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%%     Minimal Scenarios for Leptogenesis and CP Violation       %%
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%%    G.C. Branco, R. Gonzalez Felipe, F. R. Joaquim,            %%
%%       I. Masina, M.N. Rebelo and C.A. Savoy                   %%
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\def\dmsol{\Delta m^2_\odot}
\def\dmatm{\Delta m^2_@}
\def\dmthtw{\Delta m^2_{32}}
\def\dmtwon{\Delta m^2_{21}}
\def\dmthon{\Delta m^2_{31}}
\def\tgatm{\tan^2\theta_@}
\def\im{{\rm Im}}
\def\tgsol{\tan^2\theta_\odot}
\def\y{{\rm y}}
\def\nl{\nonumber \\}

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\begin{document}
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\begin{flushright}
FISIST/24\,-2002/GFP\\
SACLAY-T02/144
\end{flushright}

\title{Minimal Scenarios for Leptogenesis and CP Violation}

\author{G. C. Branco}
\email{gbranco@alfa.ist.utl.pt}%
\affiliation{Departamento de F{\'\i}sica and
Grupo de F{\'\i}sica de Part{\'\i}culas (GFP) \\
Instituto Superior T{\'e}cnico, \\ Av.Rovisco Pais, 1049-001
Lisboa, Portugal}

\author{R. Gonz{\'a}lez Felipe}
\email{gonzalez@gtae3.ist.utl.pt}%
\affiliation{Departamento
de F{\'\i}sica and Grupo de F{\'\i}sica de Part{\'\i}culas (GFP) \\
Instituto Superior T{\'e}cnico, \\ Av.Rovisco Pais, 1049-001
Lisboa, Portugal}

\author{F. R. Joaquim}
\email{filipe@gtae3.ist.utl.pt}
\affiliation{Departamento de
F{\'\i}sica and Grupo de F{\'\i}sica de Part{\'\i}culas (GFP) \\ Instituto
Superior T{\'e}cnico, \\ Av.Rovisco Pais, 1049-001 Lisboa, Portugal}

\author{I. Masina}
\email{masina@spht.saclay.cea.fr}
\affiliation{Service de Physique
Th{\'e}orique, Laboratoire de la Direction des Sciences de la
Mati{\`e}re du Commissariat {\`a} l'{\'E}nergie Atomique et Unit{\'e} de
Recherche Associ{\'e}e au CNRS (URA 2306), CEA-Saclay,\\
F-91191 Gif-sur-Yvette, France}

\author{ M. N. Rebelo}
\email{rebelo@alfa.ist.utl.pt}
\affiliation{Departamento de
F{\'\i}sica and Grupo de F{\'\i}sica de Part{\'\i}culas (GFP) \\
Instituto Superior T{\'e}cnico, \\ Av.Rovisco Pais, 1049-001
Lisboa, Portugal}

\author{C. A. Savoy}
\email{savoy@spht.saclay.cea.fr}
\affiliation{Service de Physique
Th{\'e}orique, Laboratoire de la Direction des Sciences de la
Mati{\`e}re du Commissariat
{\`a} l'{\'E}nergie Atomique et Unit{\'e} de Recherche Associ{\'e}e au
CNRS (URA 2306), CEA-Saclay,\\
F-91191 Gif-sur-Yvette, France}%

\begin{abstract}
The relation between leptogenesis and CP violation at low energies
is analyzed in detail in the framework of the minimal seesaw
mechanism. Working, without loss of generality, in a weak basis
where both the charged lepton and the right-handed Majorana mass
matrices are diagonal and real, we consider a convenient generic
parametrization of the Dirac neutrino Yukawa coupling matrix and
identify the necessary condition which has to be satisfied in
order to establish a direct link between leptogenesis and CP
violation at low energies. In the context of the LMA solution of
the solar neutrino problem, we present minimal scenarios which
allow for the full determination of the cosmological baryon
asymmetry and the strength of CP violation in neutrino
oscillations. Some specific realizations of these minimal
scenarios are considered. The question of the relative sign
between the baryon asymmetry and CP violation at low energies is
also discussed.
\end{abstract}
\pacs{13.35.Hb, 14.60.St, 14.60.Pq, 11.30.Er}%
\keywords{Leptogenesis, Neutrino masses and mixing, CP violation.}
\maketitle

\section{Introduction}
\label{intro}%
One of the most exciting recent developments in particle physics
is the discovery of neutrino oscillations pointed out by the
Super-Kamiokande experiment \cite{Fukuda:2001nj} and confirmed by
the Sudbury Neutrino Observatory \cite{Ahmad:2001an}. Neutrino
oscillations provide evidence for non-vanishing neutrino masses
and mixings, with the novel feature that large leptonic mixing
angles are required, in contrast to what happens in the quark
sector. Indeed, the combined results from these experiments
suggest that, in addition to the large mixing angle required by
the atmospheric neutrino data, another large angle should be
present in the leptonic sector. This leads to the so-called large
mixing angle (LMA) solution of the solar neutrino problem which
turns out to be presently the most favored scenario for the
explanation of the solar neutrino deficit. From a theoretical
point of view, understanding the large leptonic mixing is still an
unresolved mystery for which a considerable number of solutions
have been proposed \cite{Masina:2001pp}. On the other hand, the
appearance of neutrino masses much smaller than those of charged
leptons is elegantly explained through the seesaw mechanism
\cite{Yanagida:1979} which can be implemented by extending the
standard model (SM) particle content with right-handed neutrinos.
These can be easily accommodated in grand unified theories (GUT)
where they appear on equal footing with the other SM particles.

The heavy singlet neutrino states can also play an important role
in cosmology, namely, in the explanation of the observed
cosmological baryon asymmetry. During the last few years, the data
collected from the acoustic peaks in the cosmic microwave
background radiation~\cite{Jungman:1995bz} has allowed to obtain a
precise measurement of the baryon asymmetry of the universe (BAU).
The MAP experiment~\cite{MAP} and the PLANCK
satellite~\cite{PLANCK} planned for the near future should further
improve this result. At the present time, the measurement of the
baryon-to-entropy ratio $Y_B=n_B/s$ is
\begin{align}
\label{YBrng}%
0.7 \times 10^{-10} \lesssim Y_B\lesssim 1.0 \times 10^{-10}\,.
\end{align}

Leptogenesis is one of the most attractive mechanisms to generate
the BAU. As first suggested by Fukugita and Yanagida
\cite{Fukugita:1986hr}, the key ingredient in leptogenesis are the
heavy Majorana neutrinos which, once included in the SM, can give
rise to a primordial lepton asymmetry through their
out-of-equilibrium decays. This lepton asymmetry is subsequently
reprocessed into a net baryon asymmetry by the anomalous sphaleron
processes.

In spite of being attractive and successful, leptogenesis turns
out to be extremely difficult or even impossible to test
experimentally in a direct way. This difficulty is obviously
related to the large masses of the heavy Majorana neutrino
singlets. Nevertheless, the joint analysis of leptogenesis and
low-energy neutrino phenomenology can be viewed as an indirect way
of testing it and here the experimental results from neutrino
oscillation experiments such as those related to the search of
leptonic CP violation in the future long-baseline neutrino
experiments are extremely valuable \cite{Lindner:2002vt}.

In this paper, we will address the question of linking the amount
and sign of the BAU to low-energy neutrino experiments, namely to
the sign and strength of the CP asymmetries measured through
neutrino oscillations. Our analysis is performed in the weak basis
(WB) where the charged lepton mass matrix $m_\ell$ and the
right-handed Majorana matrix $M_R$ are both real and diagonal. In
this WB, all CP-violating phases are contained in the Dirac
neutrino mass matrix $m_D$. The matrix $m_D$ is arbitrary and
complex, but since three of its nine phases can be eliminated
through rephasing, one is left with six independent physical
CP-violating phases. In order to study the link between the BAU
generated through leptogenesis and CP violation at low energies,
it is crucial to use a convenient parametrization of $m_D$. We
shall make use of the fact that any arbitrary complex matrix can,
without loss of generality, be written as the product of a unitary
matrix $U$ and a lower triangular matrix $Y_\triangle$. We show
that $U$ contains three phases which do not contribute to
leptogenesis, while the other three phases contained in
$Y_\triangle$ contribute to both leptogenesis and low-energy CP
violation. As a result, a necessary condition for having a link
between leptogenesis and low-energy CP breaking is that the matrix
$U$ contains no phases, the simplest choice being obviously
$U=\openone$. Within this class of Dirac neutrino mass matrices,
we perform a search of the minimal scenarios where not only a good
fit of low-energy neutrino data is obtained but also a link
between the observed size and sign of the BAU and the strength of
CP violation observable at low energies through neutrino
oscillations can be established.
\section{\bf General Framework}
\label{generframe}%
We work in the framework of a minimal extension of the SM which consists
of adding to the standard spectrum one right-handed neutrino per
generation. Before gauge symmetry breaking, the leptonic couplings to the
SM Higgs doublet $\phi$ can be written as:
\begin{align}
{\cal L}_Y  = - Y_\nu\left(\bar{\ell}_L^{\;0},
\bar{\nu}_L^{\;0}\right)\widetilde {\phi}\,\nu_{R}^{\,0} -
Y_{\ell} \left(\bar{\ell}_L^{\;0}, \bar{\nu}_L^{\;0}\right)\phi \
\ell_{R}^{\,0} + {\rm H.c.}\,, \label{Lyuk}
\end{align}
where $\widetilde {\phi} = i \tau _2 \phi ^\ast $. After
spontaneous gauge symmetry breaking the leptonic mass terms are
given by
\begin{align}
{\cal L}_m  &= -\left[\,\bar{\nu}_L^{\;0} m_D \nu_{R}^{\,0} + \frac{1}{2}
\nu_{R}^{0\,T} C M_R \nu_{R}^{\,0}+
\bar{\ell}_L^{\;0} m_{\ell}\,\ell_R^{\,0}\,\right] + {\rm H.c.} \nonumber \\
&= - \left[\frac{1}{2}\,n_{L}^{T} C {\cal M}^* n_L+
\bar{\ell}_L^{\;0} m_{\ell}\,\ell_R^{\,0} \right] + {\rm H.c.}
\,,\label{Lmass}
\end{align}
where $m_D = v\,Y_\nu$ is the Dirac neutrino  mass matrix with
$v=\langle \phi^{\,0} \rangle /\sqrt{2} \simeq 174\,$GeV, $M_R$
and $m_\ell=v\,Y_{\ell}$ denote the right-handed Majorana neutrino
and charged lepton mass matrices, respectively, and $n_L=
({\nu}_{L}^0, {(\nu_R^0)}^c)$. Among all the terms, only the
right-handed neutrino Majorana mass term is SU(2) $\times$ U(1)
invariant and, as a result, the typical scale of $M_R$ can be much
above  the electroweak symmetry breaking scale $v$, thus leading
to naturally small left-handed Majorana neutrino masses of the
order $m^2_D/M_R$ through the seesaw mechanism. In terms of
weak-basis eigenstates the leptonic charged current interactions
are given by:
\begin{align}
{\cal L}_W  = -\frac{g}{\sqrt 2}W^{+}_{\mu}\,\bar{\nu}_L^{\;0} \,
\gamma ^{\mu}\,\ell_L^{\,0} + {\rm H.c.}\,. \label{Lcc1}
\end{align}
It is clear from Eqs. (\ref{Lmass}) and (\ref{Lcc1}) that it is
possible to choose, without loss of generality, a weak basis (WB)
where both $m_\ell$ and $M_R$ are diagonal, real and positive.
Note that in this WB, $m_D$ is a general complex matrix which
contains all the information on CP-violating phases. Since in the
present framework there is no $\Delta L=2$ mass term of the form
$\frac{1}{2} \nu_{L}^{0\,T} C M_L \nu_{L}^0$, the total number of
CP-violating phases for $n$ generations is given by $n(n -1)$
\cite{Endoh:2000hc} which are all contained in $m_D$ in this
special weak basis\footnote{The counting of independent
CP-violating phases for the general case, where besides $m_D$ and
$M_R$ there is also a left-handed Majorana mass term at tree level
has been discussed in Ref.~\cite{Branco:gr}.}.

We recall that the full $6 \times 6$ neutrino mass matrix $\mathcal{M}$ is
diagonalized via the transformation:
\begin{align}
V^T {\cal M}^* V = \cal D , \label{Mnudi}
\end{align}
where ${\cal D} ={\rm diag} (m_1, m_2, m_3, M_1, M_2, M_3)$, with $m_i$
and $M_i$ denoting the physical masses of the light and heavy Majorana
neutrinos, respectively. It is convenient to write $V$ and $\cal D$ in the
following form, together with the definition of $\cal M$ :
\begin{align}
V= \left (\begin{array}{cc}
K & Q \\
S & T \end{array}\right) \;\;,\;\;{\cal D}=\left(\begin{array}{cc}
d_\nu & 0 \\
0 & D_R \end{array}\right)\;\;,\;\;{\cal M}= \left (\begin{array}{cc}
0 & m_D \\
m_D^{\,T} & M_R \end{array}\right).
\end{align}
From Eq. (\ref{Mnudi}) one obtains, to an excellent approximation, the
seesaw formula:
\begin{align}
d_\nu\simeq -K^\dagger\, m_D\, M_R^{-1}\, m_D^{\,T}\, K^* \equiv
K^\dagger\,\mathcal{M}_\nu\, K^*\,, \label{ssaw}
\end{align}
where $\mathcal{M}_\nu$ is the usual light neutrino effective mass
matrix. The leptonic charged-current interactions are given by
\begin{align}
- \frac{g}{\sqrt{2}} \left( \bar{\ell}_{L} \,\gamma_{\mu} \, K
{\nu}_{L} + \bar{\ell}_{L}\, \gamma_{\mu}\, Q\, N_{L} \right)
W^{\mu} +{\rm H.c.}\,, \label{Lcc2}
\end{align}
where $\nu_i$ and  $N_i$ denote the light and heavy neutrino mass
eigenstates, respectively. The matrix $K$ which contains all
information on mixing and CP violation at low energies can then be
parametrized, after eliminating the unphysical phases, by $K= {
U_{\delta}} P$ with $P ={\rm diag}(1, e^{i\,\alpha},e^{i\,\beta})$
($\alpha$ and $\beta$ are Majorana phases) and $U_{\delta}$ a
unitary matrix which contains only one (Dirac-type) phase
$\delta$. In the limit where the heavy neutrinos exactly decouple
from the theory, the matrix $K$ is usually referred as the
Pontecorvo-Maki-Nakagawa-Sakata mixing matrix, which from now on
we shall denote as $U_\nu$.

