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\title{Leptogenesis and low energy phases}

\author{Sacha Davidson$^*$ and
Alejandro Ibarra$^{\dagger}$}


\address{ $^*$ Department of Physics, Univ. of Durham,
Durham, DH1 3LE, UK}

\address{$^{\dagger}$ Theoretical Physics, Univ. of Oxford,
 1 Keble Road, Oxford, OX1 3NP, UK}


\begin{abstract}
In this work we analyze the relevance of the different
low energy phases for the CP asymmetry of the Universe. To 
this end, we develop a parametrization of the see-saw
mechanism in terms of low energy data that will allow
us to study leptogenesis from a bottom-up perspective. 
We find that the relevant phases for leptogenesis
depend on the particular scenario, and we classify the 
different possibilities in connection to lepton flavour
violation. We find that the phase that would be measured 
at the neutrino factory is relevant for leptogenesis
over much of the parameter space where this phase
can be measured.
\end{abstract}

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\section{Introduction}

After the discovery of neutrino oscillations, leptogenesis \cite{Michael}
stands as one of the most appealing explanations for the 
observed baryon asymmetry of the Universe. It relies
on the see-saw mechanism, that assumes that there exist
three right handed neutrinos with Majorana masses in the
range $\sim 10^{10}-10^{15}$ GeV, that couple to the left-handed
neutrinos through Yukawa couplings. These simple assumptions, 
that can be naturally accommodated in a GUT theory, are enough
to explain the smallness of neutrino masses. It is 
also remarkable that the decay of the 
right-handed neutrinos in the early Universe can also generate
the observed baryon asymmetry. This is the so-called
leptogenesis scenario.

To generate a 
lepton asymmetry three conditions have to be satisfied 
in the early Universe:
deviation from thermal equilibrium, C and CP violation, and lepton 
number violation. This lepton asymmetry would be eventually 
reprocessed into the observed baryon asymmetry by sphaleron processes.
The successes of the hot Big Bang theory indicate that there were
processes out of thermal equilibrium in the early Universe. On the
other hand, it is believed that neutrinos are Majorana particles 
and have lepton number interactions, although the experiments
on neutrinoless double-beta decay have not confirmed this. Finally,
no indication has been found so far for CP violation in the leptonic
sector. For this reason, a lot of effort is being bestowed
in the design of a neutrino factory, that could possibly 
measure the leptonic version of the CKM phase. Therefore, 
it is interesting to study the interplay between the CP violation that
would be observed at low energies at the neutrino factory, and
the CP violation that appears at high energies and that could be
responsible for the baryon asymmetry. 

\section{The top-down vs. the bottom-up approach}

The lepton asymmetry is generated through the decay rate difference 
of right-handed neutrinos into lepton and Higgs doublets,
and their conjugate counterparts. These processes
occur at very high energies and are not directly testable by experiments,
so it is not straight-forward to make predictions about leptogenesis. However,
these Yukawa couplings and right-handed masses leave an imprint
at low energies that can be exploited to obtain information
about the theory at high energies. Clearly,
the neutrino masses, the mixing angles and the phases of the MNS matrix
depend on the neutrino Yukawa couplings and the right handed neutrino
masses. However, this information is not enough to reconstruct the
whole theory, since the high energy theory has
18 parameters (12 real parameters and 6 phases), while the
neutrino mass matrix only has nine (three masses, three mixing angles
and three CP violating phases). 

Fortunately there is a second window onto
the high energy physics, namely radiative corrections. In 
a supersymmetric theory, neutrino Yukawa
couplings affect the renormalization group evolution of the slepton
mass matrix. Therefore, if we were able to measure the slepton mass
matrix at low energies and we knew its structure at high energies, we 
would be able to disentangle the effects of the neutrino Yukawa couplings
and obtain additional information about them. As a matter of fact,
{\it the information encoded in radiative corrections is enough 
to reconstruct the complete
theory} \cite{di}. Of course one has to know 
which is the theory at high energies,
and this relies on theoretical assumptions. However, a common assumption
in the literature is to assume that the slepton mass matrix at high
energies is proportional to the identity. This assumption is motivated
both by phenomenological reasons (for instance, the non observation of 
rare lepton decays) as well as theoretical reasons (this is
what one obtains in several well motivated supersymmetry 
breaking scenarios, such as minimal supergravity). 
If this is the case, neutrino
Yukawa couplings spoil the diagonal form and this will give rise to clear
signatures that could be measured with low energy experiments. 
For example, the off-diagonal entries
in the Yukawa couplings would give rise to flavour violating
entries in the slepton mass matrix, that would in turn induce
rare processes like $\mu \rightarrow e \gamma$. Therefore, the observation
of these processes would give additional information about the Yukawa
couplings. Also, at tree level the three
sneutrino masses are degenerate, and this degeneracy is lifted by
radiative corrections. This non-degeneracy could also be
measured and could provide further information about the see-saw mechanism.
Although the possibility of reconstructing the high energy
theory just from low energy data is very attractive, in practice
is not attainable \cite{di}. 
Nevertheless, this parametrization opens the possibility
of studying leptogenesis from a bottom-up perspective and
provides a natural set up to address the interplay between
leptogenesis and low energy phases.

