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\begin{document}
\title{Leptogenesis-MNS Link in Unified Models with Natural Neutrino Mass
Hierarchy}
\author{S. F. King}
\affiliation{Department of Physics and Astronomy,
University of Southampton, Southampton, SO17 1BJ, U.K.}

\begin{abstract}
We discuss the relation between leptogenesis and the MNS phases in the class
of see-saw models called light sequential dominance in which the
right-handed neutrinos dominate sequentially, with the dominant
right-handed neutrino being the lightest one. 
The heaviest right-handed neutrino then decouples,
leaving effectively two right-handed neutrinos.
Light sequential dominance is motivated by
$SO(10)$ models which predict three right-handed neutrinos.
With an approximate Yukawa texture zero 
in the 11 position there are only two see-saw phases 
which are simply related to the MNS phase 
measurable at a neutrino factory and the
leptogenesis phase. The leptogenesis phase is predicted to be
equal to the neutrinoless double beta decay phase.
\end{abstract}

\maketitle

There is by now good evidence for neutrino masses and mixings
from atmospheric and solar neutrino oscillations \cite{Gonzalez-Garcia:2002sm}.
The minimal neutrino sector required to account for the
atmospheric and solar neutrino oscillation data consists of
three light physical neutrinos with left-handed flavour eigenstates,
$\nu_e$, $\nu_\mu$, and $\nu_\tau$, defined to be those states
that share the same electroweak doublet as the left-handed
charged lepton mass eigenstates.
The neutrino flavor eigenstates $\nu_e$, $\nu_\mu$, and $\nu_\tau$ are
related to the neutrino mass eigenstates $\nu_1$, $\nu_2$, and $\nu_3$
with mass $m_1$, $m_2$, and $m_3$, respectively, by a $3\times3$ 
unitary matrix called the MNS matrix $U_{MNS}$
\cite{Maki:1962mu,Lee:1977qz}
\begin{equation}
\left(\begin{array}{c} \nu_e \\ \nu_\mu \\ \nu_\tau \end{array} \\ \right)=
U_{MNS}
\left(\begin{array}{c} \nu_1 \\ \nu_2 \\ \nu_3 \end{array} \\ \right)
\; .
\end{equation}
Assuming the light neutrinos are Majorana,
$U_{MNS}$ can be parameterized in terms of three mixing angles
$\theta_{ij}$, a Dirac phase $\delta_{\rm MNS}$, together with
two Majorana phases $\beta_1,\beta_2$, as follows
\begin{equation}
U_{MNS}=R_{23}U_{13}R_{12}P_{12}
\end{equation}
where
\bea
R_{23} & = &
\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & c_{23} & s_{23} \\
0 & -s_{23} & c_{23} \\
\end{array}\right), \ \  
R_{12}  = 
\left(\begin{array}{ccc}
c_{12} & s_{12} & 0 \\
-s_{12} & c_{12} & 0\\
0 & 0 & 1 \\
\end{array}\right), \nonumber \\
U_{13} & = &
\left(\begin{array}{ccc}
c_{13} & 0 & s_{13}e^{-i\delta_{\rm MNS}} \\
0 & 1 & 0 \\
-s_{13}e^{i\delta_{\rm MNS}} & 0 & c_{13} \\
\end{array}\right), \nonumber \\
P_{12}  & = &
\left(\begin{array}{ccc}
e^{i\beta_1} & 0 & 0 \\
0 & e^{i\beta_2} & 0\\
0 & 0 & 1 \\
\end{array}\right)
\label{MNS}
\eea
where $c_{ij} = \cos\theta_{ij}$ and $s_{ij} = \sin\theta_{ij}$.

The most elegant explanation of small neutrino masses continues
to be the see-saw mechanism \cite{seesaw,Mohapatra:1979ia}. 
According to the see-saw
mechanism, lepton number is broken at high energies due to 
right-handed neutrino Majorana masses, resulting in small left-handed
neutrino Majorana masses suppressed by the heavy mass scale.
The see-saw mechanism also provides an attractive mechanism
for generating 
the baryon asymmetry of the universe 
via leptogenesis \cite{yanagida1}, \cite{luty}.

