% This is the revised version of the paper on sneutrino mass-splitting and
% the BAU
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\newcommand{\be}{\begin{eqnarray}}
\newcommand{\ee}{\end{eqnarray}}
\newcommand{\nn}{\nonumber}
\newcommand{\ovl}{\overline}
\newcommand{\ra}{\rightarrow}
\newcommand{\lra}{\longleftrightarrow}
{\newcommand{\lsim}{\mbox{\raisebox{-.6ex}{~$\stackrel{<}{\sim}$~}}}
{\newcommand{\gsim}{\mbox{\raisebox{-.6ex}{~$\stackrel{>}{\sim}$~}}}
\def\mNl{m_{\tilde{\nu}_l^N}}
\def\mM{\tilde{m}_M}
\def\0n{0\nu\beta\beta}
\def\LM{$L\hskip-2mm /\mbox{MSSM}\;$}
\def\mN{m_{\tilde{\nu}_l^N}}
\def\mD{\tilde{m}_D}
%\def\siml{\vspace{-0.1cm}$\stackrel{\leq}{\sim}$}
\def\Lbar{L\hskip-1.5mm/}

\begin{document}
\begin{center}
{\Large \bf Light Lepton Number Violating Sneutrinos and the Baryon 
Number of the Universe} \\
\vspace{0.5cm}
{\large H.V. Klapdor-Kleingrothaus $^{\dagger}$, 
St. Kolb$^{\dagger,\ddagger}$ and 
V.A. Kuzmin$^{\ddagger}$}\\
\vspace{0.3cm}
{$^{\dagger}$ Max-Planck-Institut f\"ur Kernphysik,
P.O. 10 39 80, D-69029 Heidelberg, Germany} \\ \vspace{0.3cm}
{$^{\ddagger}$ Institute for Nuclear Research of the Russian Academy
of Sciences, 60th October Anniversary Prospect 7a, 117321 Moscow, Russia}
\end{center}

\vspace{0.5cm}

\begin{abstract}
Recent results of neutrino oscillation experiments point to a non-vanishing
neutrino mass. Neutrino mass models favour Majorana-type neutrinos. In such
circumstances it is natural that the supersymmetric counterpart of the 
neutrino, the sneutrino, bears also lepton number violating properties.
If the amount of lepton number violation is large enough the sneutrino
may be the Cold Dark Matter in the universe. On the other hand, the fact 
that the universe exhibits an asymmetry in the baryon and antibaryon 
numbers poses constraints on the extent of
lepton number violation in the light sneutrino sector if the electroweak
phase transition is second or weak first order. From the requirement that 
the Baryon Asymmetry of the Universe should not be washed out by sneutrino 
induced lepton number violating interactions and sphalerons
below the critical temperature of the electroweak phase transition we find 
that the mass-splitting of the light sneutrino mass states is compatible
with the sneutrino Cold Dark Matter hypothesis only for heavy gauginos
$M_1,M_2\gsim 500 GeV$ and opposite sign gaugino mass parameters.
 
%
%\noindent
%PACS numbers: 14.80.Ly, 12.60.Jv, 14.60.Pq
\end{abstract}

\vspace{0.5cm}
%\section{Introduction}
\noindent
{\bf I \hspace{0.2cm} Introduction}\\

There are hints from neutrino oscillation experiments that the neutrino 
is massive (\cite{kamiokande} and refs. therein. For a recent overview 
see {\it e.g.} \cite{oscover}). In 
most neutrino mass models the neutrino is of Majorana-type, {\it i.e.} it
violates lepton number $L$. If this is indeed the case the next generation
of experiments searching for neutrinoless double beta ($\0n$) decay, which 
%takes place only for Majorana neutrinos, 
are the only experiments capable of deciding on the nature of the 
neutrino, possibly will be able to indeed observe a $\0n$-decay
signal (for a recent overview see {\it e.g.} \cite{klaptrento}).
%$\0n$-decay takes place only for Majorana neutrinos and such experiments
%are the only ones which are able to decide on the nature of the neutrino.