\medskip \bigskip
\textbf{CP violation in neutrino oscillations}
\medskip \bigskip

It has been shown \cite{Branco:2001pq} that the strength of CP
violation at low energies, observable for example through neutrino
oscillations, can be obtained from the following low-energy WB
invariant:
\begin{align}
{\cal T}_{CP} = {\rm Tr}\left[\,\mathcal{H}_{\nu}, H_\ell
\,\right]^3=6\,i \,\Delta_{21}\,\Delta_{32}\,\Delta_{31}\,{\rm Im}
\left[\,
(\mathcal{H}_{\nu})_{12}(\mathcal{H}_{\nu})_{23}(\mathcal{H}_{\nu})_{31}\,
\right]\,, \label{TCP}
\end{align}
where $\mathcal{H}_{\nu}=\mathcal{M}_\nu\,\mathcal{M}_\nu^{\dag}$,
$H_\ell=m_\ell\,{m_\ell}^{\dagger}$ and
$\Delta_{21}=({m_{\mu}}^2-{m_e}^2)$ with analogous expressions for
$\Delta_{31}$, $\Delta_{32}$. This relation can be computed in any
weak basis. The low-energy invariant (\ref{TCP}) is sensitive to
the Dirac-type phase $\delta$ and vanishes for $\delta=0$. On the
other hand, it does not depend on the Majorana phases $\alpha$ and
$\beta$ appearing in the leptonic mixing matrix. The quantity
${\cal T}_{CP}$ can be fully written in terms of physical
observables once
\begin{align}
{\rm Im} \left[\, (\mathcal{H}_{\nu})_{12}(\mathcal{H}_{\nu})_{23}
(\mathcal{H}_{\nu})_{31}\, \right] = - \dmtwon\,\dmthon\,\dmthtw\,{\cal
J}_{CP}\,, \label{ImHHH}
\end{align}
where the $\Delta m_{ij}^2$'s are the usual light neutrino mass
squared differences and ${\cal J}_{CP}$ is the imaginary part of
an invariant quartet appearing in the difference of the
CP-conjugated neutrino oscillation probabilities
$P(\nu_e\rightarrow\nu_\mu)-P(\bar{\nu}_e\rightarrow
\bar{\nu}_\mu)$. One can easily get:
\begin{align}
{\cal J}_{CP} &\equiv {\rm Im}\left[\,(U_\nu)_{11} (U_\nu)_{22}
(U_\nu)_{12}^\ast (U_\nu)_{21}^\ast\,\right] \nonumber \\
&= \frac{1}{8} \sin(2\,\theta_{12})
\sin(2\,\theta_{13})\sin(2\,\theta_{23}) \sin \delta\,, \label{Jgen1}
\end{align}
where the $\theta_{ij}$ are the mixing angles appearing in the standard
parametrization adopted in \cite{Hagiwara:pw}. Alternatively, one can use
Eq.~(\ref{ImHHH}) and write:
\begin{align} {\cal J}_{CP}=-\frac{{\rm Im}\left[\,
(\mathcal{H}_{\nu})_{12}(\mathcal{H}_{\nu})_{23}(\mathcal{H}_{\nu})_{31}\,
\right]}{\dmtwon\,\dmthon\,\dmthtw}\,. \label{Jfin}
\end{align}
This expression has the advantage of allowing the computation of
the low-energy CP invariant without resorting to the mixing matrix
$U_\nu$.

It is also possible to write WB invariants useful to leptogenesis
\cite{Branco:2001pq} as well as WB invariant conditions for CP
conservation in the leptonic sector relevant in specific frameworks
\cite{Branco:gr,Branco:1999bw}.

\section{{\bf CP asymmetries in heavy Majorana neutrino decays}}
\label{CPasymmetries}%
The starting point in leptogenesis scenarios is the $CP$ asymmetry
generated through the interference between tree-level and one-loop
heavy Majorana neutrino decay diagrams. In the simplest extension
of the SM, such diagrams correspond to the decay of the Majorana
neutrino into a lepton and a Higgs boson. Considering the decay of
one heavy Majorana neutrino $N_j$, this asymmetry is given by:
\begin{align} \label{epsin}
\varepsilon_j=\frac{\Gamma\,(N_j \rightarrow \ell\,\phi)-\Gamma
\,(N_j \rightarrow \bar{\ell}\,\phi^{\,\dag})}{\Gamma\,(N_j
\rightarrow \ell\,\phi)+\Gamma\,(N_j \rightarrow
\bar{\ell}\,\phi^{\,\dag})}\ .
\end{align}
In terms of the Dirac neutrino Yukawa couplings the CP asymmetry
(\ref{epsin}) is \cite{Covi:1996wh}:
\begin{align}
\label{epsj1} \varepsilon_j=\frac{1}{8\pi(Y_\nu^{\dag}
Y_\nu^{})_{jj}} \sum_{k\neq j}\,\im[\,(Y_\nu^{\dag}\,Y_\nu^{})_
{jk}^2\,]\,f\!\left (\frac{M_k^{\,2}}{M_j^{\,2}}\right)\,,
\end{align}
where $f(x)$ stands for the loop function which includes the
one-loop vertex and self-energy corrections to the heavy neutrino
decay amplitudes,
\begin{align}
\label{f} f(x)=\sqrt{x} \left[\,(1+x)\ln \left(\frac{x}
{1+x}\right)+\frac{2-x}{1-x}\,\right]\,. \end{align}%
From Eq.~(\ref{epsj1}) it can be readily seen that the CP
asymmetries are only sensitive to the CP-violating phases
appearing in $Y_\nu^\dagger Y_\nu^{}$ (or equivalently in
$m_D^\dagger m_D^{}$) in the WB where $M_R$ and $m_\ell$ are
diagonal.

\subsection{Hierarchical case: $\bm{M_1 < M_2 \ll M_3}$ }

In the hierarchical case $M_1 < M_2 \ll M_3$, only the decay of
the lightest heavy neutrino $N_1$ is relevant for leptogenesis,
provided the interactions of $N_1$ are in thermal equilibrium at
the time $N_{2,3}$ decay, so that the asymmetries produced by the
latter are erased before $N_1$ decays. In this situation, it is
sufficient to take into account the CP asymmetry $\varepsilon_1$.
Since in the limit $x \gg 1$ the function $f(x)$ can be
approximated by\footnote{This approximation can be reasonably used
for $x \gtrsim 15$.} $f(x)\simeq -3/(2\sqrt{x})$, we have from
Eq.~(\ref{epsj1})
\begin{align}
\label{eps1} \varepsilon_1=-\frac{3}{16\pi(Y_\nu^{\dag}
Y_\nu^{})_{11}} \sum_{k=2,3}\,\im[\,(Y_\nu^{\dag}\,Y_\nu^{})_
{1k}^2\,]\,\frac{M_1}{M_k}\,,
\end{align}
which can be recast in the form \cite{Buchmuller:2000nd}
\begin{align}
\varepsilon_1 \simeq -\frac{3\,M_1}{16\,\pi}\frac{ {\rm
Im}\left[\,Y_\nu^\dagger\,Y_\nu^{}\,D_R^{-1}\,Y_\nu^T\,Y_\nu^\ast\,\right]
_{11}}{(Y_\nu^{\dag}Y_\nu^{})_{11}}=\frac{3\,M_1}{16\,\pi\,v^2}\frac{
{\rm Im}\left[\,Y_\nu^\dagger\,\mathcal{M}_{\nu}\,
Y_\nu^\ast\,\right]_{11}} {(Y_\nu^{\dag} Y_\nu^{})_{11}}\,,
\label{ep1Mn}
\end{align}
using the seesaw relation given in Eq.~(\ref{ssaw}).
\subsection{Two-fold quasi-degeneracy: $\bm{M_1 \simeq M_2 \ll M_3}$}
\label{degenRH}%

In the context of thermal leptogenesis, when the observed baryon
asymmetry is generated through the decays of the lightest heavy
Majorana neutrino $N_1$, there exists an upper bound on the CP
asymmetry $\varepsilon_1$ which directly depends on the mass of
the lightest neutrino $M_1$ \cite{Davidson:2002qv}. In turn, such
a bound implies a lower bound on the lightest mass $M_1$,
typically $M_1 \gtrsim 10^{8}$~GeV. The latter bound\footnote{A
more stringent constraint, $M_1 \gtrsim 10^{10}$~GeV, is obtained
in \cite{Ellis:2002eh}.} is however barely compatible with the
reheating temperature bound $T_R \lesssim 10^{8} \sim 10^9$~GeV
required in several supergravity models in order to avoid a
gravitino overproduction \cite{Ellis:1984eq}. To overcome this
problem\footnote{Another possible solution to the gravitino
problem is to consider non-thermal production mechanisms
\cite{Giudice:1999fb}. Since in these cases the condition $M_1 <
T_R$ is not required, the gravitino problem is easily avoided once
heavy particles can be created with a relatively low reheating
temperature without threatening big bang nucleosynthesis.} one can
consider, for instance, the decays of two heavy neutrinos which
are quasi-degenerate in mass, $M_1 \simeq M_2$. In this case, the
CP asymmetries $\varepsilon_i$ are enhanced due to self-energy
contributions \cite{Flanz:1994yx} and the required baryon
asymmetry can be produced by right-handed heavy neutrinos with
moderate masses $M_1 \simeq M_2 \lesssim 10^8$~GeV. Moreover, it
has been shown that in the presence of small Dirac-type leptonic
mixing at high energies and GUT-inspired Dirac neutrino Yukawa
couplings, the heavy Majorana neutrino degeneracy is compatible
with the LMA solar solution \cite{GonzalezFelipe:2001kr}.

Let us assume that the heavy Majorana neutrinos $N_1$ and $N_2$
are quasi-degenerate. It is useful to define the parameter
$\delta_N$ which represents the degree of degeneracy between the
masses $M_1$ and $M_2$ as
\begin{align}
\label{deltN} \delta_N=\frac{M_2}{M_1}-1\,.
\end{align}
Since $M_1 \simeq M_2\ $, we expect $\delta_N \ll 1$. For the
perturbative approach to remain valid, the tree-level decay width
$\Gamma_i$ for each of the heavy Majorana neutrinos must be much
smaller than the mass difference between them. This is translated
into the relations
\begin{align}
\label{crit1} \Gamma_i = \frac{(H_\nu)_{ii}\,M_i}{8\,\pi} \ll
M_2-M_1 =\delta_N\,M_1\,, \quad i=1,2\,,
\end{align}
where $H_\nu=Y_\nu^\dagger\,Y_\nu$. From this equation we can find
the following lower bound for $\delta_N$:
\begin{align}
\label{dNlim} \delta_N \gg {\rm
max}\left\{\frac{(H_\nu)_{ii}\,M_i}{8\,\pi\,M_1}\right\}_{i=1,2}\,.
\end{align}
Assuming that this criterium is verified, the CP asymmetries
$\varepsilon_i$ can be obtained combining Eqs.~(\ref{epsj1}) and
(\ref{deltN}). We find
\begin{align}
\label{e1e2d}
\varepsilon_1&=-\frac{1}{8\,\pi\,(H_\nu)_{11}}\left\{{\rm
Im}\,[\,(H_\nu)_{21}^2\,]\,f[\,(1+\delta_N)^2\,]-\frac{3}{2}\,{\rm
Im} [\,(H_\nu)_{31}^2\,]\, \frac{M_1}{M_3} \right\}\,,\nl
\varepsilon_2&=-\frac{1}{8\,\pi\,(H_\nu)_{22}}\left\{{\rm
Im}\,[\,(H_\nu)_{12}^2\,]\,f[\,(1+\delta_N)^{-2}\,]-\frac{3}{2}\,{\rm
Im} [\,(H_\nu)_{32}^2\,]\, \frac{M_2}{M_3}\right\}\,.
\end{align}
Taking into account that for $\delta_N \ll 1$ the function
$f[\,(1+\delta_N)^{\pm 2}\,]$ can be approximated by\footnote{For
$\delta_N < 10^{-2}$ the error associated to this approximation is
less than 1\%.}:
\begin{align}
\label{fapp2} f[\,(1+\delta_N)^{\pm 2}\,]\simeq \mp
\frac{1}{2\,\delta_N}\,,
\end{align}
we obtain:
\begin{align}\label{e12d2}
\varepsilon_j&=\frac{1}{16\,\pi\,(H_\nu)_{jj}}\left[\frac{{\rm
Im}\,[\,(H_\nu)_{21}^2\,]}{\delta_N}+ 3\,{\rm Im}
[\,(H_\nu)_{3j}^2\,]\, \frac{M_j}{M_3}\right]\;\;,\;\; j=1,2\,.
\end{align}
Typically, the term proportional to $M_j/M_3$ can be neglected and
in this case $\varepsilon_1$ and $\varepsilon_2$ have the same
sign. This aspect turns out to be relevant for the discussion on
the relative sign between the BAU and low-energy leptonic CP
violation.
\section{On the connection between leptogenesis and low-energy CP violation}
\label{parametrizations}%
In this section we analyze the possible link between CP violation at low
energies, measurable for example through neutrino oscillations, and
leptogenesis. The possibility of such a connection has been previously
analyzed in the literature
\cite{Branco:2001pq,Branco:2002kt,Frampton:2002qc}. Nevertheless, we find
it worthwhile presenting here a thorough discussion on the subject. In
particular, we will address the following questions:
\begin{itemize}
\item{If the strength of CP violation at low energies in neutrino
oscillations is measured, what can one infer about the viability
or non-viability of leptogenesis?}%
\item{From the sign of the BAU, can one predict the sign of the CP
asymmetries at low energies, namely the sign of $\mathcal{J}_{CP}$?}
\end{itemize}
We will show that having an explicit parametrization of $m_D$ (or
equivalently of $Y_\nu=m_D/v$) is crucial not only to determine
which phases are responsible for leptogenesis and which ones are
relevant for leptonic CP violation at low energies, but also to
analyze the relationship between these two phenomena.