There is a large literature concerned with a possible 
link between the phases responsible for leptogenesis and those
in the low energy neutrino sector, especially the Dirac phase 
$\delta_{\rm MNS}$ which may be measured at a neutrino factory.
In general the conclusion seems to be that there is no direct link 
\cite{large}.
The leptogenesis phase appears to be quite independent of the 
neutrino factory phase. However in specific regions of parameter
space \cite{Davidson:2002em} or in specific models 
\cite{Joshipura:2001ui,2RHN,Frampton:2002qc} such a link may exist.
For example in models with only two right-handed neutrinos, 
there does appear to be a link at least in special cases 
\cite{2RHN,Frampton:2002qc}.
However many models predict three right-handed neutrinos due to
a gauged $SU(2)_R$ which may be embedded into a larger gauge
group such as $SO(10)$. Nevertheless it is well known 
\cite{King:1998jw,King:1999mb,King:2002nf} that certain classes of 
three right-handed neutrino models can behave effectively as two right-handed
neutrino models. This provides the motivation for our present study.

The purpose of the present paper is to discuss leptogenesis in
a class of models with three right-handed neutrinos in which 
a neutrino mass hierarchy arises naturally due to a
single right-handed neutrino dominantly contributing to the
heaviest neutrino mass, and a 
second right-handed neutrino dominantly contributing to the
the second heaviest neutrino mass \cite{King:1999mb}.
This mechanism called sequential dominance was discussed recently 
including phases in \cite{King:2002nf} from which many of the 
results in this paper have been derived. We shall further
focus on the case that the dominant right-handed neutrino is the
lightest one, called light sequential dominance (LSD).
In LSD the heaviest right-handed
neutrino is irrelevant for both neutrino oscillation experiments
and leptogenesis, and effectively decouples, leading to an 
effective two right-handed neutrino description consistent with $SO(10)$.

For the present discussion we shall work in the flavour basis
where the charged lepton mass matrix and the right-handed neutrino
Majorana mass matrix are both diagonal with real positive eigenvalues.
We write the latter as
\begin{equation}
M_{RR}=
\left( \begin{array}{ccc}
Y & 0 & 0    \\
0 & X & 0 \\
0 & 0 & X'
\end{array}
\right) 
\label{seq1}
\end{equation}
where we assume $Y\ll X\ll X'$.
We write the neutrino Yukawa matrix as
\begin{equation}
Y^{\nu}_{LR}=
\left( \begin{array}{ccc}
0 & a & a'    \\
e & b & b' \\
f & c & c'
\end{array}
\right).
\label{dirac}
\end{equation}
The condition for sequential dominance is \cite{King:1999mb}
\beq
\frac{|e^2|,|f^2|,|ef|}{Y}\gg
\frac{|xy|}{X} \gg
\frac{|x'y'|}{X'}
\label{srhnd}
\eeq
where $x,y\in a,b,c$ and $x',y'\in a',b',c'$, 
where all Yukawa couplings are assumed to be complex.
As we shall see this implies that the third right-handed neutrino
of mass $X'$ is irrelevant for both leptogenesis and neutrino 
oscillations. Note that realistic $SO(10)$ models 
have been proposed \cite{King:2001uz} which satisfy Eq.\ref{srhnd},
and which have an approximate texture zero in the 11 position.