On the other hand, it has been shown in \cite{theorem} that if the 
neutrino is a massive 
Majorana field the low energy effective theory of the supersymmetric 
extension of the Standard Model (for a phenomenological overview
see {\it e.g.} \cite{haberkane}) will contain mass terms 
for the sneutrino
which violate $L$ too, regardless of the mechanism which is
responsible for the generation of sneutrino masses in the unbroken 
theory. 
In \cite{susyseesaw} a model containing heavy $SU(2)$ singlet 
sneutrino fields and $L$-violating mass terms involving these 
fields has been examined (such models have 
been considered previously in connection with the generation of the 
Baryon Asymmetry of the Universe (BAU) at 
some high temperature, see {\it e.g.} \cite{baugen}). In both cases
below the electroweak symmetry breaking scale 
the weak states $\tilde{\nu},\tilde{\nu}^*$ are no longer mass
states and the resulting mass states violate $L$, exhibit a mass-splitting 
and give rise to $L$-violating processes  
%Such Lepton number violating sneutrino masses on the other hand mediate
%$L$-violating processes 
which have been analyzed {\it e.g.} for the Next Linear Collider in 
\cite{phen} and for $\0n$ decay in \cite{sndb}.

It has been pointed out in \cite{cdm} that in a scenario where the
% of the lepton number violating sneutrino masses 
%the mass-splitting of the light sneutrino states being of order $10GeV$ 
light sneutrino mass states exhibit a mass-splitting
the lightest sneutrino could account for the Cold Dark Matter (CDM) in the 
universe. This is due to the fact that the sneutrino mass states couple 
``off-diagonally'' to the $Z^0$ on grounds of Bose statistics and
angular momentum conservation so that the arguments excluding ordinary
sneutrinos from constituting a substantial fraction of the CDM are
not valid in the case of light sneutrinos exhibiting a mass difference.
However, it has been shown in \cite{cdm} that the mass difference
should be of order ${\cal O}(\mbox{few}\; GeV)$.

On the other hand sneutrino mediated $L$-violating reactions, like 
any other $L$-violating process, may be dangerous for the BAU  
due to sphalerons \cite{sph}. The constraints on the 
$L$-violating sneutrino properties stemming from the requirement that a BAU 
generated at some early epoch in the evolution of the universe 
should not be destroyed by sneutrino-induced interactions during a
later epoch are the subject of this note. 

%\section{Lepton number violation in the sneutrino sector}
\vspace{0.5cm}
\noindent
{\bf II \hspace{0.2cm} Lepton number violation in the sneutrino sector}\\

In the following the discussion will be restricted to the one-generation
case. We plan to investigate the impact of possible $CP$ violation in a
multi-generation scheme for the light sneutrino sector elsewhere.  
It has been pointed out in \cite{theorem} that the low energy ({\it i.e}
below the scale where the $SU(2)_L \times U(1)_Y$ gauge group is broken
into $U(1)_{em}$) sneutrino mass terms can be written as
%in the one generation case (we plan to investigate the 
\be \label{effective}
{\cal L}_{mass}^{eff} = 
-\frac{1}{2}(\tilde{m}_D^2 \tilde{\nu}_L \tilde{\nu}_L^* + 
             \tilde{m}_M^2 \tilde{\nu}_L \tilde{\nu}_L + h.c.).
\ee
Here $\tilde{m}_D^2$ contains as usual contributions from $V_{soft}$
and from the D-term, whereas $\tilde{m}_M^2$
violates $L$ explicitly and may have its origin at some high energy scale. 
Furthermore, loops containing Majorana neutrinos and neutralinos 
induce contributions to $\tilde{m}_M^2$ radiatively. Expression 
(\ref{effective}) is valid if no right-handed sneutrino sector
is present in the theory and independent of the mechanism which generates
the $L$-violating sneutrino mass. Writing 
$\tilde{\nu}_L=1/\sqrt{2}(\tilde{\nu}_1 + i \tilde{\nu}_2)$
where $\tilde{\nu}_{1,2}$ are real, the resulting mass states are simply
$\tilde{\nu}_{1,2}$ with masses $m_{1/2}^2=\tilde{m}_D^2 \pm 
\tilde{m}_M^2$ and the mass difference is 
$\Delta m^2 = 2 \tilde{m}_M^2$.