From the available neutrino oscillation data, one obtains some
information on the effective neutrino mass matrix
$\mathcal{M}_\nu$ which can be decomposed in the following way:
\begin{align}
U_{\nu}\,d_\nu\,U_{\nu}^{\,T}=\mathcal{M}_{\nu} \equiv
L\,L^T\;,\;L\equiv i\,m_D\,D_R^{-1/2}\,. \label{MnuLL}
\end{align}
The extraction of $L$ from $\mathcal{M}_\nu$ suffers from an
intrinsic ambiguity \cite{Casas:2001sr} in the sense that, given a
particular solution $L_0$ of Eq.~(\ref{MnuLL}), the matrix
$L=L_0\,R$ will also satisfy this equation, provided that $R$ is
an arbitrary orthogonal complex matrix, $R \in O(3, \mathbb{C})$,
i.e. $R\,R^T=\openone$. It is useful to take as a reference
solution $L_0 \equiv U_\nu\,d_\nu^{1/2}$, so that:
\begin{align}
\label{L2} L \equiv U_\nu\,d_\nu^{1/2} R\,.
\end{align}
Since three of the phases of $m_D$ can be eliminated, the matrix
$L$ has 15 independent parameters. The parametrization of $L$
given in Eq.~(\ref{L2}) has the interesting feature that all its
parameters are conveniently distributed among $U_\nu$, $d_\nu$ and
$R$, which contain 6 (3 angles + 3 phases), 3 and 6 (3 angles + 3
phases) independent parameters, respectively. Of the 18 parameters
present in the Lagrangian of the fundamental theory described by
$m_D$ and $D_R$, only 9 appear at low energy in $\mathcal{M}_\nu$
through the seesaw mechanism. To further disentangle $m_D$ from
$D_R$ in $L$, one needs the 3 remaining inputs, namely the three
heavy Majorana masses of $D_R$. As for the meaning of the
information encoded in $R$, it turns out that the pattern of this
matrix has a suggestive interpretation in terms of the different
r\^oles played by the heavy neutrinos in the seesaw mechanism. In
fact, $R$ can be viewed as a dominance matrix
\cite{Lavignac:2002gf} since it gives the weights of each heavy
Majorana neutrino in the determination of the different light
neutrino masses $m_i$ \cite{Masina:2002qh}. The fact that
$R_{ij}^2$ are weights for $m_i$ is quite obvious due to the
orthogonality of $R$:
\begin{align}
m_i  = \sum_j  m_i R_{ij}^{\,2}\,.
\end{align}
On the other hand, since $U_\nu^{\,\dagger}\,m_D =-i\,
d_\nu^{1/2}R\, D_R^{1/2}$, the single contribution $m_i
R_{ij}^{\,2}$ is also given by:
\begin{align}
m_i R_{ij}^{\,2}= -\frac{(U_\nu^\dagger m_D)^2_{ij}}{M_j} \equiv
\frac{X_{ij}}{M_j}\,.
\end{align}
Therefore, once $U_\nu$ is fixed, each weight $R_{ij}^2$ just
depends on the mass $M_j$ of the j-th heavy Majorana neutrino and
on its couplings with the left-handed neutrinos $(m_D)_{kj}$.
Thus, the contribution of each heavy neutrino to $m_i$ is well
defined and expressed by the weight ${\rm Re}(R_{ij}^2)$. One may
roughly say that the heavy Majorana neutrino with mass $M_j$
dominates\footnote{See for instance Refs.~\cite{Smirnov:af,
Altarelli:1999dg} for other approaches to the dominance
mechanism.} in $m_i$ if
\begin{align}
\frac{|{\rm Re}\,(X_{ij})|}{M_j} \gg \frac{|{\rm
Re}\,(X_{ik})|}{M_k}\;\;,\;\;k\ne j
\end{align}
which implies $|{\rm Re}(R_{ij}^{\,2})| \gg |{\rm
Re}(R_{ik}^{\,2})|\,$. So that, if one of the heavy Majorana
neutrino neutrino gives the dominant contribution to one of the
masses $m_i$, this information is encoded in the structure of $R$.
The interpretation in terms of weights is straightforward for the
rotational part of $R$. However, one has to be careful because in
the presence of the three boosts (controlled by the three phases)
the weights $R_{ij}^2$ are not necessarily real and positive.
Although this situation is more subtle, the above dominance
arguments still hold.

Coming back to the connection between leptogenesis and low-energy
data, it is important to note that $U_\nu$ does not appear in the
relevant combination for leptogenesis $Y_\nu^{\dag}Y_\nu^{}$, in
the same way as $R$ does not appear in $\mathcal{M}_\nu$. Indeed,
one has:
\begin{align}
 m_D^\dagger m_D^{} = D_R^{1/2}\,R^{\dagger}\,d_\nu\,R\,D_R^{1/2}
 \,.\label{HHdag}
\end{align}
From the above discussion, it follows that it is possible to write
$m_D$ in the form $m_D=-i\,U_\nu\,d_{\nu}^{1/2}R\,D_R^{1/2}$ in
such a way that leptogenesis and the low-energy neutrino data
(contained in $\mathcal{M}_\nu$) depend on two independent sets of
CP-violating phases, respectively those in $R$ and those in
$U_{\nu}$. In particular, one may have viable leptogenesis even in
the limit where there are no CP-violating phases (neither Dirac
nor Majorana) in $U_\nu$ and hence, no CP violation at low
energies \cite{Rebelo:2002wj}. Therefore, in general it is not
possible to establish a link between low-energy CP violation and
leptogenesis. This connection is model dependent: it can be drawn
only by specifying a particular \emph{ansatz} for the fundamental
parameters of the seesaw, $m_D$ and $D_R$, as will be done in the
following sections.

The relevance of the matrix $R$ for leptogenesis can be rendered
even more explicit \cite{Masina:2002qh} by rewriting the
$\varepsilon_1$ asymmetry by means of Eq.~(\ref{HHdag}) and
defining $R_{ij}= |R_{ij}| e^{i \varphi_{ij}/2}$, $\dmsol
\equiv\dmtwon$ and $\dmatm \equiv \dmthtw$. In the case of
hierarchical heavy Majorana neutrinos, say $M_1 \ll M_2 \ll M_3$
one obtains
\begin{align}
\varepsilon_1 \simeq \frac{3}{16\pi} \frac{M_1}{v^2} \frac {
\dmatm |R_{31}|^2 \sin \varphi_{31}  - \dmsol  |R_{11}|^2
\sin\varphi_{11}} { m_1|R_{11}|^2 +m_2|R_{21}|^2
+m_3|R_{31}|^2}\,, \label{edb}
\end{align}
and we recover what one would have expected by intuition, namely
that the physical quantities involved in determining
$\varepsilon_1$ are just $M_1$, the spectrum of the light
neutrinos, $m_i$, and the first column of $R$, which expresses the
composition of the lightest heavy Majorana neutrino in terms of
the light neutrino masses $m_i$. In the case $M_1 \simeq M_2 \ll
M_3$, similar expressions hold,
\begin{align}
\varepsilon_j \simeq \frac{1}{16 \pi v^2} \frac{M_1 M_2}{M_2-M_1}
\frac{{\rm Im}[(R^\dagger d_\nu R)_{21}^2\,]} {(R^\dagger d_\nu
R)_{jj}}\,, \quad j=1,2 \,.
\end{align}
where now also the mass $M_2$ and the second column of $R$ are involved. A
detailed study of the relevance of the matrix $R$ for leptogenesis is
under way \cite{inprep}.

As stressed before, different \emph{ans\"atze} for $R$ have no
direct impact on CP violation at low energy; the impact is in a
sense indirect because $R$ specifies if dominance of some heavy
Majorana neutrino is at work in the seesaw mechanism
\cite{Lavignac:2002gf}.

In conclusion, the link between leptogenesis and low-energy CP
violation can only be established in the framework of specific
\emph{ans\"atze} for the leptonic mass terms of the Lagrangian. We
shall derive a necessary condition for such a link to exist. In
order to obtain this connection, it is convenient to use a
triangular parametrization for $m_D$, which we describe next.

\medskip \bigskip
\textbf{Triangular parametrization}
\medskip \bigskip

It can be easily shown that any arbitrary complex matrix can be
written as the product of a unitary matrix U with a lower
triangular matrix $Y_{\triangle}$. In particular, the Dirac
neutrino mass matrix can be written as:
\begin{align}
m_D = v\,U\,Y_{\triangle}\,, \label{mDtri}
\end{align}
with $Y_{\triangle}$ of the form:
\begin{align}
Y_{\triangle}= \left(\begin{array}{ccc}
y_{11} & 0 & 0 \\
y_{21}\,e^{i\,\phi_{21}} & y_{22} & 0 \\
y_{31}\,e^{i\,\phi_{31}} & y_{32}\,e^{i\,\phi_{32}} & y_{33}
\end{array}
\right)\,, \label{Ytri1}
\end{align}
where $y_{ij}$ are real positive numbers. Since $U$ is unitary, in
general it contains six phases. However, three of these phases can
be rephased away by a simultaneous phase transformation on
${\nu}_{L}^{\,0}$, $\ell_{L}^{\,0}$, which leaves the leptonic
charged current invariant. Under this transformation, $m_D
\rightarrow P_{\xi} m_D $, with $P_{\xi}={\rm diag} \left(e^{i \,
\xi_1},e^{i \, \xi_2},e^{i \, \xi_3} \right)$. Furthermore,
$Y_{\triangle}$ defined in Eq.~(\ref{Ytri1}) can be written as:
\begin{align}
Y_{\triangle}= {P_{\beta}^\dagger}\ {\hat Y_{\triangle}}\ P_{\beta}\,,
\label{Ytri2}
\end{align}
where $P_\beta ={\rm diag} (1, e^{i \, \beta_1}, e^{i \,
\beta_2})$ with $\beta_1=-\phi_{21}$, $\beta_2=-\phi_{31}$ and
\begin{align}
{\hat Y_{\triangle}}= \left(\begin{array}{ccc}
y_{11} & 0 & 0 \\
y_{21}  & y_{22} & 0 \\
y_{31}  & y_{32}\,e^{i \, \sigma} & y_{33}
\end{array}
\right) \,,\label{Ytri3}
\end{align}
with $\sigma=\phi_{32}- \phi_{31}+ \phi_{21}$. It follows then
from Eqs.~(\ref{mDtri})~and~(\ref{Ytri2}) that the matrix $m_D$
can be decomposed in the form
\begin{align}
m_D=v\,U_{\rho}\,P_{\alpha}\,{\hat Y_{\triangle}}\,P_{\beta}\,,
\label{mDdec}
\end{align}
where $P_\alpha ={\rm diag} (1, e^{i \, \alpha_1}, e^{i \,
\alpha_2})$ and $U_\rho$ is a unitary matrix containing only one
phase $\rho$. Therefore, in the WB where $m_\ell$ and $M_R$ are
diagonal and real, the phases $\rho$, $\alpha_1$, $\alpha_2$,
$\sigma$, $\beta_{1}$ and $\beta_{2}$ are the only physical phases
characterizing CP violation in the leptonic sector. The phases
relevant for leptogenesis are those contained in $m_D^{\,\dagger}
m_D^{}$. From Eqs.~(\ref{Ytri2})-(\ref{mDdec}) we conclude that
these phases are $\sigma$, $\beta_{1}$ and $\beta_{2}$, which are
linear combinations of the phases $\phi_{ij}$. On the other hand,
all the six phases of $m_D$ contribute to the three phases of the
effective neutrino mass matrix at low energies
\cite{Branco:2001pq} which in turn controls CP violation in
neutrino oscillations. Since the phases $\alpha_1$, $\alpha_2$ and
$\rho$ do not contribute to leptogenesis, it is clear that a
necessary condition for a direct link between leptogenesis and
low-energy CP violation to exist is the requirement that the
matrix $U$ in Eq. (\ref{mDtri}) contains no CP-violating phases.
Note that, although the above condition was derived in a specific
WB and using the parametrization of Eq. (\ref{mDtri}), it can be
applied to any model. This is due to the fact that starting from
arbitrary leptonic mass matrices, one can always make WB
transformations to render $m_\ell$ and $M_R$ diagonal, while $m_D$
has the form of Eq.~(\ref{mDtri}). A specific class of models
which satisfy the above necessary condition in a trivial way are
those for which $U=\openone$, leading to $m_D=v\,Y_\triangle$.
This condition is necessary but not sufficient to allow for a
prediction of the sign of the CP asymmetry in neutrino
oscillations, given the observed sign of the BAU together with the
low-energy data. Therefore, we will consider next a more
restrictive class of matrices $m_D$ of this form and we will show
that, in an appropriate limiting case, our structures for $m_D$
lead to the ones assumed by Frampton, Glashow and Yanagida in
\cite{Frampton:2002qc}.

\section{Minimal Scenarios}
\label{minimal}

From the analysis carried out in the previous section, it becomes
clear that the computation of the cosmological baryon asymmetry
$Y_B$ in leptogenesis scenarios strongly depends on the Yukawa
structure of the Dirac neutrino mass term and on the heavy
Majorana neutrino mass spectrum. Moreover, if one assumes that the
seesaw mechanism is responsible for the smallness of the neutrino
masses, then the connection between the baryon asymmetry and
low-energy neutrino physics is unavoidable. In fact, this
constitutes an advantage for the leptogenesis mechanism when
compared to other baryogenesis scenarios. In the context of
supersymmetric extensions of the SM it is possible (although not
always simple) to combine the study of leptogenesis and neutrino
physics with other physical phenomena like flavor-violating decays
\cite{Ellis:2002eh,Ellis:2002xg}. In general, this analysis does
not give us definite answers, yet it may help to discriminate
among certain neutrino mass models.

