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UMD-PP-03-010\\SLAC-PUB-9470
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{\LARGE \bf Leptogenesis Within A Predictive G(224)/SO(10)-Framework}
\vskip.25in
{\large{Jogesh C. Pati,\\ Department of Physics, University of Maryland,
College Park  MD 20742, USA\footnote{present address},\\
and\\ Stanford Linear Accelerator Center, Stanford University, Menlo Park, CA,
94025, USA.\\
August 15, 2002.}}
\end{center}

\vskip.25in
\begin{abstract}
A G(224)/SO(10)-framework has been proposed (a few years ago) that 
successfully describes
the masses and mixings of all fermions including neutrinos. Baryogenesis
via leptogenesis is considered within this framework by allowing for 
natural phases ($\sim 1/30$-1/2) in the entries of the Dirac and
Majorana mass-matrices. It is shown that the framework leads quite naturally
to the desired magnitude for the baryon asymmetry, in full accord with the
observed features of atmospheric and solar neutrino oscillations, as well as
with those of quark and charged lepton masses and mixings. Hereby one
obtains a {\it unified description} of fermion masses, neutrino oscillations
and baryogenesis within a single predictive framework.
\end{abstract}
%----------------------------------------------INTRODUCTION--------------
\newpage
{\large

\section{Introduction}

The observed matter-antimatter asymmetry of the universe
\cite{expr} is an important clue
to physics at truly short distances. A natural understanding of its magnitude
(not to mention its sign) is thus a worthy challenge. Since the discovery of
the electroweak sphaleron effect \cite{Ruzmin}, baryogenesis via leptogenesis
\cite{Yanagida,Others} appears to be the most attractive and promising
mechanism to generate such an asymmetry. In the context of a unified theory
of quarks and leptons, leptogenesis involving decays of heavy right-handed
(RH)
neutrinos, is naturally linked to the masses of quarks and leptons, neutrino
oscillations and, of course, CP violation.

In this regard, the route to higher unification based on an effective
four-dimensional gauge symmetry of either
G(224)=SU(2)$_L\times$SU(2)$_R\times$SU(4)$^C$
\cite{PS}, or SO(10) \cite{Georgi} (that may emerge from a string
theory near the string scale and breaks spontaneously to the standard model
symmetry near the GUT scale \cite{FN1}) offers some distinct advantages, which
are directly relevant to leptogenesis. These in particular include: (a) the
existence of the RH neutrinos as a compelling feature, (b) B-L as a local
symmetry, and (c) quark-lepton unification through SU(4)-Color. These three
features, first introduced in Ref. \cite{PS}, are common to both G(224) and
SO(10), though not to SU(5) \cite{GeorgiGlashow} and [SU(3)]$^3$ \cite{SU3}.
They, together with the seesaw mechanism \cite{seesaw} and the supersymmetric
unification-scale \cite{SUSYunifscale}, help explain even quantitatively
\cite{SeeJCP} the scale of $\nu_\tau$-mass [or rather of
$\Delta m^2(\nu_\mu$-$\nu_\tau)$] as observed at SuperKamiokande 
\cite{Superk}.
Furthermore, these three features also provide just the needed ingredients -
that is superheavy $\nu_R$'s and spontaneous violation of B-L at high
temperatures - for implementing baryogenesis via leptogenesis.

Now, in a theory with RH neutrinos having heavy Majorana masses, the magnitude
of the lepton-asymmetry is known to depend crucially on both the Dirac as well
as Majorana mass matrices of the neutrinos \cite{SeeReview}. In this regard,
 a predictive SO (10)/G(224) framework, describing the masses and mixings of
all fermions, including neutrinos, has been proposed \cite{BPW} that appears
to be remarkably successful.
In particular it makes seven predictions including: $m_b(m_b)\approx 4.9$ GeV,
$m(\nu_L^\tau)\sim (1/20)$ eV(1/2-2), $V_{cb}\approx 0.044$,
$\sin^22\theta_{\nu_\mu\nu_\tau}^{\rm osc}\approx 0.9$-0.99, 
$V_{us}\approx 0.20$-0.23, $V_{ub}\approx 0.0025$-0.0032
 and $m_d\approx 8$ MeV, all 
in good accord with observations, to within 10\% (see Sec. 2). It
has been noted recently \cite{JCP} that the large angle MSW solution (LMA),
which is preferred by experiments \cite{LMA}, can arise quite plausibly
within
the same framework through SO(10)-invariant higher dimensional operators
which can contribute directly to the Majorana masses of the left-handed
neutrinos (especially to the $\nu_L^e\nu_L^\mu$ mixing mass) without
involving the familiar seesaw. 

As an additional point, it has been noted by Babu and myself
\cite{BabuJCPCascais} that the framework proposed in Ref. \cite{BPW} can
 naturally accomodate CP violation by introducing complex phases in the 
entries
of the fermion mass-matrices, which preserve the pattern of the mass-matrices
suggested in Ref. \cite{BPW} as well as its successes.

The purpose of the present paper is to estimate the lepton and thereby the
baryon excess that would typically be expected within this realistic
G(224)/SO(10)-framework for fermion masses and mixings
\cite{BPW,BabuJCPCascais}, by allowing for natural CP violating phases
($\sim$ 1/30 to 1/2, say) in the entries of the mass-matrices as in Ref.
\cite{BabuJCPCascais}. The goal would thus be to obtain a {\it unified
 description} of (a) fermion masses, (b) neutrino oscillations, and (c)
 leptogenesis within a single predictive framework \cite{FN2}.

 It should be noted that there have in fact been several attempts in the
 literature \cite{Incompletelist} at estimating the lepton and baryon
 asymmetries, many of which have actually been carried out in the context
 of SO(10) \cite{SO(10)}, though (to my knowledge) without an accompanying
 realistic framework for the masses and mixing of quarks, charged leptons as
 well as neutrinos \cite{FN3}. Also the results in these attempts as
 regards leptogenesis have not been uniformly encouraging \cite{23}.

 The purpose of this letter is to note that the G(224)/SO(10) framework, 
proposed in Ref. \cite{BPW} and \cite{BabuJCPCascais},
 leads quite naturally to the desired magnitude for baryon asymmetry, in full
 accord with the 
observed features of atmospheric and solar neutrino oscillations,
 as well as with those of quark and charged lepton masses and mixings. To
 present the analysis it would be useful to recall the salient features of
 prior works \cite{BPW,BabuJCPCascais}. This is what is done in the next 
section.

\section{Fermion Masses and Neutrino Oscillations in G(224)/SO(10): A Brief
 Review of Prior Work}