If a right-handed sector is included
the sneutrino mass terms after electroweak symmetry breaking 
without an explicit $L$-violating mass term for the $SU(2)$ doublet 
sneutrinos are
\cite{susyseesaw}
\be \label{massmatrix}
{\cal L}_{mass}=-\frac{1}{2} (\phi_1,\phi_2) 
\left( 
\begin{array}{cc} {\cal M}_+^2 & 0 \\
                                0     & {\cal M}_-^2       
\end{array}
\right)
\left( \begin{array}{c} \phi_1 \\ \phi_2 \end{array} \right)
\ee
where $\phi=(\tilde{\nu}_i,\tilde{N}_i)^T$ and 
$\tilde{N}=(\tilde{N}_1 + i \tilde{N}_2)/\sqrt{2}$ is the $SU(2)_L$
singlet field. The matrices ${\cal M}_{\pm}^2$ are defined as 
\be  
{\cal M}_{\pm}^2 =
\left( \begin{array}{cc}
m^2_{\tilde{L}} + \frac{1}{2}m^2_Z \cos 2 \beta + m_D^2 & 
m_D (A_{\nu} - \mu \cot \beta \pm M) \\
m_D (A_{\nu} - \mu \cot \beta \pm M) &
M^2 + m_D^2 + m_{\tilde{N}}^2 \pm 2 B_N M \end{array} \right) \; .
\ee
The parameters $m_{\tilde{N}}^2$ and $A_{\nu}$ are
contained in $V_{soft}$ in analogy to the charged sfermion sectors.
Furthermore $V_{soft}$ contains a term $M B_N \tilde{N} \tilde{N} + h.c.$
which violates $L$. The remaining entries of the mass matrix 
(\ref{massmatrix}) are the
$F$-terms stemming from the superpotential which in comparison to the 
MSSM contains an additional term  $M \hat{N} \hat{N}$. 
%(Check where the term $\sim m_Z^2 \cos 2 \beta$ comes from !!!)
The Dirac neutrino mass is $m_D=\lambda v_2$,
$\lambda$ being a Yukawa coupling, $v_i/\sqrt{2}$ is the vacuum 
expectation value of the neutral component of the $SU(2)$ doublet Higgs 
$H_i$, and $\tan \beta=v_2/v_1$. It is natural that $M \gg m_Z$
since $m_{\nu} \approx m_D^2/M$, 
$\mu,A_{\mu},m_{\tilde{L}} \sim {\cal O}(m_Z)$ since sfermions
with nontrivial $SU(2)_L \times U(1)_Y$ transformation properties
should not be much heavier than $100GeV$-$1TeV$, 
whereas $B_N,m_{\tilde{N}}$ may a priori be much bigger since they
pertain to the $SU(2)_L \times U(1)_Y$ singlet field $\tilde{N}$,
{\it cf.} the discussion in \cite{susyseesaw}. 
%The mass states for the Lagrangian (\ref{effective}) are simply
%$\tilde{\nu}_{1,2}$ and the mass difference is 
%$\Delta m^2 = 2 m^2_M$.
%masses 
%$m^2_{\pm}=m^2_{\tilde{L}} \pm m^2_M$ respectively, while for 

For the matrix (\ref{massmatrix}) the mass states read
\be \label{states}
\xi^l_{1,2} = \cos \Theta_{\pm} \tilde{\nu}_{1,2} + 
              \sin \Theta_{\pm} \tilde{N}_{1,2} \nn \\
\xi^h_{1,2} = - \sin \Theta_{\pm} \tilde{\nu}_{1,2} +
              \cos \Theta_{\pm} \tilde{N}_{1,2} 
\ee
where the angles $\Theta_{\pm}$ diagonalize ${\cal M}^2_{\pm}$
and the indices $l,h$ refer to light and heavy sneutrino mass states.
In leading order in $1/M$ the mass difference of the light sneutrino 
states is
\be \label{difference}
\Delta m^2 = m^2_{\xi_2^l} - m^2_{\xi_1^l} = 
4 \frac{m_D^2}{M} (A_{\nu} - \mu \cot \beta - B_N)
\ee
in accordance with \cite{susyseesaw}. 