Recently, a considerable amount of work has been done aiming at relating
viable leptogenesis to all the available low-energy neutrino data coming
from solar, atmospheric and reactor experiments \cite{Buchmuller:2001dc}.
Roughly speaking, two different approaches to the problem are to be found.
The first one is based on the computation of the baryon asymmetry as a
function of the lightest heavy Majorana neutrino mass $M_1$, the CP
asymmetry $\varepsilon_1$ and the so-called effective neutrino mass
$\tilde{m}_1=(m_D^{\dagger}\,m_D^{})_{11}/M_1$ \cite{Plumacher:1996kc}. By
solving the Boltzmann equations, this kind of analysis provides valuable
information on the ranges of these parameters that lead to an acceptable
value of $Y_B$. The weak point of this procedure lies on the fact that the
input information depends on quantities which are not sensitive to the
full structure of $Y_\nu$ and $M_R$ and, therefore, no further conclusions
can be drawn about the class of models which can lead to acceptable values
of the input parameters referred above. In fact, the values of $M_1$,
$\tilde{m}_1$ and $\varepsilon_1$ should not be taken as independent
parameters. The second approach is based upon initial assumptions on the
structure of $Y_\nu$ and $M_R$ at high energies which are fixed by
recurring to theoretical arguments like for example grand unified theories
or flavor symmetries. Although in this framework some generality is lost,
it has the advantage that one can compute the generated baryon asymmetry
and, simultaneously, perform a low-energy neutrino data analysis. It is
precisely the latter approach that we shall follow in the present work.

In this section we present a class of minimal scenarios for
leptogenesis based on the triangular decomposition of $Y_\nu$
given in Eqs.~(\ref{mDtri}) and (\ref{Ytri1}). Namely, we would
like to answer the question of how simple can the structure of the
Dirac neutrino Yukawa coupling matrix be in order not only to get
an acceptable value of the baryon asymmetry but also to
accommodate the neutrino data provided by the atmospheric, solar
and reactor neutrino experiments. In particular, we require the
non-vanishing of the CP asymmetry generated in the decays of the
lightest heavy Majorana neutrino, since the final value of the
baryon asymmetry crucially depends on this quantity. Throughout
our analysis we shall also consider the predictions on the
CP-violating effects at low energies.

In the previous section we have seen that the Dirac neutrino
Yukawa coupling matrix $Y_\nu$ can be decomposed into the product
of a unitary matrix $U$ and a lower-triangular matrix
$Y_\triangle$ (cf. Eq.~\ref{mDtri}). It was also shown that the CP
asymmetries $\varepsilon_j$ generated in the heavy Majorana
neutrino decays do not depend on the matrix $U$. In the special
case $U=\openone$ it is possible to establish the connection
between leptogenesis, low-energy CP violation and neutrino mixing,
since the same phases affect these phenomena. We classify this
scenario as a minimal scenario for leptogenesis and CP violation
in the sense that the CP-violating sources that do not contribute
to leptogenesis are neglected. On the other hand, if $U \neq
\openone$ this connection is not trivial. Therefore, from now on
we will consider the case $U=\openone$ which implies the following
simple structure for the Dirac neutrino mass matrix:
\begin{align}
m_D=v\,Y_\triangle = v\,\left(\begin{array}{ccc}
 y_{11}   & \; 0       &\; 0 \\
 y_{21}\,e^{i\,\phi_{21}}   &\; y_{22}       &\; 0 \\
y_{31}\,e^{i\,\phi_{31}}   &\; y_{32}\,e^{i\,\phi_{32}} &\; y_{33}
\end{array}\right)\label{MDmin}\,.
\end{align}
Then, from Eq.~(\ref{epsj1}) the CP asymmetry generated in the
decay of the heavy Majorana neutrino $N_j$ is
\begin{align}
\label{ejtri} \varepsilon_j=-\frac{1}{8\pi(H_\triangle)_{jj}} \sum_{i\neq
j}\,\im[(H_\triangle)_{ij}^2]\,f_{ij}\,,
\end{align}
where
\begin{align}
\label{fij}
 H_\triangle =Y^{\dag}_\triangle Y_\triangle^{} \quad,\quad
 f_{ij}=f\!\left(\frac{M_i^{\,2}}{M_j^{\,2}}\right)\,,
\end{align}
with $f(x)$ defined in Eq.~(\ref{f}).

From Eqs.~(\ref{MDmin}) and (\ref{fij}) we readily obtain
\begin{align} &\im[(H_\triangle)_{21}^2]=y_{21}^2\,
   y_{22}^2\sin
(2{{\phi}_{21}}) + 2\,y_{21}\,y_{22}\,y_{31}\,y_{32}\sin
\theta_1\,
    + y_{31}^2\,y_{32}^2\sin \theta_2\,,\nl
&\im[(H_\triangle)_{31}^2]=y_{31}^2\,y_{33}^2\,\sin (2\,{{\phi
}_{31}})\,,\nl %\quad,\quad
&\im[(H_\triangle)_{32}^2]=\,y_{32}^2\,y_{33}^2\sin (2\,{{\phi
}_{32}})\,,\label{imhij}
\end{align} with
$\theta_1={{\phi }_{21}} + {{\phi }_{31}} - {{\phi }_{32}}$ and
$\theta_2=2\,\left( {{\phi }_{31}} - {{\phi }_{32}} \right)$.

All the information about neutrino masses and mixing is fully
contained in the effective neutrino mass matrix $\mathcal{M}_\nu$
which is determined through the seesaw formula given by
Eq.~(\ref{ssaw}). In this case
\begin{align}
\label{Mntri}
\mathcal{M}_\nu=\frac{v^2}{M_1}\left(\begin{array}{ccc} y_{11}^2
&\;y_{11}y_{21}e^{i\,\phi_{21}}\; &\;y_{11}
y_{31}e^{i\,\phi_{31}}\;\\ y_{11}y_{21}e^{i\,\phi_{21}} &y_{21}^2
e^{2i\,\phi_{21}}+y_{22}^2\frac{M_1}{M_2} &y_{21}
y_{31}e^{i(\phi_{31}+\phi_{21})}+y_{22}y_{32}
\frac{M_1}{M_2}\,e^{i\phi_{32}} \\ y_{11} y_{31}e^{i\phi_{31}}
&y_{21}
y_{31}e^{i(\phi_{31}+\phi_{21})}+y_{22}y_{32}\frac{M_1}{M_2}\,
e^{i\phi_{32}}
&y_{31}^2e^{2i\phi_{31}}+y_{33}^2\frac{M_1}{M_3}+y_{32}^2\frac{M_1}{M_2}
\,e^{2i\phi_{32}}
\end{array}\right)\,.
\end{align}
It follows from Eqs.~(\ref{ejtri})-(\ref{Mntri}) that, in
principle, one can obtain simultaneously viable values for the CP
asymmetries $\varepsilon_j$ and a phenomenologically acceptable
effective neutrino mass matrix in order to reproduce the solar,
atmospheric and reactor neutrino data. This can be achieved by
consistently choosing the values of the free parameters $y_{ij}$,
$M_i$ and $\phi_{ij}\ $. Yet, a closer look at
Eqs.~(\ref{ejtri})-(\ref{imhij}) shows that there are terms
contributing to $\varepsilon_j$ which vanish independently from
the others. This means that a non-vanishing value of
$\varepsilon_j$ can be guaranteed even for simpler structures for
$Y_\nu$, which can be obtained from $Y_\triangle$ assuming
additional zero entries in the lower triangle\footnote{Notice
however that the vanishing of diagonal elements in $Y_\triangle$
would imply ${\rm det}\,(m_D)=0$ and consequently, ${\rm
det}\,(\mathcal{M}_\nu)=0$, leading to the existence of massless
light neutrinos.}. The results are given in Table~\ref{table1}
where we present the textures constructed from $Y_\triangle$ by
neglecting one (textures I-III) and two (textures IV-VI)
off-diagonal entries. The form of the effective neutrino mass
matrix $\mathcal{M}_\nu$ and the expressions for the CP
asymmetries $\varepsilon_{1,2}$ for each case are also
given\footnote{In commonly used language, textures I-III and IV-VI
belong to the classes of four and five texture zero matrices,
respectively. For a complete discussion on seesaw realization of
texture-zero mass matrices see e.g. \cite{Kageyama:2002zw} and for
its implications at low energies see e.g.
\cite{Frampton:2002yf}.}.

\begin{turnpage}
\squeezetable
\begin{table*}
\caption{Minimal textures based on the triangular decomposition of
$m_D$ and their respective light neutrino mass matrix
$\mathcal{M_\nu}$ and CP-asymmetries $\varepsilon_1$ and
$\varepsilon_2$. } \label{table1}
\medskip
%\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
%\renewcommand{\arraystretch}{1.2} % enlarge line spacing
%\scriptsize{
\begin{tabular}{clll}
%{\textheight}{@{\extracolsep{\fill}}llll} \noalign{\medskip}
\hline\noalign{\medskip} \normalsize{Texture} &\cc{
\normalsize{$Y_{\triangle}$}}
&\cc{\normalsize{$\frac{M_1}{v^2}\mathcal{M_\nu}$}}
& \cc{\normalsize{$\varepsilon_{1,2}$}}  \\
\noalign{\medskip}\hline %\hline
\noalign{\medskip} \normalsize{I} &$\left(\begin{array}{ccc}
 y_{11}   & 0       &0 \\
 y_{21}\,e^{i\,\phi_{21}}  &y_{22}       &0 \\
0   & y_{32}\,e^{i\,\phi_{32}} &y_{33}
\end{array}\right)$ &$\left(\begin{array}{ccc}
y_{11}^2                             &y_{11}\,y_{21}\,e^{i\,\phi_{21}}       &0 \\
y_{11}\,y_{21}\,e^{i\,\phi_{21}}     & y_{21}^2\,e^{2i
{{\phi}_{21}}} +
  \frac{M_1}{M_2}\,y_{22}^2      &y_{22}\,y_{32}\,\frac{M_1}{M_2}\,e^{i\,\phi_{32}} \\
0   & y_{22}\,y_{32}\,\frac{M_1}{M_2}\,e^{i\,\phi_{32}}
&y_{33}^2\,\frac{M_1}{M_3}+y_{32}^2\,\frac{M_1}{M_2}\,e^{2\,i\,\phi_{32}}
\end{array}\right)$ &$\begin{array}{l} \varepsilon_1^{\rm \, I}
=-\dfrac{y_{21}^2\,y_{22}^2\,\sin(2\,\phi_{21})}{8\pi\,(\,y_{11}^2+y_{21}^2\,)}\,
f_{21} \\ \\ \varepsilon_2^{\rm
\,I}=\dfrac{y_{21}^2\,y_{22}^2\,\sin(2\,\phi_{21})\,
f_{12}-y_{32}^2\,y_{33}^2\,\sin(2\,\phi_{32})\,f_{32}}
{8\pi\,(\,y_{22}^2+y_{32}^2\,)}\,
 \end{array}$\\
\noalign{\bigskip}%\hline \noalign{\bigskip}
\normalsize{II} &$\left(\begin{array}{ccc}
 y_{11}   & 0       &0 \\
 0  &y_{22}       &0 \\
 y_{31}\,e^{i\,\phi_{31}}   & y_{32}\,e^{i\,\phi_{32}} &y_{33}
\end{array}\right)$
&$\left(\begin{array}{ccc}
y_{11}^2   &0       &y_{11}\,y_{31}\,e^{i\,\phi_{31}} \\
0   &  y_{22}^2\,\frac{M_1}{M_2}      &y_{32}\,y_{22}\,\,
\frac{M_1}{M_2}e^{i\,\phi_{32}} \\
y_{11}\,y_{31}\,e^{i\,\phi_{31}}   &
y_{32}\,y_{22}\,\,\frac{M_1}{M_2}e^{i\,\phi_{32}}
&y_{31}^2\,e^{2\,i\,\phi_{31}}+y_{33}^2\,\frac{M_1}{M_3}+y_{32}^2\,
\frac{M_1}{M_2}\,e^{2\,i\,\phi_{32}}
\end{array}\right)$
&$\begin{array}{l} \varepsilon_1^{\rm \,II} =\dfrac{y_{31}^2
y_{32}^2\sin[2(\phi_{32}-\phi_{31})]\,f_{21}- y_{31}^2
y_{33}^2\sin(2\phi_{31})f_{31}}{8\pi(y_{11}^2+y_{31}^2)}
\\ \\ \varepsilon_2^{\rm
\,II}=\dfrac{y_{31}^2
y_{32}^2\sin[2(\phi_{31}-\phi_{32})]\,f_{12}- y_{32}^2
y_{33}^2\sin(2\phi_{32})f_{32}}{8\pi(y_{22}^2+y_{32}^2)}
\end{array}$
\\
\noalign{\bigskip} \normalsize{III} &$\left(\begin{array}{ccc}
 y_{11}   & 0       &0 \\
 y_{21}\,e^{i\,\phi_{21}}  &y_{22}       &0 \\
y_{31}\,e^{i\,\phi_{31}}   & 0 &y_{33}
\end{array}\right)$
&$\left(\begin{array}{ccc}
y_{11}^2   &y_{11}\,y_{21}\,e^{i\,\phi_{21}}       &y_{11}\,y_{31}\,e^{i\,\phi_{31}} \\
y_{11}\,y_{21}\,e^{i\,\phi_{21}}   & y_{21}^2\,e^{2i\phi_{21}}+
y_{22}^2\,\frac{M_1}{M_2}
& y_{21}\,y_{31}\,e^{i(\phi_{21}+\phi_{31})}   \\
y_{11}\,y_{31}\,e^{i\,\phi_{31}}
&y_{21}\,y_{31}\,e^{i(\phi_{21}+\phi_{31})}
&y_{31}^2\,e^{2\,i\,\phi{31}}+y_{33}^2\,\frac{M_1}{M_3}
\end{array}\right)$
&$\begin{array}{l} \varepsilon_1^{\rm \,III} =-\dfrac{y_{21}^2
y_{22}^2\sin(2\phi_{21})f_{21}+ y_{31}^2
y_{33}^2\sin(2\phi_{31})f_{31}}{8\pi(y_{11}^2+y_{21}^2+y_{31}^2)}
\\ \\ \varepsilon_2^{\rm \,III}=\dfrac{y_{21}^2
y_{22}^2\sin(2\phi_{21})}{8\pi\, y_{22}^2}f_{12}\end{array}$
\\
\noalign{\bigskip}%\hline \noalign{\bigskip}
\normalsize{IV} &$ \left(\begin{array}{ccc}
y_{11}   &0       & 0 \\ y_{21}\,e^{i\,\phi_{21}}   &y_{22}       &0 \\
0   &0 &y_{33} \end{array}\right)$ &$\left(\begin{array}{ccc}
y_{11}^2   &y_{11}\,y_{21}\,e^{i\,\phi_{21}}       & 0 \\
y_{11}\,y_{21}\,e^{i\,\phi_{21}}   & y_{21}^2\,e^{2\,i\,\phi_{21}}
+y_{22}^2\,\frac{M_1}{M_2}      & 0   \\
0   &0 &y_{33}^2\,\frac{M_1}{M_3}
\end{array}\right)$ &$\begin{array}{l} \varepsilon_1^{\rm \, IV}
=-\dfrac{y_{21}^2\,y_{22}^2\,\sin(2\,\phi_{21})}{8\pi\,(\,y_{11}^2+y_{21}^2\,)}\,
f_{21} \\ \\\varepsilon_2^{\rm \, IV}
=\dfrac{y_{21}^2\,y_{22}^2\,\sin(2\,\phi_{21})}{8\pi\,y_{22}^2}\,
f_{12} \end{array}$\\
\noalign{\bigskip}%\hline \noalign{\bigskip}
\normalsize{V} &$\left(\begin{array}{ccc}
 y_{11}   & 0       &0 \\
0   &y_{22}       &0 \\
 y_{31}\,e^{i\,\phi_{31}}   &0&y_{33}
\end{array}\right)$
&$\left(\begin{array}{ccc}
y_{11}^2   &0       &y_{11}\,y_{31}\,e^{i\,\phi_{31}} \\
0   &  y_{22}^2\,\frac{M_1}{M_2}      & 0   \\
y_{11}\,y_{31}\,e^{i\,\phi_{31}}   &0
&y_{31}^2\,e^{2\,i\,\phi_{31}}+y_{33}^2\,\frac{M_1}{M_3}
\end{array}\right)$
&$\begin{array}{l} \varepsilon_1^{\rm \, V}
=-\dfrac{y_{31}^2\,y_{33}^2\,\sin(2\,\phi_{31})}{8\pi\,(\,y_{11}^2+y_{31}^2\,)}\,
f_{31} \\ \\ \varepsilon_2^{\rm \,V}=0\end{array}$
\\
\noalign{\bigskip}%\hline \noalign{\bigskip}
\normalsize{VI} &$ \left(\begin{array}{ccc}
 y_{11}   & 0       &0 \\
 0  &y_{22}       &0 \\
0   & y_{32}\,e^{i\,\phi_{32}} &y_{33}
\end{array}\right)$
&$\left(\begin{array}{ccc}
y_{11}^2   &0       &0 \\
0   &  y_{22}^2\,\frac{M_1}{M_2}      &y_{22}\,y_{32}\,
\frac{M_1}{M_2}\,e^{i\,\phi_{32}} \\
0   & y_{22}\,y_{32}\,\frac{M_1}{M_2}\,e^{i\,\phi_{32}}
&y_{33}^2\,\frac{M_1}{M_3}+y_{32}^2\,\frac{M_1}{M_2}\,
e^{2\,i\,\phi_{32}}
\end{array}\right)$
&$\begin{array}{l} \varepsilon_1^{\rm \, VI} =0
\\ \\ \varepsilon_2^{\rm
\,VI}=-\dfrac{y_{32}^2\,y_{33}^2\,\sin(2\,\phi_{32})}{8\pi\,
(\,y_{22}^2+y_{32}^2\,)}f_{32}
\end{array}$\\\noalign{\bigskip}\hline
\end{tabular}
%}
\end{table*}
\end{turnpage}