The $3\times 3$ Dirac mass matrices for the four sectors $(u,d,l,\nu)$
proposed in Ref. \cite{BPW} were motivated in part by the notion that flavor
symmetries \cite{Hall} are responsible for the hierarchy among the elements
of these matrices (i.e., for "33"$\gg$"23"$\gg$"22"$\gg$"12"$\gg$"11", etc.),
and in part by the group theory of SO(10)/G(224), relevant to a minimal Higgs
system (see below). Up to minor variants \cite{FN4}, they are as follows:
\begin{eqnarray}
\label{eq:mat}
\begin{array}{cc}
M_u=\left[
\begin{array}{ccc}
0&\epsilon'&0\\-\epsilon'&\zeta_{22}^u&\sigma+\epsilon\\0&\sigma-\epsilon&1
\end{array}\right]{\cal M}_u^0;&
M_d=\left[
\begin{array}{ccc}
0&\eta'+\epsilon'&0\\ \eta'-\epsilon'&\zeta_{22}^d&\eta+\epsilon\\0&
\eta-\epsilon&1
\end{array}\right]{\cal M}_d^0\\
&\\
M_\nu^D=\left[
\begin{array}{ccc}
0&-3\epsilon'&0\\-3\epsilon'&\zeta_{22}^u&\sigma-3\epsilon\\
0&\sigma+3\epsilon&1\end{array}\right]{\cal M}_u^0;&
M_l=\left[
\begin{array}{ccc}
0&\eta'-3\epsilon'&0\\ \eta'+3\epsilon'&\zeta_{22}^d&\eta-3\epsilon\\0&
\eta+3\epsilon&1
\end{array}\right]{\cal M}_d^0\\
\end{array}
\end{eqnarray}
These matrices are defined in the gauge basis and are
multiplied by $\bar\Psi_L$ on left and $\Psi_R$ on right.
Note the group-theoretic up-down and quark-lepton correlations: the same 
$\sigma$ occurs in $M_u$ and $M_\nu^D$, and the same $\eta$ occurs in $M_d$
and $M_l$. It will become clear that the $\epsilon$ and $\epsilon'$ entries
are proportional to B-L and are antisymmetric in the family space 
(as shown above).
Thus, the same $\epsilon$ and $\epsilon'$ occur in both ($M_u$ and $M_d$) and
also in ($M_\nu^D$ and $M_l$), but $\epsilon\rightarrow -3\epsilon$ and
$\epsilon'\rightarrow -3\epsilon'$ as $q\rightarrow l$. Such correlations 
result in enormous reduction of parameters and thus in increased predictivity.
Such a patern for the mass-matrices can be obtained, using a minimal Higgs
system ${\bf 45}_H,{\bf 16}_H,{\bf \bar {16}}_H \mbox{ and }{\bf 10}_H $
 and a singlet S of SO(10), through effective couplings as follows \cite{FN26}:
\begin{multline}
\label{eq:Yuk}
{\cal L}_{\rm Yuk}=h_{33}{\bf 16}_3{\bf 16}_3{\bf 10}_H\\ +\left[
h_{23}{\bf 16}_2{\bf 16}_3{\bf 10}_H(S/M)+a_{23}{\bf 16}_2{\bf 16}_3{\bf 10}_H
({\bf 45}_H/M')(S/M)^p+g_{23}{\bf 16}_2{\bf 16}_3{\bf 16}_H^d
({\bf 16}_H/M'')(S/M)^q\right]\\+
\left[h_{22}{\bf 16}_2{\bf 16}_2{\bf 10}_H(S/M)^2+g_{22}{\bf 16}_2{\bf 16}_2
{\bf 16}_H^d({\bf 16}_H/M'')(S/M)^{q+1} \right]\\+
\left[g_{12}{\bf 16}_1{\bf 16}_2
{\bf 16}_H^d({\bf 16}_H/M'')(S/M)^{q+2}+
a_{12}{\bf 16}_1{\bf 16}_2
{\bf 10}_H({\bf 45}_H/M')(S/M)^{p+2}
\right]
\end{multline}
Typically we expect $M'$, $M''$ and $M$ to be of order $M_{\rm string}$
\cite{FN6}. The VEV's of $\langle{\bf 45}_H\rangle$ (along B-L),
$\langle{\bf 16}_H\rangle=\langle{\bf\bar {16}}_H\rangle$ (along standard 
model
singlet sneutrino-like component) and of the SO(10)-singlet $\langle S \rangle$
are of the GUT-scale, while those of ${\bf 10}_H$ and of the down type 
SU(2)$_L$-doublet component in ${\bf 16}_H$ (denoted by ${\bf 16}_H^d$) are
of the electroweak scale \cite{BPW,FN7}. Depending upon whether 
$M'(M'')\sim M_{\rm GUT}$ or $M_{\rm string}$ (see footnote \cite{FN6}), 
the 
exponent $p(q)$ is either one or zero \cite{FN8}.

The entries 1 and $\sigma$ arise respectively from $h_{33}$ and $h_{23}$
couplings, while $\hat\eta\equiv\eta-\sigma$ and $\eta'$ arise respectively
from $g_{23}$ and $g_{12}$-couplings. The (B-L)-dependent antisymmetric 
entries $\epsilon$ and $\epsilon'$ arise respectively from the $a_{23}$ and
$a_{12}$ couplings. [Effectively, with $\langle{\bf 45}_H\rangle\propto$ B-L,
the product ${\bf 10}_H\times{\bf 45}_H$ contributes as a {\bf 120}, whose 
coupling
is family-antisymmetric.] The small entry $\zeta_{22}^u$ arises from the 
$h_{22}$-coupling, while $\zeta_{22}^d$ arises from the joint contributions of
$h_{22}$ and $g_{22}$-couplings. As discussed in \cite{BPW}, using some of the 
observed masses as inputs, one obtains 
$|\hat\eta|\sim|\sigma|\sim|\epsilon|\sim {\cal O}(1/10)$, 
$|\eta'|\approx 4\times 10^{-3}$ and $|\epsilon'|\sim 2\times 10^{-4}$. The 
success of the framework presented in Ref. \cite{BPW} 
(which set $\zeta_{22}^u=\zeta_{22}^d=0$) in describing fermion masses and
mixings remains essentially unaltered if 
$|(\zeta_{22}^u,\zeta_{22}^d)|\leq (1/3)(10^{-2})$ (say). 

Such a hierarchical form of the mass-matrices, with $h_{33}$-term being
dominant, is attributed in part to flavor gauge symmetry(ies) that
distinguishes between the three families \cite{FN9}, and in part to higher
dimensional operators involving for example $\langle{\bf 45}_H\rangle/M'$
or $\langle{\bf 16}_H\rangle/M''$, which are supressed by 
$M_{\rm GUT}/M_{\rm string}\sim 1/10$, if $M'$ and/or 
$M''\sim M_{\rm string}$. 

To discuss the neutrino sector one must specify the Majorana mass-matrix of
the RH neutrinos as well. These arise from the effective couplings of the
form
\cite{FN30}:
\begin{eqnarray}
\label{eq:LMaj}
{\cal L}_{\rm Maj}=f_{ij}{\bf 16}_i{\bf 16}_j{\bf\bar{16}}_H{\bf\bar{16}}_H/M
\end{eqnarray}
where the $f_{ij}$'s include appropriate powers of $\langle S \rangle/M$, in
accord with flavor charge assignments of ${\bf 16}_i$ (see \cite{FN9}). For
the $f_{33}$-term to be leading, we must assign the charge $-a$ to 
${\bf\bar{16}}_H$. This leads to a hierarchical form for the Majorana
mass-matrix
\cite{BPW}:
\begin{eqnarray}
\label{eq:MajMM}
M_R^\nu=\left[
\begin{array}{ccc}
x&0&z\\0&0&y\\z&y&1
\end{array}
\right]M_R
\end{eqnarray}
Following the flavor-charge assignments given in footnote \cite{FN9}, we 
expect $|y|\sim
\langle S/M\rangle\sim 1/10$, 
$|z|\sim (\langle S/M\rangle)^2\sim 10^{-2}$(1 to 
1/2, say),  $|x|\sim (\langle S/M\rangle)^4\sim (10^{-4}$-$10^{-5})$ (say).
The "22" element (not shown) is $\sim (\langle S/M\rangle)^2$ and its
magnitude is
taken to be $< |y^2/3|$, while the "12" element (not shown) is
$\sim (\langle S/M\rangle)^3$.. We expect $M_R=f_{33}\langle {\bf \bar{16}}_H
\rangle^2/M_{\rm string}\approx (10^{15}$ GeV)(1/2-2) for 
$\langle{\bf \bar{16}}_H\rangle\approx 2\times 10^{16}$ GeV, 
$M_{\rm string}\approx 4\times 10^{17}$ GeV \cite{Kaplunovsky} and 
$f_{33}\approx 1$. Allowing for 2-3 mixing, this value of $M_R$ [together with
the SU(4)-color relation $m(\nu_\tau^{\rm Dirac})=m_t(M_{\rm GUT})\approx 110$
 GeV] 
leads to $m(\nu_L^\tau)\approx (1/20$ eV)(1/2-2) \cite{BPW,SeeJCP}, in good 
accord with the SuperK data.