In the following it will not be distinguished further between the 
models eq. (\ref{effective}) and eq. (\ref{massmatrix}) and the light
sneutrino mass states will be denoted by $\xi^l_{1,2}$. Note that
the mass difference $\Delta m$ of the light states is related to 
$\Delta m^2$ by 
\be \label{relation}
\Delta m=m_{\xi^l_2}-m_{\xi^l_1}=\Delta m^2/(m_{\xi^l_1}+m_{\xi^l_2}) \ .
\ee


\vspace{0.5cm}
\noindent
{\bf \hspace{0.2cm} III Baryon number depletion by sneutrinos below 
the critical temper-\-
\hspace*{1cm} ature}\\
%\section{Baryon number depletion by sneutrinos below the critical
%temperature}

In both scenarios eqs. (\ref{effective}),(\ref{massmatrix}) sneutrinos 
give rise to $L$-violating processes,
{\it e.g.} sneutrino decays or 2$\leftrightarrow$2 scatterings.
%In both scenarios sneutrinos give rise to $L$-violating processes, {\it
%e.g.} sneutrino oscillations, $\0n$ and 2$\leftrightarrow$2 
%$L$-violating scatterings \cite{phen,sndb,susyseesaw}. 
These scatterings have the potential to erase an asymmetry in the number 
of baryons and
antibaryons in the early universe. Such an asymmetry has to be generated
somewhen during the evolution of the universe in order to  
explain the absence of antimatter in the universe observed (for an 
overview see {\it e.g.} \cite{kotu}). In the following it is assumed
that at some high temperature above the electroweak symmetry breaking
scale the BAU has been generated by
some mechanism in the right amount. One example of such a mechanism is
the decay of heavy right-handed neutrinos and/or sneutrinos which 
incorporates the three basic conditions for the generation of the BAU:
baryon or lepton number violation, $CP$-violation and out of thermal 
equilibrium circumstances. All three conditions may be satisfied by 
the decay of
$SU(2)_L$ singlet sneutrinos, in particular the out of equilibrium condition
requires them to be very massive (${\cal O}(10^{10}GeV)$) in accordance
with the considerations mentioned above ($M \gg {\cal O}(m_Z)$).


However, at temperatures below the electroweak symmetry breaking scale 
$T \sim {\cal O}(100GeV)$ the light sneutrino states violate $L$ too 
and sneutrino interactions 
%mediate scatterings which 
may bring the distributions of leptons and antileptons
into equilibrium. An estimation of interactions induced by Majorana 
neutrinos below $T_C$ has been given in \cite{utpal}.
The approximate criterium for equilibration is that
the rate of the process in question should be bigger than the expansion
rate of the universe $H(T)=1.7 \sqrt{g_*}\;T^2/M_{Pl}$ (in a radiation 
dominated universe) where $g_*\approx 200$ in the MSSM and 
$M_{Pl} \approx 10^{19} GeV$. 


As long as sphaleron-mediated processes are operative the lepton and 
baryon numbers are both proportional to the combination $B-L$ \cite{sph}.
As a consequence the asymmetry in baryons vanishes if the asymmetry
in leptons is somehow erased. Hence, if there is a temperature range
between the critical temperature of the $SU(2)_L \times
U(1)_Y \ra U(1)_{em}$ phase transition and the temperature $T_{out}$
at which sphaleron-mediated processes drop out of equilibrium ({\it i.e.} 
$\Gamma_{Sph}(T_{out}) < H(T_{out})$ where $\Gamma_{Sph}(T)$ is the sphaleron 
rate) so that
\be \label{range}
T_{out} < T < T_C
\ee
then $L$-violating interactions are potentially dangerous for the BAU. 
The presence of a temperature range satisfying (\ref{range}) is not
possible in models where the BAU is generated during the electroweak
phase transition itself (Electroweak Baryogenesis) since in such models 
the sphaleron-mediated processes must be switched off immediately below
$T_C$. Therefore any constraints on the $L$-violating properties of light
sneutrinos do {\it a priori} not hold in the context of Electroweak 
Baryogenesis or any other model where sphaleron-mediated processes are out
of equilibrium immediately after the electroweak phase transition. 