Let us first discuss textures IV-VI. For these three textures the
effective neutrino mass matrix $\mathcal{M}_\nu$ predicts a complete
decoupling of one light neutrino from the other two. This is in
disagreement with the available neutrino data which indicates that, in the
framework of the LMA solution, only one neutrino mixing angle should be
small, namely $\theta_{13}$, instead of two as predicted by textures
IV-VI. Furthermore, texture VI predicts a vanishing value for the CP
asymmetry in the decay of the lightest heavy Majorana neutrino, implying
$Y_B=0$ in the case of hierarchical heavy Majorana neutrinos. Texture III
can also be excluded on the grounds that it cannot account for the large
solar angle. To illustrate this, let us write for this texture the matrix
$L$ given in Eq.~(\ref{MnuLL}) as
\begin{equation}
L = i\,m_D\,D_R^{-1/2} \equiv\left(\begin{array}{ccc}
z_1 &\;\; 0 &\;\;0 \\ z_2 &\;\;y_2 &\;\;0 \\
z_3 &\;\; 0 &\;\;x_3 \end{array}\right)\,,
\end{equation}
where $z_1$, $y_2$ and $x_3$ are real and positive \cite{Lavignac:2002gf}.
We also take $z_2$ and $z_3$ real to simplify the discussion. Considering
first the hierarchical case $\dmatm \simeq  m_3^2\ $, a large atmospheric
angle and $m_2 < m_3$ naturally arise if $z_2 \sim z_3
> y_2,x_3\ $. More precisely, $\tan \theta_{23} \simeq z_2/z_3\ $.
In addition, this implies
\begin{equation}
\tan \theta_{13} \simeq \frac{z_1}{\sqrt{z_2^2+z_3^2}} \equiv
t_{13}\,,
\end{equation}
which has to satisfy the CHOOZ bound, $t_{13} \le 0.2$
\cite{Apollonio:1999ae}. The solar angle is approximately given by
\begin{equation}
\tan( 2 \theta_{12}) \simeq 2\,t_{13}\,\frac{y_2^2+x_3^2}{
\frac{z_3}{z_2}\,y_2^2 + \frac{z_2}{z_3}\,x_3^2 - t_{13}
(\frac{z_2}{z_3}\,y_2^2 + \frac{z_3}{z_2}\,x_3^2) } \le
\frac{2\,t_{13}\,\beta }{1-t_{13}\,\beta^2} \equiv B_{12}(\beta,
t_{13})\,,
\end{equation}
where $\beta \equiv {\rm max}(\tan \theta_{23}, \cot \theta_{23})$. The
upper bound $B_{12}(\beta, t_{13}=0.2)$ is an increasing function of
$\beta$. Experimentally  $0.7 \le \beta \le 1.7$, so that $B_{12}^{\rm
max} = B_{12}(1.7,0.2) = 4$. This corresponds to $\tan^2\theta_{12} \le
0.3$, while the fitted LMA solution requires $\tan^2\theta_{12} > 0.3$. We
thus conclude that type III cannot account for the observed large solar
angle in the hierarchical case. Moreover, it turns out that it cannot also
account for the pattern of ${\cal M}_\nu$ required by inverse hierarchy
since, in this case, the atmospheric oscillation fit would require $z_1
z_2 \simeq z_1 z_3 \simeq \sqrt{\dmatm}$ and $z_1^2, z_2^2, z_3^2 \ll
\sqrt{\dmatm}$. Finally, the degenerate case can be accommodated only by
tuning the elements of $L$. In view of this, from now on we will focus our
analysis only on textures I and II.

Another interesting fact which comes out from the observation of
Table~\ref{table1} is that the phase content for textures I-II can
be further reduced. Indeed, it is straightforward to show that
with only one phase in $Y_\triangle$ it is possible to obtain a
non-vanishing CP asymmetry $\varepsilon_1$.

As it has been stated in Section~\ref{generframe}, the strength of CP
violation at low energies is sensitive to the value of the CP invariant
$\mathcal{J}_{CP}$ given by Eq.~(\ref{Jfin}). For the textures I and II we
get:
\begin{align}
\label{JI}  \mathcal{J}_{CP}^{\rm \, I}=&\frac{y_{11}^2\,
y_{21}^2\, y_{32}^2\,y_{22}^2\,v^{12}}{M_1^3 M_2^3\dmtwon \dmthon
\dmthtw}\left[(y_{21}^2\, y_{32}^2 +y_{11}^2\,y_{22}^2+y_{11}^2\,
y_{32}^2)\sin(2\phi_{21})-y_{22}^2\,y_{33}^2
\frac{M_1}{M_3}\sin(2\phi_{32})\right.\nl & +\left.y_{33}^2
(y_{11}^2+ y_{21}^2)\frac{M_2}{M_3}\sin\left[2\left(\phi_{21} -
\phi_{32} \right)\right] \right]  \,,\nonumber \\
 \mathcal{J}_{CP}^{\rm \, II}=&\frac{y_{11}^2\, y_{22}^2\,
y_{31}^2\, y_{32}^2\,v^{12}}{M_1^3 M_2^3\dmtwon \dmthon
\dmthtw}\left[(y_{22}^2\,y_{31}^2+y_{11}^2\,y_{22}^2+y_{11}^2\,
y_{32}^2)\,\sin\left[2\,(\phi_{32} -\phi_{31})\right]\right.\nl &
\left.+\,y_{22}^2\,y_{33}^2
\frac{M_1}{M_3}\,\sin(2\phi_{32})-y_{11}^2\,y_{33}^2\,
\frac{M_2}{M_3}\,\sin(2\phi_{31})
 \right]\,.
\end{align}%
 From Table~\ref{table1} and Eqs.~(\ref{JI}) it is clear that
 we can have both $\varepsilon_1 \neq 0$ and $\mathcal{J}_{CP} \neq 0$
 with a single non-vanishing phase.

 \bigskip \medskip
{\bf On the relative sign between \bm{$Y_B$} and \bm{$\mathcal{J
}_{CP}$}} %\label{sign}
\bigskip \medskip

It has recently been pointed out in \cite{Frampton:2002qc} that
the relative sign between CP violation and the baryon asymmetry
can be predicted in a specific class of models. In our framework,
it is clear from Table~\ref{table1} and Eqs.~(\ref{JI}) that the
relative sign between the low-energy invariant $\mathcal{J}_{CP}$
and the CP asymmetries $\varepsilon_j$ cannot be predicted without
specifying the values of the heavy neutrino masses $M_i$ and the
Dirac Yukawa couplings\footnote{For the sake of more generality,
we do not assume the mass-ordering $M_1 < M_2$ in this
discussion.}. This is mainly due to the fact that at least one of
these quantities depends on both phases appearing in the Dirac
neutrino Yukawa coupling matrix for textures I and II. Therefore,
in order to establish a direct connection between the sign of the
CP asymmetries $\varepsilon_j$ and the low-energy CP invariant
$\mathcal{J}_{CP}$ further assumptions are required. For instance,
considering the case $\phi_{32}=0$ and $M_1,M_2 \ll M_3$ so that
the terms proportional to $f_{31}$ can be safely neglected in
$\varepsilon_{1}^{\rm II}$ (see Table \ref{table1}), we obtain
from Eqs.~(\ref{JI}) and Table \ref{table1} the relative signs
given in Table \ref{table2}.
\begin{table*}
 \caption{Relative sign between the CP asymmetries
$\varepsilon_{1,2}$ and the low-energy CP invariant
$\mathcal{J}_{CP}$ for textures I and II considering the different
heavy Majorana mass regimes. We assume $M_1,M_2 \ll M_3$ and
$\phi_{32}=0$. The parameter $x \simeq 0.4$ corresponds to the
zero of the loop function $f(x)$ defined in
Eq.~(\ref{f}).}\label{table2}
\medskip
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{0.7pc}
\begin{tabular}{lcccc}
\hline\noalign{\smallskip}&$\dfrac{M_1}{M_2}< x$
&$x<\dfrac{M_1}{M_2}<1$ &$\dfrac{M_2}{M_1}< x$ &$x<\dfrac{M_2}{M_1}<1$ \\
\noalign{\smallskip} \hline
%Texture I   &           &          &          & \\
sgn$\,(\varepsilon_1^{\rm I}\cdot\mathcal{J}_{CP}^{\rm \, I})$ &$+$ &$+$ &$+$ &$-$\\
sgn$\,(\varepsilon_2^{\rm I}\cdot\mathcal{J}_{CP}^{\rm \, I})$   &$-$   &$+$ &$-$  &$-$ \\
\hline\noalign{\smallskip}
sgn$\,(\varepsilon_1^{\rm II}\cdot\mathcal{J}_{CP}^{\rm \, II})$   &$-$    &$-$  &$-$ &$+$ \\
sgn$\,(\varepsilon_2^{\rm II}\cdot\mathcal{J}_{CP}^{\rm \, II})$
&$+$ &$-$  &$+$ &$+$ \\ \hline\noalign{\smallskip}
\end{tabular} \end{table*}

It is well known that in the case of hierarchical heavy neutrinos, the
sign of $Y_B$ is fixed by the sign of the CP asymmetry generated in the
decay of the lightest heavy neutrino, let us say $\varepsilon_L$ . For
$Y_B$ to be positive, as required by the observations, one must have
$\varepsilon_L < 0$. Thus, the measurement of the sign of CP violation at
low-energy could in principle exclude some of the heavy Majorana neutrino
mass regimes presented in Table~\ref{table2}. As an example, let us assume
that $\mathcal{J}_{CP}< 0$. In this case, the requirement $\varepsilon_L <
0$ would exclude the mass regime $M_2/M_1 < x$ for texture I and $M_1/M_2
< x$ for texture II.