Ignoring possible phases in the parameters and thus the source of CP violation
for a
moment, as was done in Ref. \cite{BPW}, the parameters $(\sigma,\eta,
\epsilon, \epsilon',\eta', {\cal M}_u^0, {\cal M}_D^0,\mbox{ and } y)$ can be 
determined
by using, for example, $m_t^{\rm phys}=174$ GeV, $m_c(m_c)=1.37$ GeV,
$m_S(1\mbox{ GeV})=110$-116 MeV, $m_u(1\mbox{ GeV})=6$ MeV, the observed
masses of $e$, $\mu$, and $\tau$ and
$m(\nu_L^\mu)/m(\nu_L^\tau)\approx 1/15$-1/8 (as suggested by a combination
of atmospheric and solar neutrino data, including SMA and LMA solutions) as
inputs. One is thus led, for this CP conserving case, to the following fit for
 the parameters, and the
associated predictions \cite{BPW}. [In this fit, we drop 
$|\zeta_{22}^{u,d}|\lesssim (1/3)(10^{-2})$ and leave the small quatities 
$x$ and
$z$ in $M_R^\nu$ undetermined and proceed by assuming that they have the
magnitudes suggested by flavor symmetries
(i.e., $x\sim (10^{-4}$-$10^{-5})$ and $z\sim 10^{-2}$(1 to 1/2)
(see remarks below Eq. \eqref{eq:MajMM})]:
\begin{eqnarray}
\label{eq:fit}
\begin{array}{c}
\sigma\approx 0.110, \quad \eta\approx 0.151, \quad \epsilon\approx -0.095,
 \quad |\eta'|\approx 4.4 \times 10^{-3},\\
\begin{array}{cc}
\epsilon'\approx 2\times 10^{-4},& {\cal M}_u^0\approx m_t(M_X)\approx 110
\mbox{ GeV}, \\{\cal M}^0_D\approx m_b(M_X)\approx 1.5 \mbox{ GeV}, & 
y\approx -(\mbox{1/20 to 1/17}).
\end{array}
\end{array}
\end{eqnarray}
These in turn lead to the following predictions for the quarks and
light neutrinos \cite{BPW}:
\begin{eqnarray}
\label{eq:pred}
\begin{array}{l}
m_b(m_b)\approx(4.7\mbox{-}4.9)\mbox{ GeV},\\
V_{cb}\approx\left|\sqrt{\frac{m_s}{m_b}}\left|\frac{\eta+\epsilon}
{\eta-\epsilon}\right|^{1/2}- \sqrt{\frac{m_c}{m_t}}\left|\frac{\sigma
+\epsilon}{\sigma-\epsilon}\right|^{1/2}\right|\approx 0.044,\\
\left\{ \begin{array}{l}
\theta^{\rm osc}_{\nu_\mu\nu_\tau}\approx\left|\sqrt{\frac{m_\mu}{m_\tau}}
\left|\frac{\eta-3\epsilon}{\eta+3\epsilon}\right|^{1/2}+
\sqrt{\frac{m_{\nu_\mu}}{m_{\nu_\tau}}}\right|\approx |0.437+(0.258-0.353)|,\\
\mbox{Thus, } \sin^2 2\theta^{\rm osc}_{\nu_\mu\nu_\tau}\approx 0.92\mbox{-}
0.99, \mbox{   (for $m(\nu_\mu)/m(\nu_\tau)\approx 1/15$-1/8),}\\
\end{array}\right.\\
V_{us}\approx \left|\sqrt{\frac{m_d}{m_s}}-\sqrt{\frac{m_u}{m_c}}\right|
\approx 0.20,\\
\left|\frac{V_{ub}}{V_{cb}} \right|\approx \sqrt{\frac{m_u}{m_c}}\approx
0.07,\\
m_d(\mbox{1 GeV})\approx \mbox{8 MeV}, m(\nu_L^{\tau})\approx
\mbox{(1/20 eV)(1/2-2)},\\
\theta^{\rm osc}_{\nu_e\nu_\mu}\approx 0.06
\mbox{  (ignoring non-seesaw contributions).}
\end{array}
\end{eqnarray}

The Majorana masses of the RH neutrinos ($N_{iR}\equiv N_i$) are given by:
\begin{eqnarray}
\label{eq:MajM}
M_{3}& \approx & M_R\approx 10^{15}\mbox{ GeV (1/2-2)},\nonumber\\
M_{2}& \approx & |y^2|M_{3}\approx \mbox{(2.5$\times 10^{12}$ GeV)(1/2-2)},\\
M_{1}& \approx & |x-z^2|M_{3}\sim (1/2\mbox{-}2)10^{-5}M_{3}\sim
\mbox{$10^{10}$ GeV(1/4-4)}.\nonumber
\end{eqnarray}

Leaving out the $\nu_e$-$\nu_\mu$ oscillation angle for a moment, it seems
remarkable that the first seven predictions in Eq. \eqref{eq:pred} agree
with observations, to within 10\%. Particularly intriguing is the
{\it group-theoretic correlation} between the contribution from the first
term in $V_{cb}$ and that in $\theta^{\rm osc}_{\nu_\mu\nu_\tau}$, which
explains simultaneously why one is small ($V_{cb}$) and the other is
large ($\theta^{\rm osc}_{\nu_\mu\nu_\tau}$). That in turn provides some
degree of confidence in the gross structure of the mass-matrices.

As regards $\nu_e$-$\nu_\mu$ and $\nu_e$-$\nu_\tau$ oscillations, the
standard seesaw mechanism would typically lead to rather small angles as in 
Eq. \eqref{eq:pred}, within the framework presented above \cite{BPW}.
It has, however, been noted recently \cite{JCP} that small intrinsic 
(non-seesaw) masses $\sim 10^{-3}$ eV of the LH neutrinos can arise quite
plausibly through higher dimensional operators of the form \cite{FN32}:
$W_{12}\supset \kappa_{12}{\bf 16}_1{\bf 16}_2{\bf 16}_H{\bf 16}_H{\bf 10}_H
{\bf 10}_H/M_{\rm eff}^3$, without involving the standard seesaw mechanism
\cite{seesaw}.
One can verify that such a term would lead to an intrinsic Majorana mixing
mass term of the form $m_{12}^0\nu_L^e\nu_L^\mu$, with a strength given by
$m_{12}^0\approx \kappa_{12}\langle{\bf 16}_H \rangle^2(175\mbox{ GeV})^2/
M_{\rm eff}^3\sim (1.5\mbox{-}6)\times 10^{-3}$ eV, for 
$\langle{\bf 16}_H \rangle\approx (1\mbox{-}2)M_{\rm GUT}$ and 
$\kappa_{12}\sim 1$, if $M_{\rm eff}\sim M_{\rm GUT}\approx 2\times 10^{16}$
GeV \cite{FN33}. Such an intrinsic Majorana and $\nu_e\nu_\mu$ mixing mass
$\sim $ few$\times 10^{-3}$ eV, though small compared to $m(\nu_L^\tau)$, is
still much larger than what one would generically get for the corresponding
term from the standard seesaw mechanism [as in Ref. \cite{BPW}]. Now, the
diagonal ($\nu_L^\mu\nu_L^\mu$) mass-term, arising from standard seesaw can 
naturally be $\sim$ (3-8)$\times 10^{-3}$ eV for $|y|\approx 1/20$-1/15, say
\cite{BPW}. Thus, taking the net values of $m_{22}\approx (6$-$7)\times 
10^{-3}$ eV, $m_{12}^0\sim 3\times 10^{-3}$ eV  as above and 
$m_{11}^0\ll 10^{-3}$ eV (as in \cite{BPW}), which are all plausible, we
obtain $m_{\nu_\mu}\approx(6$-$7)\times 10^{-3}$ eV, $m_{\nu_e}\sim 
\mbox{(1 to few)}
\times 10^{-3}$ eV, so that $\Delta m^2_{12}\approx (3.6\mbox{-}5)\times 
10^{-5}$ eV$^2$ and $\sin^22\theta_{12}^{\rm osc}\approx 0.6$-0.7. These go
well with the LMA MSW solution of the solar neutrino puzzle.