In order to estimate the temperature range (\ref{range})
%where $T_C$ is the critical temperature of the $SU(2)_L \times
%U(1)_Y \ra U(1)_{em}$ phase transition and $T_{sp}$ is the temperature
%at which sphaleron mediated processes drop out of equilibrium 
the following assumptions are made: 
%at temperatures not far below $T_C$
the evolution of the vacuum expectation value ({\it vev}) of the Higgs 
fields $\langle v (T)\rangle$ 
%(check what happens when two Higgs fields are involved, we are working 
%in a SUSY context) 
is described by
\be \label{vacvalue}
\langle v (T) \rangle = \langle v (0) \rangle (1 - T^2/T_C^2)^{1/2}\;\;,
\langle v (0) \rangle = 246 GeV\;.
\ee
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%(Question: what can be said about this formula, when does it hold, %
%it obviously does not hold for strong first order transitions since%
%there it is required $\langle v (T_C) \rangle / T_C \geq 1$)       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This behaviour of the {\it vev} is valid for a second order phase 
transition and approximately valid for a weak first order phase 
transition. The exact behaviour of $\langle v (T) \rangle$  
depends on the SUSY parameters, but for our phenomenological
purposes we will satisfy ourselves with eq. (\ref{vacvalue}). Lattice
simulations suggest that $T_C \approx 150 GeV$ for a Higgs mass of
$70 GeV$ \cite{valuetc} and that it should be higher for larger values of the
Higgs mass, but in order to keep the results general we will vary $T_C$ 
freely between $50 GeV$ and $250GeV$. In models of electroweak
baryogenesis sphalerons have to drop out of equilibrium immediately
after the phase transition which translates into the condition 
\cite{sphrate} $\langle v (T=T_C) \rangle / T_C > 1$ in contrast to 
(\ref{vacvalue}), see the comment above. 
%(Include some comment on the validity of this approximation).
%Mention already here something about electroweak baryogenesis,
%especially the relation <v(T=T_C)>/T_C > 1 

The sphaleron rate below $T_C$
is described by (\cite{sphrate} and refs. therein)
\be \label{sphrate}
\Gamma_{Sph} \approx 2.8 \cdot 10^5 \; T^4 \; \kappa \; 
            \left(\frac{\alpha_W}{4 \pi}\right)^4
            \left(\frac{2 m_W (T)}{\alpha_W T}\right)^7
            \exp\left(-\frac{E_{sp}(T)}{T}\right)
\ee
where 
\be
m_W (T) = \frac{1}{2} g_2 \langle v (T) \rangle \ ,
\ee
the free energy of the sphaleron configuration is given by
\be
E_{Sph} (T) = \frac{2 m_W (T)}{\alpha_W} B \left(\frac{m_H}{m_W}\right),
\ee
%
% New New New: the numerical value of kappa is introduced
%
%
$B(0)=1.52,\; B(\infty)=2.72$ and $\kappa$=exp(-3.6) \cite{moore}.
In figure \ref{tempdiff} the temperature range $T_C-T_{out}$ is
plotted as a function of $T_C$. It becomes smaller than one
for small values of $T_C \lsim 100 GeV$ but is of order 
${\cal O}(10 GeV)$ for $T_C \sim 200 GeV$. 

\begin{figure}[t]
\vspace*{-1cm}
\hspace*{3cm}
\epsfxsize65mm
\epsfbox{toutfig.eps}
\vspace{-0.5cm}
\caption{The difference of the critical temperature $T_C$ and 
the sphaleron 
freezing out temperature $T_{out}$ determined by 
$\Gamma_{sph}(T_{out})=H(T_{out})$ in dependence of $T_C$.}
\label{tempdiff}
\end{figure}

For $L$-violating sneutrino decays and $L$-violating sneutrino 
mediated scatterings
the damping of the preexisting Lepton number (per comoving volume)
$L_i$ is described by the Boltzmann equation for the evolution
of $L$ and is given by (see {\it e.g.} \cite{kotu})
\be \label{evolution}
L(z)=L_i \exp \left[ 
      - \int\limits_{z_c}^{z_{out}} d z' z' 
                 [g_* \frac{n_{\tilde{\nu}}}{s} % \gamma_D 
                  \Gamma_D(z')+ n_{\gamma} 
                  \langle \sigma |v| \rangle)]/H(T=m_{\tilde{\nu}}) \right]
\ee
where $z=m_{\tilde{\nu}}/T$, $\Gamma_D$ is the $L$-violating sneutrino 
decay width, $\langle \sigma |v| \rangle$ is the thermally averaged
cross-section of sneutrino mediated $L$-violating scatterings and $n_i$
is the number density of particle $i$ (in the scattering term the
number density of the initial state particles has been assumed to be the
photon number density). In order to derive bounds
on $L$ violation we will allow that during
the epoch $T_{out}<T<T_C$ $L$ may be depleted by a factor $k < 1$,
that is we assume the preliminary $L$ density to be 
$n_L/n_{\gamma}\approx 10^{-10}/k$. 
%The resulting constraints depend only very weakly on the assumption
%on the initial asymmetry and hence on $k$, $\Delta m \sim (ln k)^{1/4}$.
%
%In order to proceed it is convenient to distinguish between different
%mass ratios of the SUSY particles. First consider the case that
%all SUSY particles but one or several gauginos have decayed away
%(conserving $L$) at temperatures $T>T_C$. 