Finally, it is interesting to note that the Dirac neutrino Yukawa
matrices considered in \cite{Frampton:2002qc} correspond to our
textures I and II with $y_{33}=0$ and have the remarkable feature
that the number of real parameters equals the number of masses and
mixing angles. In this limit the heavy Majorana neutrino $N_3$
completely decouples, rendering this situation phenomenologically
equivalent to the two heavy neutrino case considered in
\cite{Frampton:2002qc}. Namely, it can be easily seen that for
$y_{33}=0$ the phase $\phi_{32}$ can be rephased away and
consequently the CP asymmetries $\varepsilon_{1,2}$ and the
low-energy CP invariant $\mathcal{J}_{CP}$ will only depend on the
phases $\phi_{21}$ and $\phi_{31}$ for textures I and II,
respectively. This means that the examples considered in
\cite{Frampton:2002qc} are special cases of our framework which is
motivated by the condition that $m_D$ does not contain any phase
that would only contribute to low-energy CP violation. It is clear
from Table~\ref{table2} that, in general, even by keeping only one
phase the sign of $\varepsilon_j\cdot\mathcal{J}_{CP}$ depends on
the particular Yukawa texture and hierarchy between the heavy
Majorana neutrinos.
\section{Examples}
\label{example}

In this section we present some examples of the minimal textures discussed
in Section~\ref{minimal} and proceed to the study of their implications to
low-energy physics as well as to the computation of the baryon asymmetry
$Y_B$ through the numerical solution of the Boltzmann equations as
described in the Appendix. We will only consider cases that lead to the
LMA solution of the solar neutrino problem, which means that the neutrino
mixing angles and the squared mass differences lie in the typical ranges:
\begin{align}
\label{LMAdt} 2.5 \times 10^{-5}\,\text{eV}^2 & \lesssim \dmsol
\lesssim 3.4 \times 10^{-4}\,\text{eV}^2 \quad,\quad 0.24 \lesssim
\tgsol \lesssim
0.89\,,\nonumber\\
1.4 \times 10^{-3}\,\text{eV}^2 & \lesssim \dmatm \lesssim 6.0 \times
10^{-3}\,\text{eV}^2 \quad,\quad 0.40 \lesssim \tgatm \lesssim 3.0\,,
\end{align}
with the solar and atmospheric mixing angles being identified as $
\theta_\odot \equiv \theta_{12} $, $ \theta_ @ \equiv \theta_{23}
$, respectively, in the standard parametrization
\cite{Hagiwara:pw}. The $\theta_{13}$ mixing angle is at present
constrained by reactor neutrino experiments to satisfy $|\sin
\theta_{13}| \equiv |U_{e3}| \lesssim 0.2$
\cite{Apollonio:1999ae}.
\subsection{Hierarchical heavy Majorana neutrinos $\bm{(M_1 < M_2 \ll M_3)}$ }
\label{hierarchical}%

The first example is a realization of the texture~I given in
Table~\ref{table1} with $\phi_{32}=0$. The entries of the Dirac
neutrino Yukawa coupling and the right-handed neutrino mass
matrices are chosen to be of order:
\begin{align}
\label{YDex1} Y_\nu = \frac{\Lambda_D}{v}\left(\begin{array}{ccc}
 \epsilon^6   & \;\;0       &\;\; 0 \\
 \epsilon^6\,e^{i\,\phi_{21}}   &\;\;\epsilon^4 &\;\; 0 \\
 0   &\;\; \epsilon^4       &\;\; 1
\end{array}\right)\;\;,\;\;D_R=\Lambda_R\,{\rm
diag}\,(\epsilon^{12},\epsilon^{10},1)\,,
\end{align}
where $\epsilon < 1$ is a small parameter. For our numerical estimates we
consider the typical Dirac neutrino mass scale to be $\Lambda_D \simeq
100$~GeV, which corresponds approximately to the top quark mass at the GUT
scale \cite{Fusaoka:1998vc}. The neutrino mass matrix $\mathcal{M}_\nu$ is
then given by:
\begin{align}
\label{mnex1} \mathcal{M}_\nu=\frac{\Lambda_D^2}{\epsilon^2
\Lambda_R}\left(\begin{array}{ccc}
\epsilon^2                   &\quad\epsilon^2\,e^{i\,\phi_{21}}     &\quad0 \\
\epsilon^2\,e^{i\,\phi_{21}} &\quad 1+\epsilon^2\,e^{2i\,\phi_{21}} &\quad 1 \\
 0                           &\quad 1                    &\quad 1+\epsilon^2
\end{array}\right)\,.
\end{align}
 In this particular case, considering for the moment $\phi_{21}=0$,
one gets\footnote{We have checked that the light neutrino masses
$m_i$ and mixing angles $\theta_{ij}$ are not sensitive to the
phase $\phi_{21}$.}: \vspace{0.2cm}
\begin{align} \label{msex1} m_1\simeq
\frac{\Lambda_D^2}{2\Lambda_R}(2- \sqrt{2}) \quad,\quad m_2\simeq
\frac{\Lambda_D^2}{2\Lambda_R}(2+\sqrt{2}) \quad , \quad m_3\simeq
\frac{\Lambda_D^2}{\epsilon^2 \Lambda_R}(2+\epsilon^2)\,,
\end{align}
leading to
\begin{align}
\label{dmex1} \dmtwon \simeq
\frac{2\,\sqrt{2}\,\Lambda_D^4}{\Lambda_R^2}\quad,\quad \dmthtw \simeq
\frac{4\,\Lambda_D^4}{\epsilon^4\,\Lambda_R^2}\,.
\end{align}
The requirement
\begin{align}
\label{rsol} \left. \frac{\dmsol}{\dmatm} \right|_{\rm
LMA}\!\!\!\,=\,\frac{\dmtwon}
 {\dmthtw}\simeq \frac{\epsilon^4}{\sqrt{2}}\,,
\end{align}
forces $\epsilon$ to be in the range $0.3 \lesssim \epsilon
\lesssim 0.7$. For the neutrino mixing angles we have:
\begin{align}
\label{angl} \tan^2\theta_{12}\simeq 1-\frac{\epsilon^2}
{2\sqrt{2}} \quad,\quad \tan^2\theta_{23}\simeq 1 \quad,\quad
|U_{e3}| \simeq \frac{\epsilon^2}{2\,\sqrt{2}}\,,
\end{align}
which are in good agreement with the data taking into account the
range of $\epsilon$. The matrix $V_L$ which corresponds to the
left-handed rotation involved in the diagonalization of $m_D$ can
be viewed as the equivalent in the leptonic sector of the quark
mixing matrix $V_{CKM}$. This matrix is obtained by diagonalizing
the Hermitian matrix $Y_\nu Y_\nu^{\,\dag}$,
\begin{align}
\label{VLex1} V_L \simeq \left(\begin{array}{ccc}
1-\epsilon^8/2              &\quad\epsilon^4-\epsilon^{12}/2 &\quad0 \\
-\epsilon^4+\epsilon^{12}/2   &\quad1-\epsilon^8/2           &\quad\epsilon^8 \\
 \epsilon^{12}                &\quad-\epsilon^8              &\quad 1
\end{array}\right)\,.
\end{align}
This is in fact an interesting result in the sense that we are in the
presence of a typical \emph{large mixing out of small mixing} situation,
where the large neutrino mixing is generated through the seesaw mechanism
\cite{Altarelli:1999dg,Akhmedov:2000yt}. The scale $\Lambda_R$, or
equivalently the mass $M_3$ of the heaviest Majorana neutrino, is
determined by requiring $\dmthtw$ to be in the experimental range given in
(\ref{LMAdt}). We find
\begin{align}
\label{LR} M_3 = \Lambda_R \,\simeq \, \frac{2\,\Lambda_D^2}{\epsilon^2\,
\sqrt{\dmatm}}\, \simeq\,
\frac{2^{5/4}\,\Lambda_D^2}{\sqrt{\dmsol}}\,\simeq \, 8\times
10^{15}\,\,{\rm GeV}\,,
\end{align}
and Eq.~(\ref{YDex1}) implies $M_1 \simeq 1.3 \times 10^{11}$~GeV and $M_2
\simeq 8.4 \times 10^{12}$~GeV, for $\epsilon=0.4$.

From the expression of the CP asymmetry $\varepsilon_1^{\rm I}$ given in
Table~\ref{table1}, we obtain
\begin{align}
\label{e1ex1} \varepsilon_1
=\dfrac{3\,y_{21}^2\,y_{22}^2\,\sin(2\,\phi_{21})}{16\pi\,(\,
y_{11}^2+y_{21}^2\,)}\,\frac{M_1}{M_2} \simeq
\frac{3\,\Lambda_D^2\,\epsilon^{10}}{32\pi\,v^2} \sin(2\,
\phi_{21})\simeq 10^{-6}\sin(2\,\phi_{21})\,,
\end{align}
which is maximized for $\phi_{21}=\pi/4\,.$ The value of the CP invariant
$\mathcal{J}_{CP}$ can be computed from the expression
\begin{align}\label{Jex1} \mathcal{J}_{CP}=&\frac{y_{11}^2\, y_{21}^2
\, y_{32}^2\,y_{31}^2\,v^{12}}{M_1^3 M_2^3\dmtwon \dmthon
\dmthtw}\left[y_{21}^2\, y_{32}^2+y_{11}^2(\,y_{22}^2+
y_{32}^2)+y_{33}^2 (y_{11}^2+ y_{21}^2)\frac{M_2}{M_3}\right]
\sin(2\phi_{21}) \,,
 \end{align}
which is obtained from Eq.~(\ref{JI}) setting $\phi_{32}=0$. The result is
straightforward in this case
\begin{align}
\label{Jap1} \mathcal{J}_{CP} \simeq
\frac{3\,\epsilon^2\sin(2\phi_{21})}{32\,\sqrt{2}} \simeq 1.1
\times 10^{-2} \sin(2\,\phi_{21})\,,
\end{align}
rendering visible CP-violating effects in the next generation of neutrino
factories for $\phi_{21}\simeq \pi/4$. It is interesting to note from
Eqs.~(\ref{e1ex1}) and (\ref{Jap1}) that the dependence of $\varepsilon_1$
and $\mathcal{J}_{CP}$ on the phase $\phi_{21}$ is such that both
quantities are simultaneously maximized. Notice also that $\varepsilon_1$
and $\mathcal{J}_{CP}$ have the same sign, as it should be according to
Table~\ref{table2}.

All the results presented above are confirmed by the full
numerical computation presented in Fig.~\ref{fig1}, where we have
randomly included $\mathcal{O}$(1) coefficients in the range
$[0.9,1.3]$ for the non-vanishing entries of $Y_\nu$ and taken
$\phi_{21}=\pi/4$. The spreading of the points in the figure, due
to the random variation of the coefficients, shows that the
textures are
stable under these perturbations.\\
\begin{figure*}
$$\includegraphics[width=12cm]{fig1col.eps}$$
\caption{Results of the numerical computation for the textures
presented in Eq.~(\ref{YDex1}) for $Y_\nu$ and $M_R$ with the
CP-violating phase $\phi_{21}=\pi/4$. The heavy Majorana neutrino
masses $M_i$ and the CP asymmetries $\varepsilon_1$,
$\varepsilon_2$ are plotted as functions of the texture parameter
$\epsilon$. The dependence of $|U_{e3}|$ and $\mathcal{J}_{CP}$ on
$\epsilon$ is also shown. The dotted areas satisfy the neutrino
constraints (\ref{LMAdt}) for the LMA solar solution.}
\label{fig1}
\end{figure*}
\begin{figure*}
$$\includegraphics[width=10cm]{fig2col.eps}$$
\caption{Results for the distributions $Y_{N_1}$, $Y_{N_2}$ and
for the baryon asymmetry $Y_B$ as functions of $z=M_1/T$, obtained
from the numerical solution of the Boltzmann equations. The plot
refers to the texture for $Y_\nu$ considered in Eq.~(\ref{YDex1})
with $\phi_{21}=\pi/4$ and $\epsilon=0.5$. The values of the heavy
Majorana neutrino masses $M_i$, the CP asymmetries
$\varepsilon_{1,2}$ and the lightest neutrino mass $m_1$ are
consistent with the ones presented in Fig.~\ref{fig1} for this
value of $\epsilon$. The value predicted for the final baryon
asymmetry is $Y_B \simeq 9 \times 10^{-11}$.} \label{fig2}
\end{figure*}
In order to compute the value of the baryon asymmetry we proceed
to the numerical solution of the Boltzmann equations as described
in the Appendix, taking the initial conditions: $Y^{\,0}_{N_i}=0
\,,\, Y_{B-L}^{\,0}=0$. The results are presented in
Fig.~\ref{fig2} where we have plotted $Y_{N_1}$, $Y_{N_2}$ and
$Y_B$ as functions of the parameter $z=M_1/T$ for given values of
the CP asymmetries, heavy Majorana neutrino masses and the
lightest neutrino mass. The predicted value for the final baryon
asymmetry is $Y_B \simeq 9\times 10^{-11}$, which is inside the
observational range (\ref{YBrng}).