In summary, the intrinsic non-seesaw contribution to the Majorana masses of the
LH neutrinos can possibly have the right magnitude for $\nu_e$-$\nu_\mu$
mixing so as to lead to the LMA solution within the G(224)/SO(10)-framework,
without upsetting the successes of the first seven predictions 
in Eq. \eqref{eq:pred}. [In contrast to the near maximality 
of the 
$\nu_\mu$-$\nu_\tau$ oscillation angle, however, which emerges as a compelling
prediction of the framework \cite{BPW}, the LMA solution, as obtained above, 
should, be regarded only as a consistent possibility within this framework.]

Before discussing leptogenesis, we need to discuss the origin of CP 
violation within the G(224)/SO(10)-framework presented above. The discussion 
so far
has ignored, for the sake of simplicity, possible CP violating phases in the
parameters ($\sigma$, $\eta$, $\epsilon$, $\eta'$, $\epsilon'$,
$\zeta_{22}^{u,d}$, $y$, $z$, and $x$) of the Dirac and Majorana mass matrices
[Eqs. \eqref{eq:mat}, and \eqref{eq:MajMM}]. In general, however, these
parameters can and generically will have phases \cite{FN34}. Some combinations
of these phases enter into the CKM matrix and define the Wolfenstein parameters
$\rho_W$ and $\eta_W$ \cite{Wolfenstein}, which in turn induce CP violation by
utilizing the standard model interactions. As observed in Ref. 
\cite{BabuJCPCascais}, an additional and potentially important source of CP 
and flavor violations (as in $K^0\leftrightarrow\bar K^0$, 
$B_{d,s}\leftrightarrow\bar B_{d,s}$, $b\rightarrow s\bar s s$, etc. 
transitions)
arise in the model through supersymmetry \cite{FN36}, involving squark and
gluino loops (box and penguin), simply because of the embedding of MSSM within
a string-unified G(224) or SO(10)-theory near the GUT-scale, and the 
assumption that primordial SUSY-breaking occurs near the string scale
($M_{\rm string}>M_{\rm GUT}$) \cite{FN37}.
 It is shown
that complexification of the parameters ($\sigma$, $\eta$, $\epsilon$,
$\eta'$, $\epsilon'$, etc.), through introduction of phases $\sim 1/30$-1/2
(say) in them, still preserves the successes of the predictions as regards
fermion masses and neutrino oscillations shown in Eq. \eqref{eq:pred}, as
long as one maintains nearly the magnitudes of the real parts of the
parameters and especially their relative signs as obtained in Ref. \cite{BPW}
and shown in Eq. \eqref{eq:fit} \cite{FN38}. Such a picture is also in accord
with the observed features of CP and flavor violations in $\epsilon_K$, 
$\Delta m_{Bd}$, and asymmetry parameter in
 $B_d\rightarrow J/\Psi+K_s$, while predicting observable new effects in
 processes such as $B_s\rightarrow \bar B_s$ and $B_d\rightarrow \Phi+K_s$
 \cite{BabuJCPCascais}.

 We therefore proceed to discuss leptogenesis concretely within the framework
 presented above by adopting the Dirac and Majorana fermion mass matrices
 as shown in Eqs. \eqref{eq:mat} and  \eqref{eq:MajMM} and assuming that the
 parameters appearing in these matrices can have natural phases
 $\sim 1/30$-1/2 (say) with either sign up to addition of $\pm \pi$, {\it 
while their real
 parts have the relative signs and nearly the magnitudes given in Eq.
 \eqref{eq:pred}.}

\section{Leptogenesis}

In the context of an inflationary scenario \cite{d} with a reheat 
temperature $T_{RH}\sim (1 \mbox{ to few})10^{9}$ GeV (say), one can avoid the
well known gravitino problem if $m_{3/2}\sim (1\mbox{ to }2)$ TeV 
\cite{gravitino} and
yet produce the lightest heavy neutrino $N_1$ efficiently from the thermal bath
for $M_{1}\sim\mbox{(3 to 5)} 
\times 10^{9}$ GeV [see Eq. \eqref{eq:MajM}].
Given lepton number violation (through the Majorana mass of $N_1$) and
CP violating phases in the fermion mass-matrices as mentioned above, the
out-of-equilibrium decays of $N_1$ (produced from the thermal bath) into
$l+\Phi_H$ and $\bar l+\bar\Phi_H$ systems would produce a lepton asymmetry.
We will assume this commonly adopted scenario to discuss leptogenesis.
(We will comment later, however, on an interesting alternative possibility
proposed in Ref. \cite{41}.) The lepton asymmetry of the universe 
[$Y_L\equiv(n_L-n_{\bar L})/s$] arising from decays of $N_1$ into $(l+\Phi_H)$
and $(\bar l+\bar\Phi_H)$ is given by:
\begin{eqnarray}
\label{eq:YL}
Y_L=d\epsilon_1/g^*
\end{eqnarray}
where $\epsilon_1$ is the lepton-asymmetry produced per $N_1$ decay
(see below), $d$ is a
dilution factor that represents washout effects due to inverse decay and lepton
number violating  scattering, and $g^*\approx 228$ is the number of light
degrees of freedom for MSSM. 

The lepton asymmetry $Y_L$ is converted to baryon asymmetry, by the sphaleron
effects, which is given by:
\begin{eqnarray}
\label{eq:YB}
Y_B=\frac{n_B-n_{\bar B}}{s}\approx C\,Y_L,
\end{eqnarray}
where, for MSSM, $C=-8/15\approx -1/2$.
Taking into account the inteference between the tree and loop-diagrams which 
induce $N_1\rightarrow (l+\Phi_H)$ and 
$N_1\rightarrow (\bar l+\bar\Phi_H)$-decays, the lepton-asymmetry parameter 
$\epsilon_1$ is given by \cite{SeeReview,epsilon1}
\begin{eqnarray}
\label{eq:epsilon1}
\epsilon_1=\frac{1}{8\pi v^2(M_D^\dagger M_D)_{11}}\sum_{j=2,3}
{\rm Im} \left[(M_D^\dagger M_D)_{j1} \right]^2 f(M_j^2/M_1^2)
\end{eqnarray}
where $M_D$ is the Dirac neutrino mass matrix evaluated in a basis in which 
the Majorana mass matrix of the RH neutrinos 
$M_R^\nu$ [see Eq. \eqref{eq:MajMM}] is diagonal, $v=(\mbox{174 GeV})
\sin\beta$ and the function $f\approx -3(M_1/M_j)$ for the case of SUSY with
$M_j\gg M_1$. The dilution factor appearing in Eq. \eqref{eq:YL} is obtained
by solving Boltzmann equations and is approximately given by 
\cite{d,variant}:
\begin{eqnarray}
\label{eq:d}
d\approx \left\{\begin{array}{cc}
\frac{0.3}{k(\ln\,k)^{0.6}}& (10\lesssim k\lesssim 10^6)\\
\frac{1}{2k}& (1\lesssim k\lesssim 10)\\
1&(0\lesssim k\lesssim 1)\\
\end{array}
\right.
\end{eqnarray}
where $k\equiv [\Gamma(N_1)/(2H)]_{T=M_1}$ is given by:
\begin{eqnarray}
\label{eq:k}
k=\frac{M_{P1}}{1.66\sqrt{g^*}(8\pi v^2)}\frac{(M_D^\dagger M_D)_{11}}{M_1}.
\end{eqnarray}
Here $M_{P1}=$ Planck mass $\approx 1.2\times 10^{19}$ GeV.

Given the Dirac and Majorana mass matrices of the neutrinos [Eqs. 
\eqref{eq:mat} and \eqref{eq:MajMM}], we are now ready to evaluate lepton
assymetry by using Eqs. \eqref{eq:YL}-\eqref{eq:k}. 