Assuming Eq.\ref{srhnd} the neutrino masses
are given to leading order in $m_2/m_3$ by 
\bea
m_1 & \sim & O(\frac{x'y'}{X'}v_2^2) \label{m1} \\
m_2 & \approx &  \frac{|a|^2}{Xs_{12}^2} v_2^2 \label{m2} \\
m_3 & \approx & \frac{|e|^2+|f|^2}{Y}v_2^2 
\label{m3}
\eea
where $v_2$ is a Higgs vacuum expectation value (vev) associated with
the (second) Higgs doublet that couples to the neutrinos
and $s_{12}=\sin \theta_{12}$ given below. 
Note that with LSD each neutrino mass is generated
by a separate right-handed neutrino, and the sequential dominance condition
naturally results in a neutrino mass hierarchy $m_1\ll m_2\ll m_3$.
The neutrino mixing angles are given to leading order in $m_2/m_3$ by, 
\bea
\tan \theta_{23} & \approx & \frac{|e|}{|f|} \label{23}\\
\tan \theta_{12} & \approx &
\frac{|a|}
{c_{23}|b|
\cos(\tilde{\phi}_b)-
s_{23}|c|
\cos(\tilde{\phi}_c)} \label{12} \\
\theta_{13} & \approx &
e^{i(\tilde{\phi}+\phi_a-\phi_e)}
\frac{|a|(e^*b+f^*c)}{[|e|^2+|f|^2]^{3/2}}
\frac{Y}{X}
\label{13}
\eea
where we have written some (but not all) complex Yukawa couplings as
$x=|x|e^{i\phi_x}$. The phase $\delta_{\rm MNS}$
is fixed to give a real angle
$\theta_{12}$ by,
\beq
c_{23}|b|
\sin(\tilde{\phi}_b)
\approx
s_{23}|c|
\sin(\tilde{\phi}_c)
\label{chi1}
\eeq
where 
\bea
\tilde{\phi}_b &\equiv & 
\phi_b-\phi_a-\tilde{\phi}+\delta_{\rm MNS}, \nonumber \\ 
\tilde{\phi}_c &\equiv & 
\phi_c-\phi_a+\phi_e-\phi_f-\tilde{\phi}+\delta_{\rm MNS}.
\label{bpcp}
\eea
The phase $\tilde{\phi}$
is fixed to give a real angle
$\theta_{13}$ by,
\beq
\tilde{\phi} \approx  \phi_e-\phi_a -\phi_{\rm COSMO}
\label{phi2dsmall}
\eeq
where
\beq
\phi_{\rm COSMO}=\arg(e^*b+f^*c)
\label{lepto0}
\eeq
is the leptogenesis phase 
corresponding to the interference diagram involving the
lightest and next-to-lightest right-handed neutrinos \cite{King:2002nf}.
Eq.\ref{lepto0} may be expressed as
\beq
\tan \phi_{\rm COSMO} \approx 
\frac{|b|s_{23}s_2+|c|c_{23}s_3}{|b|s_{23}c_2+|c|c_{23}c_3}.
\label{phi121}
\eeq
Inserting $\tilde{\phi}$ in Eq.\ref{phi2dsmall}
into Eqs.\ref{chi1},\ref{bpcp},
\bea
&&c_{23}|b|\sin(\eta_2+\phi_{\rm COSMO}+\delta_{\rm MNS}) \nonumber \\
& \approx &
s_{23}|c|
\sin(\eta_3+\phi_{\rm COSMO} +\delta_{\rm MNS}).
\label{chi12}
\eea
Eq.\ref{chi12} may be expressed as
\beq
\tan (\phi_{\rm COSMO}+\delta_{\rm MNS}) \approx 
\frac{|b|c_{23}s_2-|c|s_{23}s_3}{-|b|c_{23}c_2+|c|s_{23}c_3}
\label{phi12del}
\eeq
where we have written $s_i=\sin \eta_i, c_i=\cos \eta_i$
where
\beq
\eta_2\equiv \phi_b-\phi_e, \ \ \eta_3\equiv \phi_c-\phi_f
\label{eta}
\eeq
are invariant under charged lepton phase redefinitions.
The reason that the see-saw parameters only
involve two invariant phases $\eta_2, \eta_3$ rather than the usual six
is due to the LSD 
assumption which has the effect of decoupling the  
heaviest right-handed neutrino, which removes three phases, 
together with the assumption
of a 11 texture zero, which removes another phase.