Our next example is a particular case of the texture II presented
in Table~\ref{table1} with $\phi_{32}=0$. The Dirac neutrino
Yukawa coupling and heavy neutrino mass matrices are chosen in
this case to be of order
\begin{align}\label{YDex2}
Y_\nu = \frac{\Lambda_D}{v}\left(\begin{array}{ccc}
 \epsilon^5                       &\quad 0            &\quad 0 \\
 0                                &\quad \epsilon^3   &\quad 0 \\
 \epsilon^5 e^{i\,\phi_{31}}   &\quad \epsilon^3      &\quad 1
\end{array}\right)\;\;,\;\;D_R=\Lambda_R\,{\rm
diag}\,(\epsilon^{10},\epsilon^{8},1)\,,
\end{align}
leading to the following light neutrino mass matrix
\begin{align}
\label{mnex2} \mathcal{M}_\nu=\frac{\Lambda_D^2}{\epsilon^2
\Lambda_R}\left(\begin{array}{ccc}
\epsilon^2                    &\quad 0 &\quad\epsilon^2\,e^{i\,\phi_{31}} \\
0                             &\quad 1 &\quad 1 \\
 \epsilon^2\,e^{i\,\phi_{31}} &\quad 1 &\quad 1+ \epsilon^2(1+e^{2\,i\,\phi_{31}})
\end{array}\right)\,.
\end{align}
The predictions for the $\Delta m_{ij}^2$'s and neutrino mixing
angles are similar to the ones given in Eqs.~(\ref{dmex1}) and
(\ref{angl}). Moreover, from Eqs.~(\ref{LR}) and (\ref{YDex2}) and
for $\epsilon=0.4$, we get the following heavy Majorana neutrino
masses: $M_1 \simeq 9 \times 10^{11} \, {\rm GeV} \,,\, M_2 \simeq
6 \times 10^{12}\, {\rm GeV} \,,\, M_3 \simeq 8 \times 10^{15}\,
{\rm GeV}$. The left-handed matrix $V_L$ is given in this case by:
\begin{align}
\label{VLex2} V_L \simeq \left(\begin{array}{ccc}
-1               &\quad -\epsilon^{10}  &\quad \epsilon^{10} \\
-\epsilon^{10}   &\quad 1               &\quad\epsilon^6 \\
\epsilon^{10}    &\quad-\epsilon^6      &\quad 1
\end{array}\right)\,.
\end{align}

Finally, the CP asymmetry can be obtained from the expression of
$\varepsilon_1^{\rm \,II}$ presented in Table~\ref{table1} with
$\phi_{32}=0$. Taking into account the form of $Y_\nu$ and $D_R$
as in Eqs.~({\ref{YDex2}}), we obtain
\begin{figure*}
$$\includegraphics[width=12cm]{fig3col.eps}$$
\caption{The same as in Fig.~\ref{fig1} for the textures
considered in Eq.~(\ref{YDex2}) and $\phi_{31}=\pi/4$.}
\label{fig3}
\end{figure*}
\begin{align}
\label{e1ex2} \varepsilon_1
=\dfrac{3\,y_{31}^2\,\sin(2\phi_{31})}{16\,\pi(y_{11}^2+y_{31}^2)}\left(
y_{32}^2\,\frac{M_1}{M_2}+y_{33}^2\,\frac{M_1}{M_3}\right) \simeq
\frac{3\,\Lambda_D^2\,\epsilon^8}{32\,\pi\,v^2}\,\sin(2\,\phi_{31})
\simeq 6.5 \times 10^{-6}\,\sin(2\,\phi_{31})\,,
\end{align}
where the last estimate has been obtained assuming $\epsilon=0.4$.
It can also be shown that the CP invariant $\mathcal{J}_{CP}$
reads (see Eq.~(\ref{JI})),
\begin{align}
\label{Jex2}
 \mathcal{J}_{CP}=-\frac{y_{11}^2\, y_{22}^2\,
y_{31}^2\, y_{32}^2\,v^{12}}{M_1^3 M_2^3\dmtwon \dmthon
\dmthtw}\left[y_{22}^2\,y_{31}^2+y_{11}^2\,(y_{22}^2+
y_{32}^2)+y_{11}^2\,y_{33}^2\,\frac{M_2}{M_3}
 \right]\,\sin(2\phi_{31})\,,
\end{align}
which leads to the approximate result
\begin{align}
\mathcal{J}_{CP} \simeq -
\frac{3\,\epsilon^2\sin(2\phi_{31})}{32\,\sqrt{2}} \simeq -1.1
\times 10^{-2} \sin(2\,\phi_{31})\,.
\end{align}
Notice that $\varepsilon_1$ and $\mathcal{J}_{CP}$ have opposite signs in
this case, once again in agreement with Table~\ref{table2}.

In Fig.~\ref{fig3} we present the same numerical analysis as in
Fig.~\ref{fig1}, but for the case where $Y_\nu$ and $D_R$ are
defined through Eqs.~(\ref{YDex2}). We find good agreement between
our approximate analytical results and the numerical ones. The
integration of the Boltzmann equations is plotted in
Fig.~\ref{fig4}. The value for the final baryon asymmetry is $Y_B
\simeq 8 \times 10^{-11}$.
\begin{figure*}
$$\includegraphics[width=10cm]{fig4col.eps}$$
\caption{The same as in Fig.~\ref{fig2} for the example considered
in Eq.~(\ref{YDex2}) with $\phi_{31}=\pi/4$ and $\epsilon=0.4$.
The values of the heavy Majorana neutrino masses $M_i$, the CP
asymmetries $\varepsilon_{1,2}$ and the lightest neutrino mass
$m_1$ are consistent with the ones presented in Fig.~\ref{fig3}
for this value of $\epsilon$. The predicted value for the final
baryon asymmetry is $Y_B \simeq 8 \times 10^{-11}$. } \label{fig4}
\end{figure*}
\subsection{Two-fold quasi-degeneracy $\bm{(M_1 \simeq M_2 \ll M_3)}$}
\label{degeneracy}%

As an example, let us consider the texture of type I given in
Table~\ref{table1} and assume that $M_1 \simeq M_2 \ll M_3$. The
Hermitian matrix $H_\triangle =Y^\dag_\triangle Y_{\triangle}^{}$
is given by
\begin{align}\label{HDex3}
H_\triangle=\left(\begin{array}{ccc} y_{11}^2+y_{21}^2
&y_{21}\,y_{22}\,e^{-i\phi_{21}}
&\quad 0 \\
y_{21}\,y_{22}\,e^{i\phi_{21}}   & y_{22}^2+y_{32}^2
&y_{32}\,y_{33}\,e^{-i\phi_{32}} \\
0 &y_{32}\,y_{33}\,e^{i\phi_{32}} &y_{33}^2
\end{array}\right)\,.
\end{align}

Taking into account the requirement of Eq.~(\ref{dNlim}) we can
get the range of validity of the parameter $\delta_N$:
\begin{align}
\label{dNex3} \delta_N \gg \frac{1}{8 \pi} {\rm max}
\left\{y_{11}^2+y_{21}^2\;,\; y_{22}^2 +y_{32}^2\right\}\,.
\end{align}
Using Eqs.~(\ref{e12d2}), the CP asymmetries are given in this case by
\begin{align}
\label{esex3} \varepsilon_1 \simeq
\frac{y_{21}^2\,y_{22}^2\,\sin(2\,\phi_{21})}{16\pi\,
\delta_N\,(\,y_{11}^2+y_{21}^2\, )}\;\;,\;\; \varepsilon_2 &\simeq
\frac{1}{16\,\pi}\left[\frac{y_{21}^2\,y_{22}^2\,\sin(2\,\phi_{21})}{
\delta_N\,(\,y_{22}^2+y_{32}^2\,
)}+\frac{3\,y_{32}^2\,y_{33}^2}{y_{22}^2+y_{32}^2}\,\frac{M_2}{M_3}\sin(2\,
\phi_{32})\right]\,.
\end{align}

Let us now consider the following simple realization of $Y_\nu$
and $D_R$:
\begin{align}\label{YDex3}
Y_\nu =\frac{\Lambda_D}{v}\left(\begin{array}{ccc}
 \epsilon^{10}                    &\quad 0               &\quad 0 \\
 \epsilon^{10}\,e^{i\,\phi_{21}}   &\quad \epsilon^9      &\quad 0 \\
 0                             &\quad \epsilon^9      &\quad 1
\end{array}\right)\;\;,\;\;D_R=\Lambda_R\,{\rm
diag}\,(\epsilon^{20},(1+{\delta_N})\,\epsilon^{20},1)\,,
\end{align}
with $\delta_N \ll 1$. From Eq.~(\ref{dNex3}) we get in this case
for $\epsilon=0.3$:
\begin{align}
\label{dNrng} \delta_N \gg
\frac{\Lambda_D^2\,\epsilon^{18}}{4\,\pi\,v^2} \simeq 10^{-11}\,.
\end{align}
The effective neutrino mass matrix is now given by
\begin{align}
\label{mnex3} \mathcal{M}_\nu=\frac{\Lambda_D^2}{\epsilon^2\,(1+\delta_N)
\Lambda_R}\left(\begin{array}{ccc} \epsilon^2\,(1+\delta_N) & \quad
\epsilon^2\,(1+\delta_N)\,e^{i\,\phi_{21}}
&\quad0 \\
\epsilon^2\,(1+\delta_N)\,e^{i\,\phi_{21}}   & \quad 1+
\epsilon^2\,(1+\delta_N)\,e^{2i\,\phi_{21}}   &\quad 1 \\
 0                                             & \quad 1
 &\quad 1+\epsilon^2\,(1+\delta_N)
\end{array}\right)\,.
\end{align}
\begin{figure*}
$$\includegraphics[width=10cm]{fig5col.eps}$$
\caption{The same as Figs.~\ref{fig2} and \ref{fig4} for the
quasi-degenerate case $M_1 \simeq M_2$ considered in
Eq.~(\ref{YDex3}) with $\phi_{21}=\pi/4$, $\epsilon=0.3$ and
$\delta_N=5 \times 10^{-7}$. The predicted value of the baryon
asymmetry is $Y_B \simeq 10^{-10}$.} \label{fig5}
\end{figure*}
In the limit $\delta_N \rightarrow 0$, Eq.~(\ref{mnex1}) is
recovered. Since $\delta_N \ll 1$ the results for the neutrino
masses and mixing parameters are practically the same as the ones
given in Eqs.~(\ref{msex1})-(\ref{angl}). The same is expected for
the estimate of the CP invariant $\mathcal{J}_{CP}$ defined in
Eq.~(\ref{esex3}). The main differences reside obviously in the
heavy Majorana neutrino mass spectrum and on the values of the CP
asymmetries, since now the relation $M_1 \ll M_2$ is no longer
valid. From Eq.~(\ref{e12d2}) we obtain
\begin{align}
\label{esdeg} \varepsilon_1 &\simeq
\frac{\epsilon^{18}\,\Lambda_D^2\,\sin(2\,\phi_{21})}{32\,\pi\,\delta_N\,v^2}
\simeq \frac{1.3\times 10^{-12}}{\delta_N}\, \sin(2\,\phi_{21})
\,,\nl
\medskip \varepsilon_2 &\simeq\frac{\epsilon^{20}\, \Lambda_D^2\,\sin(2\,
\phi_{21})}{32\,\pi\,\delta_N\,v^2} \simeq \frac{1.1\times
10^{-13}}{\delta_N}\,\sin(2\,\phi_{21})\,.
\end{align}
The results of the numerical integration of the Boltzmann
equations for this case are presented in Fig.~\ref{fig5}. A
realistic value for $Y_B$ is also generated in this case.

Before we end this section, it is worthwhile to comment on the
possible effects of quantum corrections to the effective neutrino
mass matrix $\mathcal{M}_\nu$. This discussion turns out to be
relevant since in the examples considered above we have taken the
effective neutrino mass matrix at the heavy neutrino decoupling
scale. Although a detailed treatment would require a
renormalization group analysis for the effective neutrino mass
operator, one can employ the simple analytical treatment
considered by many authors in the literature (see for example
Refs.~\cite{GonzalezFelipe:2001kr,Casas:1999tp}). Following this,
we recall that the effective neutrino mass matrix at the
electroweak scale $m_Z$ can be related to the one at the heavy
neutrino decoupling scale $\Lambda_R$ as
\begin{align}
\label{rge1} \mathcal{M}_\nu(m_Z) \propto {\rm
diag}(1,1,1+\epsilon_\tau)\,\mathcal{M}_\nu(\Lambda_R)\,{\rm
diag}(1,1,1+\epsilon_\tau)\,,
\end{align}
where, neglecting the running of the charged lepton Yukawa
couplings,
\begin{align}
\label{rge2} \epsilon_\tau \simeq
\frac{3\,y_\tau^2}{32\pi^2}\ln\!\left(\frac{\Lambda_R}{m_Z}\right)\,.
\end{align}
Taking $y_\tau=m_\tau/v$ and $\Lambda_R \lesssim 10^{16}$ GeV, we
obtain $\epsilon_\tau \lesssim 10^{-5}$. Considering
$\mathcal{M}_\nu(\Lambda_R)$ as given in Eq.~(\ref{mnex1}) we
would get from Eq.~(\ref{rge1}):
\begin{align}
\label{rge3} &\frac{\dmtwon}
 {\dmthtw}\simeq
 \frac{(1-\epsilon_\tau)\,\epsilon^4}{\sqrt{2}}\quad ,\quad \tan^2\theta_{12}
 \simeq 1-\frac{\epsilon^2}{2\sqrt{2}}-\sqrt{2}\,\epsilon_\tau
 \,,\nonumber \\
&\tan^2\theta_{23}\simeq 1-2\,\epsilon_\tau\quad,\quad |U_{e3}|
\simeq\frac{\epsilon^2}{2\,\sqrt{2}}\,\left(1-\frac{3}{2}
 \,\epsilon_\tau\right).
\end{align}
Since $\epsilon_\tau \ll 1$, the results given by
Eqs.~(\ref{rsol}) and (\ref{angl}) are not significantly altered.
The same conclusions are drawn for the example considered in
Eq.~(\ref{mnex2}). Thus, we conclude that the effects of quantum
corrections due to the renormalization of the neutrino mass
operator can be safely neglected in our case.