The Majorana mass matrix [Eq. \eqref{eq:MajMM}] describing the mass-term 
$\nu^T_RCM_R^\nu\nu_R$ is diagonalized by the transformation 
$\nu_R=U_R^{(1)}U_R^{(2)}N_R$,
where (to a good approximation)
\begin{eqnarray}
\label{eq:UR}
U_R^{(1)}\approx\left[\begin{array}{ccc}1&0&z\\0&1&y\\-z&-y&1\end{array}
\right],
\end{eqnarray}
and $U_R^{(2)}={\rm diag}(e^{i\phi_1},e^{i\phi_2},e^{i\phi_3})$ is a diagonal
phase matrix that ensures real positive eigenvalues. The phases $\phi_i$ can
of course be derived from those of the parameters in $M_R^\nu$ [see Eq. 
\eqref{eq:MajMM}].
Applying this transformation to the neutrino Dirac mass-term 
$\bar\nu_LM_\nu^D\nu_R$ given by Eq. \eqref{eq:mat}, we obtain 
$M_D=M_\nu^DU_R^{(1)}U_R^{(2)}$, which appears in Eqs.
\eqref{eq:epsilon1} and \eqref{eq:k}. In turn, this yields:
\begin{eqnarray}
\label{eq:MDMD21}
\frac{(M_D^\dagger M_D)_{21}}{\left({\cal M}_u^0\right)^2}&=&
e^{i(\phi_1-\phi_2)}\{
\left(-3{\epsilon'}^*
-\zeta^*_{13}y^*\right)\left(\zeta_{11}-z\zeta_{13}\right)
\nonumber\\&+&
\left[\zeta_{22}^{u*}-y^*\left(\sigma^*-3\epsilon*\right)\right]
\left[3\epsilon'-z\left(\sigma-3\epsilon\right)\right]+
\left(\zeta_{31}-z\right)\left[\left(\sigma^*+3\epsilon^*\right)-y^*\right]\}\\
\label{eq:MDMD11}
\frac{(M_D^\dagger M_D)_{11}}{\left({\cal M}_u^0\right)^2}&=&
\left|3\epsilon'-z(\sigma-3\epsilon)\right|^2+\left|\zeta_{31}-z\right|^2
\end{eqnarray}
In writing Eqs. \eqref{eq:MDMD21} and \eqref{eq:MDMD11}, we have allowed, for 
the sake of generality, the relatively small ``11'', ``13'', and ``31''
elements in the Dirac mass-matrix $M_\nu^D$, denoted by $\zeta_{11}$,
$\zeta_{13}$ and $\zeta_{31}$ respectively, which are not exibited in Eq.
\eqref{eq:MajMM}. Guided by considerations of flavor symmetry 
(see footnote \cite{FN9}), we would expect 
$|\zeta_{11}|\sim(\langle S\rangle/M)^4\sim 10^{-4}$-$10^{-5}$, and
$|\zeta_{13}|\sim|\zeta_{31}|\sim(\langle S\rangle/M)^2\sim 10^{-2}$(1 to
1/3) (say). These small elements (neglected in \cite{BPW}) would not, 
of course, have any noticeable effects on the predictions of the fermion
masses and mixings given in Eq. \eqref{eq:pred}, except possibly on $m_d$.

We now proceed to make numerical estimates of lepton and baryon-asymmetries
by taking the magnitudes and the relative signs of the real parts of the
parameters ($\sigma$, $\eta$, $\epsilon$, $\eta'$, $\epsilon'$, and $y$)
approximately the same as in Eq. \eqref{eq:fit}, but allowing in general
for natural phases in them. As mentioned before [see for example the fit
given in footnote \cite{FN38} and Ref. \cite{BabuJCPCascais} (to appear)]
such a procedure introduces CP violation in accord with observation, while
preserving the successes of the framework as regards its predictions for
fermion masses and neutrino oscillations \cite{BabuJCPCascais, BPW}.

Given the magnitudes of the parameters (see Eqs. \eqref{eq:fit} and 
Ref. \cite{FN38}), which are obtained from considerations of fermion masses
and neutrino oscillations \cite{BPW,BabuJCPCascais} -- that is 
$|\sigma|\approx |\epsilon|\approx 0.1$, $|y|\approx 0.06$, 
$|\epsilon'|\approx 2\times 10^{-4}$, $|z|\sim (1/200)(1\mbox{ to }1/2)$,
$|\zeta_{22}^u|\sim 10^{-3}(1\mbox{ to }3)$, $|\zeta_{13}|\sim|\zeta_{31}|
\sim(1/200)(1\mbox{ to }1/2)$, with  the real parts of ($\sigma$, $\epsilon$ 
and $y$) having the signs (+, -, -) respectively, we would expect the typical
magnitudes of the three terms of Eq. \eqref{eq:MDMD21} to be as follows:
\begin{eqnarray}
\label{eq:terms21}
|\mbox{$1^{st}$ Term}|&=&\left|\left(-3{\epsilon'}^*-\zeta_{13}^*y^*\right)
\left(\zeta_{11}-z\zeta_{13}\right)\right|\nonumber\\
&\approx& \left[(6\mbox{ to }8)\times 10^{-4}\right]\left[(2.5\times 10^{-5})
(1\mbox{ to }1/4)\right]\sim 10^{-8}\nonumber\\
|\mbox{$2^{nd}$ Term}|&=&\left|\left\{\zeta_{22}^{u*}-y^*\left(\sigma^*-
3\epsilon^*\right)\right\} \left\{3\epsilon'-z(\sigma-3\epsilon)
\right\}\right|\\
&\approx& \left(2\times 10^{-2}\right)\left[2\times 10^{-3}(1\mbox{ to }1/2)
\right]\approx  \left(4\times 10^{-5}\right)(1\mbox{ to }1/2)\nonumber\\
|\mbox{$3^{rd}$ Term}|&=&\left|\left(\zeta_{31}-z\right)\left\{
\left(\sigma^*+3\epsilon^*\right)-y^*\right\}\right|\nonumber\\
&\approx& [(1/200)(1\mbox{ to }1/3)](0.14)\approx\left(0.7\times 10^{-3}
\right)(1\mbox{ to }1/3)\nonumber
\end{eqnarray}
Thus, assuming that the phases of the different terms are roughly comparable,
the third term would clearly dominate. The RHS of Eq. \eqref{eq:MDMD11} is 
similarly estimated to be:
\begin{eqnarray}
\label{eq:terms11}
\frac{\left( M_D^\dagger M_D\right)_{11}}{\left({\cal M}_u^0\right)^2}&=&
|3\epsilon'-z(\sigma-3\epsilon)|^2+\left|\zeta_{31}-z\right|^2\nonumber\\
&\approx &\left|6\times 10^{-4}\mp 2\times 10^{-3} (1\mbox{ to }1/2)\right|^2+
\left|5\times 10^{-3}(1\mbox{ to }1/3)\right|^2\\
&\approx & 2.5\times 10^{-5}(1\mbox{ to }1/9)\nonumber
\end{eqnarray}
Since $|\zeta_{31}|$ and $|z|$ are each expected to be of order 
(1/200)(1 to 1/2), we have allowed for a possible mild cancellation between 
their contributions to $|\zeta_{31}-z|$ by putting 
$|\zeta_{31}-z|\approx$ (1/200)(1 to 1/3) (say). Note that this combination 
enters into the dominant terms of both 
$(M_D^\dagger M_D)_{21}/({\cal M}_u^0)^2$
[see the third term in Eq. \eqref{eq:terms21}] and 
$(M_D^\dagger M_D)_{11}/({\cal M}_u^0)^2$ [see the second  term in Eq. 
\eqref{eq:terms11}]. As a result, to a good approximation, the 
lepton-asymmetry parameter $\epsilon_1$ [given by Eq. \eqref{eq:epsilon1}] 
becomes
independent of the magnitude of $|\zeta_{31}-z|^2$ and thereby of the 
uncertainty in it. It is given by:
\begin{eqnarray}
\label{eq:epsilon1_}
\epsilon_1\approx \frac{1}{8\pi}\left(\frac{{\cal M}_u^0}{v}\right)^2
|(\sigma+3\epsilon)-y|^2\sin\left(2\phi_{21}\right)(-3)
\left(\frac{M_1}{M_2}\right)\approx -\left(2.4\times 10^{-6}\right)
\sin\left(2\phi_{21}\right),
\end{eqnarray}
where, $\phi_{21}={\rm arg}[(\zeta_{31}-z)(\sigma^*+3\epsilon^*-y^*)]+
(\phi_1-\phi_2)$, and
we have put $({\cal M}_u^0/v)^2\approx 1/2$, $|\sigma+3\epsilon-y|\approx 0.14$
(see Eq. \eqref{eq:fit} and Ref. \cite{FN38}), and
for concreteness $M_1/M_2\approx (4\times 10^9\mbox{ GeV})/(2\times 10^{12}
\mbox{ GeV})\approx 2\times 10^{-3}$ [see Eq. \eqref{eq:MajM}]. The parameter
$k$, given by Eq. \eqref{eq:k}, is (approximately) proportional to 
$|\zeta_{31}-z|^2$ [see Eqs. \eqref{eq:terms21} and \eqref{eq:terms11}].
It is given by:
\begin{eqnarray}
\label{eq:k_}
k\approx \frac{\left(M_{Pl}/M_1\right)}{1.66\sqrt{g^*}(8\pi)}
\left(\frac{{\cal M}_u^0}{v}\right)^2\left|\zeta_{31}-z\right|^2\approx
 60(1\mbox{ to }1/9),
\end{eqnarray}
where, as before, we have put $M_1=4\times 10^9$ GeV and 
$|\zeta_{31}-z|\approx (1/200)(1\mbox{ to }1/3)$. The corresponding dilution
factor $d$ [given by Eq. \eqref{eq:d}], lepton and baryon-asymmetries
$Y_L$ and $Y_B$ [given by Eqs. \eqref{eq:YL} and \eqref{eq:YB}] and the 
requirement on the phase-parameter $\phi_{21}$ are listed below:

\begin{center}
\noindent\begin{tabular}{|c|c|c|c|c|}
\hline
&\multicolumn{3}{c|}{$|\zeta_{31}-z|$}\\
\hhline{|~|-|-|-|}
&1/200&(1/200)(1/1.7)&(1/200)(1/3)\\ \hline\hline
$k$&60&20&7\\ \hline
$d$&1/466&1/129&1/14 \\ \hline
$Y_L/\sin(2\phi_{21})$&$-2.26\times 10^{-11}$&$-8.2\times 10^{-11}$&
$-7.5\times 10^{-10}$ \\ \hline
$Y_B/\sin(2\phi_{21})$&$1.13\times 10^{-11}$&$4.1\times 10^{-11}$&
$3.7\times 10^{-10}$ \\ \hline
$\phi_{21}$&$\sim \pi/4$& $\gtrsim \pi/15$& $\sim\pi/100$-$\pi/25$\\ \hline
\end{tabular}\\
\vskip10mm
{\bf Table 1}\end{center}

The constraint on $\phi_{21}$ is obtained from considerations of Big-Bang
nucleosynthesis, which requires $1.7\times 10^{-11}\lesssim Y_B\lesssim 
9\times 10^{-11}$ \cite{expr}. We see that the first case $|\zeta_{31}-z|
\approx 1/200$ leads to a baryon asymmetry $Y_B$ that is 
on the borderline even for a maximal $\sin(2\phi_{21})\approx 1$. 
The other cases with
$|\zeta_{31}-z|\approx (1/200)(1/1.7\mbox{ to }1/3)$, which are of course
perfectly plausible, lead to the desired magnitude of the baryon asymmetry
for natural values of the phase parameter $\sin (2\phi_{21})\sim 
(1/5\mbox{ to }1/30)$ \cite{new45}. 

We now comment briefly on the scenario proposed in Ref. \cite{41},
in which the inflaton decays directly into a pair of heavy RH neutrinos,
which in turn decay into $l+\Phi_H$ and $\bar l+\bar\Phi_H$ and thereby
generate lepton asymmetry, {\it during the process of reheating}. 
Confining to the
fermion mass-pattern in Sec. 2 [Eqs. \eqref{eq:mat}, \eqref{eq:MajMM} and 
\eqref{eq:MajM}], a very similar conclusion as above as regards leptogenesis
can be reached also within this alternative scenario. It turns out that this
scenario goes well with the mass-pattern of Sec. 2 [especially Eq. 
\eqref{eq:MajM}], in full accord with the gravitino-constraint and observed
baryon-asymmetry, provided $2M_2>m_{\rm infl}>2M_1$, so that the inflaton 
decays into $2N_1$ rather than into $2N_2$ (contrast this from the case
proposed in Ref. \cite{41}). In this case, defining the superpotential 
$W=\kappa S(-M^2+\bar\Phi\Phi)+\mbox{(non-ren. terms)}$, as in Ref. \cite{41},
where $\Phi$ and $\bar\Phi$ are the (1, 2, 4) and (1, 2, $\bar 4$) Higgs
fields and $S$ is a singlet field, 
one obtains \cite{41}: $m_{\rm infl}=\sqrt{2}\kappa M$, where
$M=\langle\mbox{(1, 2, 4)}_H\rangle\approx 2\times 10^{16}$ GeV,
$\Gamma_{\rm infl}\approx [1/(8\pi)](M_1/M)^2m_{\rm infl}$ and 
$T_{RH}\approx (1/7)(\Gamma_{\rm infl}M_{\rm Pl})^{1/2}\approx (1/7)(M_1/M)
[m_{\rm infl}M_{\rm Pl}/(8\pi)]^{1/2}$. For concreteness, take 
$M_2\approx 2\times 10^{12}$ GeV, $M_1\approx 10^{10}$ GeV (1 to 2) [in accord
with Eq. \eqref{eq:MajM}], and $m_{\rm infl}\approx 3\times 10^{12}$ GeV
(choosing $\kappa\approx 10^{-4}$). We then get: $T_{RH}\approx 
\mbox{(1 to 2)}(0.8\times 10^8\mbox{ GeV})$, and thus (see e.g., Sec. 8 of
Ref. \cite{d}), $Y_B\approx -Y_L/2\approx (-1/2)
(\epsilon_1 T_{RH}/m_{\rm infl})\approx \mbox{(1 to 2)}^2(8\times 10^{-11}
\sin 2\phi_{21})$, where we have used Eq. \eqref{eq:epsilon1_} with 
appropriate $(M_1/M_2)$, as above. This agrees with the observed value of
$Y_B\approx 4\times 10^{-11}$ (say), again for a natural value of the phase
parameter $\phi_{21}\approx (1/4)$(1 to 1/4), where the second factor
 corresponds to $M_1\approx 10^{10}$ GeV (1 to 2) \cite{FN45}.

To conclude, we have considered two alternative scenarios for inflation and
leptogenesis.
We see that the G(224)/SO(10) framework provides a simple and 
{\it unified description} of not only fermion masses and neutrino oscillations
(consistent with maximal atmospheric and large solar oscillation angles) but 
also of baryogenesis via leptogenesis, treated within either scenario. 
The existence of the right-handed neutrinos, B-L as a 
local symmetry, quark-lepton unification through SU(4)-color, the seesaw
mechanism and the magnitude of the supersymmetric unification-scale play 
crucial roles in
realizing this unified and successful description. 
These features in turn point to the relevance of either G(224) or SO(10)
symmetry being effective between the string and the GUT scales, in four 
dimensions.  While the observed magnitude of the baryon 
asymmetry seems to emerge naturally from within the framework, understanding
its observed sign (and thus the relevant CP violating phases) remains a
challenging task.\\

{\bf \Large Acknowledgements}\\

I would like to thank Kaladi S. Babu for many collaborative discussions on
CP violation, as it arises within the G(224)/SO(10)-framework, which is 
directly relevant to the present work. I have benefitted
from discussions with Gustavo Branco, Tsutumo Yanagida, and especially
Qaisar Shafi on several aspects of this work. The
sabbatical support by the University of Maryland during the author's visit to
SLAC, as well as the hospitality 
of the Theory Group of SLAC, where this work was carried out, are gratefully
acknowledged. The work is supported in part by DOE grant no. 
DE-FG02-96ER-41015.

