 & f_t^L v_L \cr f_t^R v_R & M_U}.
\label{eq14}
\end{equation}
For $v_L << v_R, M_U$, we then have
\begin{equation}
m_t = {f_t^L f_t^R v_L v_R \over M_U} \left[ 1 + {(f_t^R v_R)^2 \over M_U^2} 
\right]^{-{1 \over 2}},
\label{eq15}
\end{equation}
which is in the form of Eq.~(14) even if $v_R/M_U \sim 1$.

Since we already have dimension-five operators, we should also consider 
dimension-six operators.  In that case, we can invoke the Bardeen-Hill-Lindner 
(BHL) dynamical mechanism \cite{bhl} with a cutoff scale equal to $\Lambda_D$. 
We may assume that 
the effective dynamical BHL Higgs doublet [call it $\Phi_1 = (\phi_1^+,
\phi_1^0)$] couples only to the top quark, whereas our fundamental $\Phi_L$ 
[call it $\Phi_2$] couples to all quarks and leptons.  We thus have a 
specific two-Higgs-doublet model \cite{daskao} with experimentally verifiable 
phenomenology, as described below.

Since the BHL model predicts $m_t = 226$ GeV for $\Lambda_D = 10^{16}$ GeV, 
the effective Yukawa coupling of $\bar t_L t_R$ to $\bar \phi_1^0$ is
\begin{equation}
f_t^{(1)} = (226~{\rm GeV}) (2\sqrt 2 G_F)^{1 \over 2} = 226/174 = 1.3,
\label{eq16}
\end{equation}
and for $\tan \beta = v_2/v_1$, we have
\begin{equation}
m_t = (1.3 \cos \beta + f_t^{(2)} \sin \beta)(174~{\rm GeV}).
\label{eq17}
\end{equation}
This shows that, with a second Higgs doublet, the correct value of $m_t$ may 
be obtained.  Furthermore, $f_t^{(2)}$ may be assumed to be small, say of 
order $10^{-2}$.  This allows $v_R/\Lambda_D \sim 10^{-2}$ in Eq.~(14) and 
thus $v_R \sim 10^{14}$ GeV, so that $m_N \sim v_R^2/\Lambda_M$ is of order 
$10^9$ GeV, which may be more effective for leptogenesis, even with the 
reheating of the Universe after inflation.  At the same time, using 
Eq.~(\ref{eq17}), this fixes
\begin{equation}
\tan \beta \simeq 0.83
\label{eq18}
\end{equation}
for the phenomenology of the two-doublet Higgs sector.

Since the $d,s,b$ quarks receive masses only from $v_2$, there is no 
tree-level flavor-changing neutral currents in this sector.  This explains the 
suppression of $K_L-K_S$ mixing and $B - \bar B$ mixing.  On the other hand, 
both $v_1$ and $v_2$ contribute to the $u,c,t$ quarks, so our model does 
predict tree-level flavor-changing neutral curents in this sector.  Suppose 
the Yukawa interaction $f_t^{(1)} \bar \phi_1^0 \bar t_L t_R$ is replaced 
by $f_t^{(1)} (v_1/v_2) \bar \phi_2^0 \bar t_L t_R$, then the resulting 
mass matrix would be exactly proportional to the Yukawa matrix.  This means 
that there would not be any flavor-nondiagonal interactions.  Hence the 
term which contains all the flavor-changing interactions is given by 
\cite{ma01}
\begin{equation}
f_t^{(1)} \left( \bar \phi_1^0 - {v_1 \over v_2} \bar \phi_2^0 \right) \bar 
t'_L t'_R + H.c.,
\label{eq19}
\end{equation}
where $t'_{L,R}$ are the original entries in the $u,c,t$ mass matrix before 
diagonalization to obtain the mass eigenstates.  We thus expect contributions 
to, say $D - \bar D$ mixing, beyond that of the standard model.  Let
\begin{equation}
t'_{L,R} \simeq t_{L,R} + \epsilon_{tc}^{L,R} c_{L,R} + \epsilon_{tu}^{L,R} 
u_{L,R},
\label{eq20}
\end{equation}
where the $\epsilon$ parameters are at most of order $f_t^{(2)}/f_t^{(1)} \sim 
10^{-2}$, then \cite{ma01}
\begin{equation}
{\Delta m_{D^0} \over m_{D^0}} \simeq {B_D f_D^2 [f_t^{(1)}]^2 \over 3 
m^2_{eff} \sin^2 \beta} |\epsilon^L_{tc} \epsilon^L_{tu} \epsilon^R_{tc} 
\epsilon^R_{tu}|,
\label{eq21}
\end{equation}
where $m^2_{eff}$ is the effective normalized contribution from $\phi_1^0 - 
(v_1/v_2) \phi_2^0$.  Using $f_D = 150$ MeV, $B_D = 0.8$, and the present 
experimental upper bound \cite{pdg} of $2.5 \times 10^{-14}$ on this fraction, 
we then obtain
\begin{equation}
{|\epsilon^L_{tc} \epsilon^L_{tu} \epsilon^R_{tc} \epsilon^R_{tu}| \over 
10^{-8}} \left( {100~{\rm GeV} \over m_{eff}} \right)^2 < 1.
\label{eq22}
\end{equation}
This shows that $D - \bar D$ mixing may be observable in this model, in 
contrast to the negligible expectation of the standard model.