\section{Leptogenesis from the bottom-up}

In this talk we will concentrate in the step of leptogenesis
that is most closely related to neutrino physics, namely the
CP asymmetry. Using a bottom-up approach we were
able to reformulate leptogenesis in terms of low energy data:
 neutrino and sneutrino masses, 
mixing angles and CP violating phases \cite{di3}. 
We found that the CP asymmetry
generated in the decay of the lightest right-handed neutrino
depends on five phases: the three phases in the MNS matrix 
and two combinations of phases in the sneutrino mass matrix. 
We also found that leptogenesis does not depend on a single phase,
but instead on combinations of them, being the particular
combination rather model dependent.

We have been able to classify all the
possible different scenarios in terms of low energy observables, and
identify the relevant phases for leptogenesis in each scenario. 
The different scenarios can be classified essentially 
depending on the rates of rare lepton decays,
 like $\mu \rightarrow
e \gamma$ and $\tau \rightarrow e \gamma$ (the asymmetry produced
in the decay of the lightest right-handed neutrino 
is not very sensitive to $\tau \rightarrow \mu \gamma$).
Here, we report the main results (for details, see \cite{di3})
and concentrate on the phases that have
better prospects to be measured experimentally: the ``neutrino factory
phase'' (the $\nu$-fact phase) and the ``neutrinoless double-beta decay phase''
(the $(\beta \beta)_{0 \nu}$-decay phase):
\begin{itemize}
%
\item {\bf Vanishing rates.} In this case, leptogenesis depends 
exclusively on the phases that appear in the MNS matrix, and furthermore,
depends only on phase differences. 
We find that the  $(\beta \beta)_{0 \nu}$-decay 
phase is always important, whereas the $\nu$-fact phase is relevant 
when the CHOOZ angle is 
$\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 0.01$ 
(for the hierarchy of light neutrino masses 0.01:0.1:1). It is 
interesting to note that for these values of the CHOOZ angle, 
the neutrino factory should be able to observe CP violation.
%
\item {\bf Small rates.} Now all the five phases become 
relevant, although this number can be reduced in particular
limits. If all the mixing angles in the
sneutrino sector are smaller than the CHOOZ angle, then
the  $\nu$-fact phase is relevant for leptogenesis. Otherwise,
if one mixing angle in the sneutrino sector is larger,  the  $\nu$-fact phase 
becomes irrelevant. As before,
the  $(\beta \beta)_{0 \nu}$-decay is always important.
%
\item {\bf Large rates.} In this case, the $\nu$-fact phase is 
essentially irrelevant, whereas the $(\beta \beta)_{0 \nu}$-decay phase 
is still important. However, leptogenesis depends not only on the 
$(\beta \beta)_{0 \nu}$-decay phase, but also on some other 
phases associated to sneutrino physics that have 
little prospects to be measured.
\end{itemize}

\section{Conclusions}

We have developed a bottom-up parametrization of the see-saw
mechanism that has allowed us to formulate leptogenesis just in 
terms of low energy observables: neutrino masses and mixing
angles, branching ratios for rare lepton decays, sneutrino masses, etc.,
and phases that could be in principle measured at low energies. We
have classified the possible scenarios by the rates for the
rare lepton processes and we have identified the relevant
low energy phases for leptogenesis in each scenario. We
have found that the ``neutrinoless double-beta decay phase'' is 
always important for leptogenesis, whereas the ``neutrino factory
phase'' is important over much of the parameter space
where the neutrino factory can measure CP violation (as long
as the rates for the rare lepton decays are not too large). Also, we have
found that leptogenesis depends on combinations of phases
rather than on a single phase. Therefore, even if we could 
measure CP violation at the neutrino factory, little could be 
said about leptogenesis. However, and since one does not expect
cancellations among the different phases, the observation
of CP violation at the neutrino factory would give further
support to leptogenesis as the actual mechanism to generate
the observed baryon asymmetry in the Universe.

\section*{References}
\begin{thebibliography}{99}
%
\bibitem{Michael} See the contribution by Michael Pl\"umacher to these
proceedings for a nice review of leptogenesis.

\bibitem{di} S. Davidson and A. Ibarra, JHEP {\bf 0109} (2201) 013.

\bibitem{di3} S. Davidson and A. Ibarra, hep-ph/0206304.

\end{thebibliography}

\end{document}