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\bibitem{FN4} The zeros in "11", "13" and "31" elements signify that they
are relatively small quantities (specified below). While the "22" elements
were set to zero in Ref. \cite{BPW}, because they are meant to be
$<$"23""32"/"33"$\sim 10^{-2}$ (see below), and thus unimportant for purposes
of Ref. \cite{BPW}, they are retained here, because such small
$\zeta_{22}^u$ and $\zeta_{22}^d$ [$\sim (1/3)\times 10^{-2}$ (say)] can
still be important for CP violation and thus baryogenesis.

\bibitem{FN26} For G(224), one can choose the corresponding sub-multiplets --
that is (1, 1, 15)$_H$, (1, 2, $\bar{4}$)$_H$, (1, 2, 4)$_H$, (2, 2, 1)$_H$
-- together with a singlet $S$, and write a superpotential analogous to Eq. 
\eqref{eq:Yuk}.

\bibitem{FN6} If the effective non-renormalizable operator like 
${\bf 16}_2{\bf 16}_3{\bf 10}_H{\bf 45}_H/M'$ is induced through exchange
of states with GUT-scale masses involving renormalizable couplings, rather
than through quantum gravity, $M'$ would, however, be of order GUT-scale. 
In this case $\langle {\bf 45}_H\rangle/M'\sim 1$, rather than 1/10.

\bibitem{FN7} While ${\bf 16}_H$ has a GUT-scale VEV along the SM singlet, it 
turns out that it also has a VEV of EW scale along the ``$\tilde\nu_L$'' 
direction due to its mixing  with ${\bf 10}_H^d$, so that the $H_d$ of 
MSSM is a mixture of  ${\bf 10}_H^d$ and  ${\bf 16}_H^d$ (See Ref. \cite{BPW}).

\bibitem{FN8}  The flavor charge(s) of ${\bf 45}_H$(${\bf 16}_H$) would
get determined depending upon whether $p$($q$) is one or zero (see below).

\bibitem{FN9} The basic presumption here is that effective dimensionless
couplings allowed by SO(10)/G(224) and flavor symmetries are of order unity
[i.e., $(h_{ij},g_{ij},a_{ij})\approx 1/3$-3 (say)]. The need for appropriate
powers of $(S/M)$ with $\langle S\rangle/M\sim M_{\rm GUT}/M_{\rm string}\sim
(1/10$-1/20) in the different couplings leads to a hierarchical structure. 
As an 
example, consider just one U(1)-flavor symmetry with one singlet S. The
hierarchical form of the Yukawa couplings exibited in Eqs.
\eqref{eq:mat} and  \eqref{eq:Yuk} would
be allowed, for the case of $p=1$, $q=0$, if (${\bf 16}_3$, ${\bf 16}_2$, 
${\bf 16}_1$,
${\bf 10}_H$, ${\bf 16}_H$, ${\bf 45}_H$ and S) are assigned U(1)-charges of
($a$, $a+1$, $a+2$, $-2a$, $-a-1/2$, 0, -1). 
It is assumed that other fields are present that would make the U(1) symmetry 
anomaly-free. With this assignment of charges, one
would expect $|\zeta_{22}^{u,d}|\sim (\langle S\rangle/M)^2$; one may thus take
$|\zeta_{22}^{u,d}|\sim (1/3)\times 10^{-2}$ without upsetting the success of
Ref. \cite{BPW}. In the same spirit, one would expect $|\zeta_{13},
\zeta_{31}|\sim (\langle S\rangle/M)^2\sim 10^{-2}$ and $|\zeta_{11}|\sim 
(\langle S\rangle/M)^4\sim 10^{-4}$ (say). The value of ``a'' would get fixed
by the presence of other operators (see later).
\bibitem{FN30} These effective non-renormalizable couplings can of course
arise through exchange of (for example) ${\bf 45}$ in the string tower, 
involving
renormalizable ${\bf 16}_i{\bf \bar{16}}_H{\bf 45}$ couplings. In this case, 
one would expect $M\sim M_{\rm string}$.
\bibitem{Kaplunovsky} P. Ginsparg, Phys. Lett. {\bf B197}, 139 (1987); 
V. S. Kaplunovsky, Nucl. Phys. {\bf B307}, 145 (1988); Erratum: ibid
{\bf B382}, 436 (1992).
\bibitem{FN32} Note that such an operator would be allowed by the flavor 
symmetry defined in Ref. \cite{FN9} if one sets $a=1/2$. In this case, 
operators such as $W_{23}$ and $W_{33}$ that would contribute to 
$\nu_L^\mu\nu^\tau_L$ and $\nu_L^\tau\nu^\tau_L$ masses would be suppressed 
relative to $W_{12}$ by flavor symmetry. As pointed out by other authors
(see e.g., S. Weinberg, Phys. Rev. Lett. {\bf 43}, 1566 (1979) and Proc. 
XXVI Int'l Conf. on High Energy Physics, Dallas, TX, 1992; E. Akhmedov, Z.
Berezhiani and G. Senjanovic, Phys. Rev. {\bf D47}, 3245 (1993).), 
non-seesaw Majorana masses of the LH neutrinos 
can arise directly, even in the standard model, through operators of the form
$L_iL_j\Phi_H\Phi_H/M$, by utilizing quantum gravity. [For SO(10), two 
${\bf 16}_H$'s are needed additionally to violate B-L by two units.] In the
case of the standard model, ordinarily, one would expect 
$M\sim M_{\rm Planck}$. Thus one would still
need to find a reason (in the context of the standard model) why (a)
$M\sim M_{\rm GUT}$ and also (b) why $L_1L_2\Phi_H\Phi_H/M$ is the leading 
operator in its class, rather than being suppressed (due to flavor symmetries)
relative to $L_3L_3\Phi_H\Phi_H/M$ (for example). Both (a) and (b) are needed
for this direct non-seesaw mass to be relevant to the LMA MSW solution.

\bibitem{FN33} A term like $W_{12}$ can be induced in the presence of, for 
example, a singlet $\hat S$ and a ten-plet (${\bf\hat{10}}$), possessing 
effective renormalizable couplings of the form $a_i{\bf 16}_i{\bf 16}_H
{\bf\hat{10}}$, $b{\bf\hat{10}}{\bf 10}_H\hat S$ and mass terms $\hat M_S\hat S
\hat S$ and $\hat M_{10}{\bf\hat{10}}{\bf\hat{10}}$. In this case 
$\kappa_{12}/M_{\rm eff}^3\approx a_1a_2b^2/(\hat M_{10}^2\hat M_S)$. Setting
the charge $a=1/2$ (see Ref. \cite{FN9} and \cite{FN32}), and assigning charges
(-3/2, 5/2) to $({\bf\hat{10}}, \hat S)$, the couplings $a_1$, and $b$ would 
be
flavor-symmetry allowed, while $a_2$ would be suppressed but so also would be
the mass of ${\bf\hat{10}}$ compared to the GUT-scale. One can imagine that 
$\hat S$ on the other hand acquires a GUT-scale mass through for example the 
Dine-Seiberg-Witten mechanism, violating the U(1)-flavor symmetry. One can
verify that in such a picture, one would obtain $\kappa_{12}/M_{\rm eff}^3
\sim 1/M_{\rm GUT}^3$.