\section{Concluding remarks} \label{conclusion}

We have analyzed, in the context of the minimal seesaw mechanism,
the link between leptogenesis and CP violation at low energies. In
particular, it was shown that, in order to present a thorough
discussion on this question, it is convenient to work in the WB
where both the charged lepton and right-handed Majorana neutrino
mass matrices are diagonal and real, and to write, without loss of
generality, the Dirac neutrino mass matrix as the product of a
unitary matrix and a lower triangular matrix. From the analysis of
the phases that contribute to leptogenesis and low-energy CP
violation, we have identified a necessary condition which is
required in order to establish a link between these two phenomena.
We have studied a class of models which satisfy the above
necessary condition in the simplest way, namely those where the
Dirac neutrino mass matrix is of the triangular form. By choosing
this structure the number of physical parameter in the theory is
reduced then enhancing its predictability. In this case there are
only three CP-violating phases which contribute both to
leptogenesis and CP violation at low energies. We have then
studied the minimal scenarios where a correct value of the
baryon-to-entropy ratio can be generated, while accounting for all
the low-energy neutrino data in the context of the LMA solution.
Moreover, the examples considered in Section~\ref{example} predict
the existence of low-energy CP-violating effects within the range
of sensitivity of the future long-baseline neutrino oscillation
experiments. In fact, it is a remarkable feature of these
scenarios that the solutions viable for leptogenesis are precisely
those which maximize $\mathcal{J}_{CP}$. The question of relating
the observed sign of the baryon asymmetry to the sign of the
leptonic CP violation, measurable at low energies through neutrino
oscillations, was also considered. Namely, we have concluded that,
within the minimal scenarios presented, this relation crucially
depends on the heavy Majorana neutrino mass spectrum. We remark
that a full discussion of this aspect requires the computation of
the BAU since, besides the prediction of the relative sign between
the BAU and $\mathcal{J}_{CP}$, the determination of $Y_B$ is of
extreme importance to infer about the viability of a given model.

\begin{acknowledgements}
We are grateful to Sacha Davidson for useful discussions. M.N.R.
is grateful to P. Frampton and T. Yanagida for interesting
conversations. This work was partially supported by {\em Funda\c{c}{\~a}o
para a Ci{\^e}ncia e a Tecnologia} (FCT, Portugal) through the
projects CERN/FIS/43793/2001 and CFIF - Plurianual (2/91). The
work of R.G.F. and F.R.J. was supported by FCT under the grants
SFRH/BPD/1549/2000 and \mbox{PRAXISXXI/BD/18219/98}, respectively.
\end{acknowledgements}

\appendix*
\section{Boltzmann equations}
\label{App1}

The computation of the cosmological baryon asymmetry $Y_B$
involves the solution of the full set of Boltzmann equations which
governs the time evolution of the right-handed neutrino number
densities $Y_{N_j}$ and the generated lepton asymmetry $Y_{L}$.
These quantities depend not only on the physics occurring in the
thermal bath but also on the universe expansion. In the SM
framework extended with heavy right-handed neutrinos the physical
processes relevant to the generation of the baryon asymmetry are
typically the $N_i$ decays and inverse decays into Higgs bosons
and leptons, the $\Delta\,L=1$ scatterings involving the top quark
and $\Delta\,L=2$ scatterings with virtual $N_i$ and $N_i-N_j$
scatterings \cite{Luty:un,Plumacher:1997ru}. The initially
produced lepton asymmetry $Y_L$ is converted into a net baryon
asymmetry $Y_B$ through the $(B+L)$-violating sphaleron processes.
One finds the relation
\begin{align} \label{YBYL}
Y_B=\xi\,Y_{B-L}=\frac{\xi}{\xi-1}\,Y_L\,.
\end{align}
For the SM with three heavy Majorana neutrinos $\xi \simeq 1/3$
and therefore $Y_B \simeq Y_{B-L}/3 \simeq -Y_L/2\,$.

The Boltzmann equations for the $N_i$ number densities and the
$Y_{B-L}$ asymmetry in terms of the dimensionless parameter
$z=M_1/T$, where $T$ is the temperature, can be written in the
form:
\begin{align}
 \label{boltz} \frac{dY_{N_j}}{dz} & =-\frac{z}{H s(z)}\left[
\left(\gamma_j^{D}+\gamma_j^{Q}\right)\left( \frac{Y_{N_j}}{Y^{\rm
eq}_{N_j}}-1\right)+\sum_{i=1}^{3}\gamma^{NN}_{ij}\left(
\frac{Y_{N_i}Y_{N_j}}{Y^{\rm eq}_{N_i}Y^{\rm eq}_{N_j}}-1\right)
\right]\ , \nl \noalign{\medskip}
 \frac{dY_{B-L}}{dz} & =-\frac{z}{H
 s(z)}\left[\sum_{j=1}^{3}\varepsilon_j\,
\gamma_j^{D}\left( \frac{Y_{N_j}}{Y^{\rm eq}_{N_j}}-1
\right)+\gamma^{W} \frac{Y_{B-L}}{Y^{\rm eq}_\ell} \right]\ ,
\end{align}
where $H$ is the Hubble parameter evaluated at $z=1$ and $s(z)$ is
the entropy density given by
\begin{align}
\label{Hs} H = \sqrt{\frac{4 \pi^3 g_{\ast}}{45}}\,
\frac{M_1^2}{M_P}\quad,\quad s(z) =\frac{2 \pi^2 g_{\ast}}{45}
\frac{M_1^3}{z^3}\ ,
\end{align}
respectively. Here, $g_{\ast}\simeq106.75$ is the effective number
of relativistic degrees of freedom and $M_P \simeq 1.2\times
10^{19}$ GeV is the Planck mass. The equilibrium number density of
a particle $i$ with mass $m_i$ is given by
\begin{align}
\label{Yeq} Y^{\rm eq}_i(z)=\frac{45}{4\pi^4}\frac{g_i}{g_{\ast}}
\left(\frac{m_i}{M_1}\right)^2 z^2
K_2\left(\frac{m_i\,z}{M_1}\right)\,,
\end{align}
where $g_i$ denote the internal degrees of freedom of the
corresponding particle ($g_{N_i}=2$, $g_\ell=4$) and $K_n(x)$ are
the modified Bessel functions. The $\gamma$'s are the reaction
densities for the different processes. For the decays one has
\begin{align}
\label{gmdec} \gamma_j^{D}&=\frac{M_1 M_j^3}{8
\pi^3z}\left(Y_\nu^\dag Y_\nu\right)_{jj}K_1\!\left(\frac{z
M_j}{M_1}\right)\,.\end{align}

The reaction densities for the $\Delta L=1$ processes with the top
quark and for the $\Delta L=2$ scatterings $N_i-N_j$ can be
written in the following way:
\begin{align}
\label{gmQNN}
\gamma_j^{Q}&=2\gamma_{t_j}^{(1)}+4\gamma_{t_j}^{(2)}\quad,\quad
\gamma_{ij}^{NN}=\gamma_{N_i N_j}^{(1)}+\gamma_{N_i N_j}^{(2)}\,,
\end{align}
respectively. Finally, $\gamma^W$ accounts for the washout
processes:
\begin{align}
\label{gamW}
\gamma^{W}&=\sum_{j=1}^{3}\left(\frac{1}{2}\gamma_j^{D}+\frac{Y_{N_j}}{Y^{\rm
eq}_{N_j}}\gamma_{t_j}^{(1)}+2\gamma_{t_j}^{(2)}\right)+2\,\gamma_N^{(1)}+2\,
\gamma_N^{(2)}\,.
\end{align}
Each of the above reaction densities can be computed through the
corresponding reduced cross section $\hat{\sigma}^{(i)}(x)$:
\begin{align}
\label{gdef} \gamma^{(i)}(z)=\frac{M_1^4}{64\,\pi^4 z}
\!\!\!\!\operatornamewithlimits{\int}_{(m_a^2+m_b^2)/M_1^2}^{\;\;\;\infty}
\!\!\!\!\hat{\sigma}^{(i)}(x)\,\sqrt{x}\,K_1(z\,\sqrt{x})\,dx\,,
\end{align}
where $x=s/M_1^2\ $, with $s$ being the center-of-mass energy
squared and $m_{a,b}$ are the masses of the initial particle
states. All the relevant reduced cross sections are summarized
below. For a more detailed presentation the reader is addressed to
Ref.~\cite{Plumacher:1997ru}.

We write the reduced cross sections as functions of the parameter
$x$, the Hermitian matrix $H_\nu=Y_\nu^\dag Y_\nu$ and the
quantities $a_j$ and $c_j$ defined as
\begin{align}
\label{Aaj} a_j=\left(\frac{M_j}{M_1}\right)^2\quad,\quad c_j=
\left(\frac{\Gamma_j}{M_1}\right)^2 =
\frac{a_j\,(H_\nu)_{jj}^2}{64\, \pi^2}\,,
\end{align}
where $\Gamma_j$ is the decay rate defined in Eq.~(\ref{crit1}).

For the $\Delta L=1$ processes involving interactions with quarks we have
\begin{align}
\label{Atj1} \hat{\sigma}_{t_j}^{(1)}(x)&\equiv
\hat{\sigma}(N_j+\ell\leftrightarrow
q+\bar{u})=3\,\alpha_u\,(H_\nu)_{jj}\left(\frac{x-a_j}{x}\right)^2\,,\\
\noalign{\medskip} \hat{\sigma}_{t_j}^{(2)}(x)& \equiv
\hat{\sigma}(N_j+u\leftrightarrow
\bar{\ell}+q)=\hat{\sigma}(N_j+\bar{q}
\leftrightarrow \bar{\ell}+\bar{u}) \nonumber \\
&=3\,\alpha_u\,(H_\nu)_{jj}\,\frac{x-a_j}{x}\,\left[\,1+\frac{a_h-a_j}
{x-a_j+a_h}+\frac{a_j-2
a_h}{x-a_j}\ln\left(\frac {x-a_j+a_h}{a_h}\right) \right]\,,
\end{align}
where $a_h$ (introduced to regularize the infrared divergencies) and
$\alpha_u$ are given by
\begin{align}
\label{Aahau} a_h=\left(\frac{\mu}{M_1}\right)^2\,,\quad
\alpha_u=\frac{{\rm Tr}(Y_u^\dag Y_u^{})}{4\pi}\simeq \frac{m_t^2}{4\pi
v^2}\,,
\end{align}
respectively. The mass parameter $\mu$ is chosen to be $\mu=800$~GeV as in
Refs.~\cite{Luty:un,Plumacher:1997ru}.

The reduced cross sections relative to the $N_i-N_j$ scatterings are:
\begin{align}
\label{ANiNj} \hat{\sigma}_{N_i N_j}^{(1)}&\equiv
\hat{\sigma}(N_i+N_j \leftrightarrow \ell+\bar{\ell})
\nl&=\frac{1}{2\pi}\left\{\,(H_\nu)_{ii}(H_\nu)_{jj}\left[\,\frac{\sqrt{\lambda_{ij}}}{x}
+\frac{a_i+a_j}{2x}\,L_{ij}\,\right]-{\rm
Re}\left[(H_\nu)_{ji}^2\right]
\frac{\sqrt{a_i a_j}}{x-a_i-a_j}\,L_{ij}\right\} \,,\\
\hat{\sigma}_{N_i N_j}^{(2)}&\equiv \hat{\sigma}(N_i+N_j
\leftrightarrow \phi+\phi^\dagger)\nl
&=\frac{1}{2\pi}\left\{\left|\,(H_\nu)_{ji}\right|^{\,2}\left[\,
\frac{L_{ij}}{2}-\frac{\sqrt{\lambda_{ij}}}{x} \,\right]-{\rm
Re}\left[(H_\nu)_{ji}^2\right] \frac{\sqrt{a_i
a_j}\,(a_i+a_j)}{x\,(x-a_i-a_j)}\,L_{ij}\right\}\,,
\end{align}
where the functions $L_{ij}$ and $\lambda_{ij}$ read as:
\begin{align}
\label{Lij} L_{ij}=2
\ln\!\left(\frac{|x+\sqrt{\lambda_{ij}}-a_i-a_j|}{2\,\sqrt{a_i
a_j}}\right)\;\;,\;\;\lambda_{ij}=x^2-2x\,(a_i+a_j)+(a_i-a_j)^2\,.
\end{align}
Since the heavy Majorana neutrinos are unstable particles, one has to
replace the usual fermion propagators by the off-shell propagators
$D_j(x)$ in the computation of the amplitudes for the right-handed
neutrino mediated processes:
\begin{align}
\label{AD}
\frac{1}{D_j}\equiv\frac{1}{D_j(x)}=\frac{x-a_j}{(x-a_j)^2+a_jc_j}\,.
\end{align}
For the $\Delta L= 2$ processes the reduced cross sections read then
\begin{align}
\label{AN1N2} \hat{\sigma}_N^{(1)}&\equiv \hat{\sigma}(\ell+\phi
\leftrightarrow
\bar{\ell}+\phi^\dagger)=\sum_{j=1}^{3}\,\left(H_\nu\right)_{jj}^2
\mathcal{A}_{jj}^{(1)}+ \sum_{\overset{n,j=1}{j<n}}^{3}{\rm
Re}\left[\left(H_\nu\right)_{nj}^2\right]\mathcal{B}_{nj}^{(1)}\,,
\nl \hat{\sigma}_N^{(2)}& \equiv \hat{\sigma}(\ell+\ell
\leftrightarrow
\phi^\dagger+\phi^\dagger)=\sum_{j=1}^{3}\,\left(H_\nu\right)_{jj}^2
\mathcal{A}_{jj}^{(2)}+ \sum_{\overset{n,j=1}{j<n}}^{3}{\rm
Re}\left[\left(H_\nu\right)_{nj}^2\right]\mathcal{B}_{nj}^{(2)}\,,
\end{align}
where
\begin{align}
\label{ACA}
\mathcal{A}_{jj}^{(1)}&=\frac{1}{2\pi}\left[1+\frac{a_j}{D_j}+\frac{a_j
\,x}{2\,D_j^2}-\frac{a_j}{x}\left(1+\frac{x+a_j}{D_j}\right)
\ln\left(\frac{x+a_j}{a_j}\right)\right]\,,\nl\noalign{\medskip}
\mathcal{B}_{nj}^{(1)}&=\frac{\sqrt{a_n
a_j}}{2\pi}\left[\frac{1}{D_j}+\frac{1}{D_n}+\frac{x}{D_jD_n}+\left(1+\frac{a_j}
{x}\right)\left(\frac{2}{a_n-a_j}-\frac{1}{D_n}\right)
\ln\left(\frac{x+a_j}{a_j}\right)\right.\nl &
\left.+\left(1+\frac{a_n}{x}\right)\left(
\frac{2}{a_j-a_n}-\frac{1}{D_j}\right)
\ln\left(\frac{x+a_n}{a_n}\right)\right]\,,\nl\noalign{\medskip}
\mathcal{A}_{jj}^{(2)}&=\frac{1}{2\pi}\left[\frac{x}{x+a_j}+\frac{a_j}{x+2a_j}
\ln\left(\frac{x+a_j}{a_j}\right)\right]\,,\nl\noalign{\medskip}
\mathcal{B}_{nj}^{(2)}&=\frac{\sqrt{a_n
a_j}}{2\pi}\left\{\frac{1}{x+a_n+a_j}\,\ln\left[\frac{(x+a_j)(x+a_n)}{a_j\,
a_n}\right] + \frac{2}{a_n-a_j}\,
\ln\left[\frac{a_n\,(x+a_j)}{a_j\,(x+a_n)}\right]\right\}\,.
\end{align}
In our analysis we have computed numerically the reaction densities
through Eq.~(\ref{gdef}) and the above definitions of the reduced cross
sections. Nevertheless, useful analytical approximations can be obtained
for specific temperature regimes \cite{Plumacher:1997ru}.

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\end{thebibliography}

\end{document}