Rare top decays such as $t \to c$ (or $u$) + neutral Higgs boson are now 
possible if kinematically allowed.  Their branching fractions are of order 
$|\epsilon|^2 \sim 10^{-4}$.  Once a neutral Higgs boson is produced at a 
future collider, its decay will also be a test of this model.  Its dominant 
decay is still $b \bar b$, but its subdominant decays will not just be 
$c \bar c$ and $\tau^- \tau^+$, but also $c \bar u$ and $u \bar c$.  There 
should be observable $D^\pm \pi^\mp$ final states, for example.

In conclusion, we have proposed a truly minimal $SU(3)_C \times SU(2)_L \times 
SU(2)_R \times U(1)_{B-L}$ gauge model, with the simplest possible Higgs 
sector.  All fermion masses, be they Majorana or Dirac, have a common origin, 
i.e. dimension-five operators.  Whereas Dirac fermions have masses at the 
electroweak scale, the observed neutrinos have naturally small Majorana 
masses from a double seesaw mechanism.  The existence of singlet right-handed 
neutrinos with masses in the range $10^9$ to $10^{13}$ GeV are required, and 
their decays establish a lepton asymmetry which is converted at the 
electroweak phase transition to the present observed baryon asymmetry of 
the Universe.

Since our proposed model is identical to the standard model below $10^{16}$ 
GeV (except for the $N_R$'s), the usual predictions of the latter also 
apply, including the expected occurrence of proton decay and 
neutron-antineutron oscillations from higher-dimensional operators due to 
new physics at or above $10^{16}$ GeV.

In the presence of dimension-six operators, we may invoke the 
Bardeen-Hill-Lindner mechanism to generate a dynamical Higgs doublet which 
renders the $top$ quark massive.  Since we also have a fundamental Higgs 
doublet, this allows us to have a realistic $m_t$ (which is not possible 
in the minimal BHL model) and an effective two-doublet Higgs sector at 
the electroweak scale with distinctive and experimentally verifiable 
flavor-changing phenomena.\\[5pt]


The work of EM was supported in part by the U.~S.~Department of Energy
under Grant No.~DE-FG03-94ER40837. BB would like to thank the organisers
of the flavour@cern series of talks, where some aspects of the 
double seesaw mechanism were discussed. 



\newpage
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q_R         q_R
   >.......<
q_R .     . q_R
    .     .
q_R .     .  .. <R>
   >....... <..<R>
q_R         

The amplitude will have a factor <R>^2/M_P^3. However, since the Higgs
scalars S are all heavy, the amplitude will be highly suppressed by these
masses 1/M_S^4. 

\newpage







%\newpage
\bibliographystyle{unsrt}
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