\bibitem{FN34} For instance, consider the superpotential for ${\bf 45}_H$ only:
$W({\bf 45}_H)=M_{45}{\bf 45}_H^2+\lambda {\bf 45}_H^4/M$, which yields
(setting $F_{{\bf 45}_H}=0$), either $\langle{\bf 45}_H\rangle=0$, or 
$\langle{\bf 45}_H\rangle^2=-[2M_{45}M/\lambda]$. Assuming that
``other physics'' would favor $\langle{\bf 45}_H\rangle\neq 0$, we see that
$\langle{\bf 45}_H\rangle$ would be pure imaginary, if the square bracket is
positive, with all parameters being real. In a coupled system, it is
conceivable that $\langle{\bf 45}_H\rangle$ in turn would induce phases (other
than "0" and $\pi$) in some of the other VEV's as well, and may itself become
complex rather than pure imaginary.

\bibitem{Wolfenstein} L. Wolfenstein, Phys. Rev. Lett. {\bf 51}, 1945 (1983).

\bibitem{FN36} Within the framework developed in Ref. \cite{BabuJCPCascais}, 
the CP violating phases entering into the SUSY contributions (for example
those entering into the squark-mixings) also arise entirely through phases
in the fermion mass matrices.

\bibitem{FN37} An intriguing feature is the prominence of the 
$\delta_{RR}^{23}(\tilde b_R\rightarrow \tilde s_R)$-parameter which
gets enhanced in part because of the largeness of the $\nu_\mu$-$\nu_\tau$
oscillation angle. This leads to large departures from the
predictions of the standard model, especially in transitions such as
$B_s\rightarrow \bar B_s$ and 
$B_d\rightarrow\Phi K_s\,(b\rightarrow s\bar s s)$ \cite{BabuJCPCascais}. This
feature has independently been noted recently by D. Chang, A. Massiero, 
and H. Murayama .
\bibitem{FN38} As an example, one such fit with complex parameters assigns
\cite{BabuJCPCascais}: $\sigma=0.10-0.012\,i$, $\eta=0.12-0.05\,i$,
$\epsilon=-0.095$, $\eta'=4.0\times 10^{-3}$, $\epsilon'=1.54\times 10^{-4}
e^{i\pi/4}$, $\zeta_{22}^u=1.25\times 10^{-3}e^{i\pi/9}$ and
$\zeta_{22}^d=4\times 10^{-3}e^{i\pi/2}$, ${\cal M}_u^0\approx 110$ GeV,
${\cal M}_D^0\approx 1.5$ GeV, $y\approx -1/17$ (compare with Eq. 
\eqref{eq:fit}
for which $\zeta_{22}^u=\zeta_{22}^d=0$). One obtains as outputs:
$m_{b,s,d}\approx(\mbox{5 GeV, 132 MeV, 8 MeV})$, $m_{c,u}\approx
(\mbox{1.2 GeV, 4.9 MeV})$, $m_{\mu, e}\approx(\mbox{102 MeV, 0.4 MeV})$
with $m_{t,\tau}\approx(\mbox{167 GeV, 1.777 GeV})$, $(V_{us}, V_{cb},
|V_{ub}|, |V_{td}|)\approx(0.217, 0.044, 0.0029, 0.011)$, while preserving
the predictions for neutrino masses and oscillations as in
Eq. \eqref{eq:pred}. The above serves to demonstrate that complexification 
of parameters of the sort presented above can
preserve the successes of Eq. \eqref{eq:pred} (\cite{BPW}). This particular
case leads to
$\eta_W=0.29$ and $\rho_W=-0.187$ \cite{BabuJCPCascais}, to be compared with
the corresponding standard model values (obtained from $\epsilon_K$, $V_{ub}$ 
and $\Delta m_{Bd}$) of $(\eta_W)_{\rm SM}\approx 0.33$ and $(\rho_W)_{\rm SM}
\approx +0.2$. The consistency of
such values for $\eta_W$ and $\rho_W$ (especially reversal of the sign of
$\rho_W$ compared to the SM value), in the light of having both standard
model and SUSY-contributions to CP and flavor-violations, and their
distinguishing tests, are discussed in Ref. \cite{BabuJCPCascais}.

\bibitem{d} For reviews, see chapters 6 and 8 in E. W. Kolb and M. S. Turner, 
``The Early Universe'', Addison-Wesley, 1990.

\bibitem{gravitino} J. Ellis, J. E. Kim and D. Nanopoulos, Phys. Lett.
{\bf 145B}, 181 (1984); M. Yu. Khlopov and A. Linde, Phys. Lett. {\bf 138B},
265 (1984); E. Holtmann, M. Kawasaki, K. Kohri and T. Moroi, .

\bibitem{41} For a specific scenario of inflation and leptogenesis
in the context of SUSY G(224), 
see R. Jeannerot, S. Khalil, G. Lazarides and Q. Shafi, JHEP {\bf 010}, 
012 (2000) , and references therein. 
As noted in this paper, with the VEV's of
(1, 2, 4)$_H$ and (1, 2, $\bar 4$)$_H$ breaking G(224) to the standard model,
and also driving inflation, just the COBE measurement of 
$\delta T/T\approx 6.6\times 10^{-6}$, interestingly enough, implies that the
relevant VEV should be of order $10^{16}$ GeV. 
In this case, the inflaton made of two complex scalar fields (i.e.,
$\theta=(\delta\tilde\nu_H^c+\delta\tilde{\bar\nu}_H^c)/\sqrt{2}$, given by 
the fluctuations of the Higgs fields, and a singlet $S$), each 
 with a mass $\sim 10^{12}$-$10^{13}$ GeV, would decay {\it directly} into
a pair of heavy RH neutrinos -- that is into $N_2N_2$ (or  $N_1N_1$) if
$m_{\rm infl}>2M_2$ (or $2M_1$).
The subsequent decays of $N_2$'s (or $N_1$'s), thus produced, 
into $l+\Phi_H$ and 
$\bar l+\bar\Phi_H$ would produce lepton-asymmetry {\it during the process of
reheating}.
I will comment later on the consistency of this possibility with the fermion
mass-pattern exhibited in Sec. 2. 
I would like to thank Qaisar
Shafi for a discussion on these issues.

\bibitem{epsilon1} L. Covi, E. Roulet and F. Vissani, in Ref. \cite{Others}.

\bibitem{variant} A variant expression $[d=1/(2\sqrt{k^2+9})]$ for the dilution
factor has also been used in the literature for lower values of $k$ 
$(0\leq k\leq 10)$ [see e.g., H. B. Nielsen and Y. Takanishi, Phys. Lett.
{\bf B507}, 241 (2001)]. This would yield a value for $d$ about (2 to 6)
times lower than that obtained from Eq. \eqref{eq:d}, for 
$k\approx (2$ to 0.5). This should be kept in mind in viewing the results
especially in the last column of Table 1.

\bibitem{new45} Because of supersymmetry, lepton asymmetry should of course
receive contributions from out-of-equilibrium decays of heavy sneutrinos
($\tilde N_1$'s), and one must include the ``light'' sleptons in the decay
and scattering processes, as well. These have not been included in our 
considerations for the sake of simplicity. We do not, however, expect them
to alter the lepton asymmetry obtained as above by more than a factor of two.

\bibitem{FN45} Note that for this scenario (with the inflaton decaying into
$2N_1$) the gravitino constraint is very well satisfied even for 
$m_{3/2}\sim 300$ GeV, because the reheating temperature is rather low
($\sim 10^8$ GeV). At the same time, one can allow $N_1$ to be heavier 
(like $2\times 10^{10}$ GeV) than the case considered before 
(like $4\times 10^{9}$ GeV) because it can be produced directly by the decay
of the inflaton rather than from the thermal bath. Since $Y_B\propto 
\epsilon_1 T_{RH}$, $Y_B$ increases as $M_1^2$ for a given $M_2$. Thus, somewhat 
higher values of $M_1$ compatible with the range shown in Eq. \eqref{eq:MajM},
are prefered.

\end{thebibliography}
}
\end{document}