Eq.\ref{phi12del} shows that
$\delta_{\rm MNS}$ is a function of 
the two see-saw phases
$\eta_2 , \eta_3$ that also determine $\phi_{\rm COSMO}$ in Eq.\ref{phi121}.
If both the phases $\eta_2 , \eta_3$ are zero,
then both $\phi_{\rm COSMO}$ and $\delta_{\rm MNS}$ are necessarily zero.
This feature is absolutely crucial. It means that,
barring cancellations, measurement of a non-zero value for 
the phase $\delta_{\rm MNS}$ at a neutrino factory will be a signal of a
non-zero value of the leptogenesis phase $\phi_{\rm COSMO}$.
If a second Yukawa element were to be 
set to zero, then another invariant see-saw phase could be removed, and then 
both $\phi_{\rm COSMO}$ and $\delta_{\rm MNS}$ would be determined
in terms of a single see-saw phase. In this case there would be 
a direct relation between $\phi_{\rm COSMO}$ and $\delta_{\rm MNS}$.
For example if we set $b=0$ then the only remaining see-saw phase is $\eta_3$
and Eqs.\ref{phi121},\ref{phi12del} would imply that 
\beq
\delta_{\rm MNS}=-2\phi_{\rm COSMO}.
\label{signs}
\eeq
With $b=0$ our results
reduce to those in \cite{Frampton:2002qc} based on two right-handed neutrinos
and two Yukawa zeroes. They also also found that 
the sign of CP asymmetry at a neutrino factory is anticorrelated
with the CP asymmetry of the universe, as in Eq.\ref{signs}
\cite{Frampton:2002qc}.

So far we have not discussed the Majorana phases $\beta_1$, $\beta_2$ 
which appear in Eq.\ref{MNS}. In LSD the lightest neutrino mass $m_1$
is very small which effectively makes $\beta_1$ unmeasurable.
The remaining Majorana phase is given from the relation 
\beq
\delta_{\rm MNS}+\beta_2= \frac{\phi_2}{2}-\frac{\phi_3}{2}
\label{beta2}
\eeq
where $\phi_2$, $\phi_3$ are the phases of the complex neutrino masses 
$m_2'=m_2e^{i\phi_2}$, $m_3'=m_3e^{i\phi_3}$, after the neutrino mass
matrix has been diagonalised. 
In our conventions these phases 
are removed to leave the positive neutrino masses in Eqs.\ref{m2},\ref{m3}
resulting in Eq.\ref{beta2}. We find
\beq
\phi_2=2\phi_a, \ \ \phi_3=2(\phi_a+\phi_{\rm COSMO}).
\label{phi}
\eeq
From Eqs.\ref{beta2},\ref{phi}, we find 
\beq
\phi_{\rm COSMO}=-(\delta_{\rm MNS}+\beta_2).
\label{remarkable1}
\eeq
The combination of phases
in Eq.\ref{beta2} is just the phase $\phi_{\beta \beta 0\nu}$ responsible
for CP violation in neutrinoless double beta decay, whose magnitude is
\beq
|\phi_{\beta \beta 0\nu}|=|\delta_{\rm MNS}+\beta_2|.
\label{beta}
\eeq
From Eq.\ref{remarkable1} and \ref{beta} we find the remarkable result
\beq
|\phi_{\rm COSMO}|=|\phi_{\beta \beta 0\nu}|.
\label{remarkable2}
\eeq
In the bottom-up analysis of \cite{Davidson:2002em}, 
they also found a relation like Eq.\ref{remarkable2} in a particular region of 
their low energy parameter space.

To conclude, we have discussed the relation between leptogenesis 
and the MNS phases in a well motivated class of models consistent with
$SO(10)$ in which the neutrino mass hierarchy arises naturally from the
LSD mechanism. With an approximate 11 texture zero
the two see-saw phases $\eta_2$, $\eta_3$ 
are simply related to $\delta_{\rm MNS}$ and $\phi_{\rm COSMO}$
according to Eqs.\ref{phi121},\ref{phi12del}.
The leptogenesis phase is predicted to be
equal to the neutrinoless double beta decay phase as in Eq.\ref{remarkable2}.

\vspace{0.25in}

I thank Alejandro Ibarra for enlightening discussions and for
critically reading the manuscript. I also thank John Ellis, Graham Ross
and Liliana Velasco-Sevilla for discussions.
I would also like to thank the CERN Theory Division for its hospitality
and PPARC for a Senior Research Fellowship.


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\end{document}