%\section{Constraints on the sneutrino mass-splitting}

\vspace{0.5cm}
\noindent
{\bf IV \hspace{0.2cm} Constraints on the sneutrino mass-splitting}\\

In the following two different cases will be distinguished: 
{\bf I)} 
%one or more gauginos are lighter than $T_C$, that is those or still
%present in the plasma at temperatures $T < T_C$; 
at least one gaugino (see below) or slepton other than the sneutrino
are lighter than $T_C$, that is the number density of particles 
satisfying this condition is approximately the photon number density;
{\bf II)} all gauginos and sleptons besides the sneutrino are 
(substantially) heavier than $T_C$ so that the number density of these
particles is exponentially suppressed and one sneutrino is the lightest 
supersymmetric particle (LSP). The discussion will be focused on the
case of sneutrino CDM, that is on a mass difference of order $5GeV$
and a sneutrino LSP mass of $m_{LSP}\sim70 GeV$ \cite{cdm}.  

%\noindent
{\bf I)} In order to keep the results as general as possible only 
scatterings will be taken into account in what follows since these
depend to a smaller extent on the exact mass relations of the
particles involved than $L$-violating decays do. Then relevant processes
capable of depleting $L$ are $2 \leftrightarrow 2$ scatterings as for 
example scattering of neutralinos or charginos into leptons mediated
by $L$-violating sneutrinos (see figure \ref{annihilation}) 
\be \label{reaction}
\chi_i \chi_j \longleftrightarrow l_l l_m \ \ , \ \ 
\chi_i l_l \longleftrightarrow \ovl{\chi}_j \ovl{l}_m 
\ee
where $\chi_k$ represents any of the chargino or neutralino
mass fields present in the plasma at $T<T_C$ 
and $l_n$ is a charged or uncharged lepton. In the one generation case 
$l$=$m$. However, since the
entries of the neutralino mixing matrix connecting gauginos with
higgsinos are proportional to the $vev$ this mixing will be 
suppressed by $\langle v(T=T_{out})\rangle/\langle v(T=0)\rangle$ 
and therefore $\chi$ will
be assumed to be $\tilde{B}$ with associated mass $M_1$ and
$\tilde{W}^3,\tilde{W}^{\pm}$ with associated mass $M_2$.

The following discussion will be restricted to the process eq.
(\ref{reaction}), but for other examples as 
$\tilde{l}^{\pm}\tilde{l}^{\pm} \longleftrightarrow W^{\pm} W^{\pm}$ 
the same arguments hold.
%Since it has been assumed that at least one of the 
%gauginos is still present in the plasma at temperatures below $T_C$
The constraint on the mass-splitting follows from the relation
\be \label{boundeq}
\int\limits_{z_C}^{z_{out}} d z' z' \frac{1}{H(m_{\tilde{\nu}})}
                  n_{\gamma} \langle \sigma |v| \rangle > \ln k \ .
\ee
\begin{figure}[t]
\hspace*{0.3cm}
\epsfxsize130mm
\epsfbox{channels.eps}
\caption{Graphs for $2\leftrightarrow2$ scatterings of gauginos 
$\chi_i$ and leptons $l$
mediated by the light sneutrino states $\xi^l_{1,2}$.} 
\label{annihilation}
\end{figure}
The zero-temperature cross-section of
this reaction is proportional to $(\Delta m^2)^2$ since \cite{theorem}
\be \label{propagator}
\langle 0|T[\tilde{\nu}(x) \tilde{\nu}(y)]|0 \rangle &=&
\Delta m^2 \frac{i}{2} \int \frac{d^4 k}{(2 \pi)^4} 
\frac{e^{-ik(x-y)}}
     {(m^2_{\xi^l_1} - k^2)(m^2_{\xi^l_2} - k^2) + i \epsilon}
\ee
where for model eq. (\ref{massmatrix}) the contributions of the heavy
states have been neglected. Therefore
the final bound on $\Delta m$ depends only weakly on the precise
value of $\langle \sigma |v| \rangle$ (and on $k$) and for our 
purposes it is sufficient to approximate the thermally averaged 
scattering cross section by
\be \label{twotwocross}
\langle \sigma |v| \rangle \approx 
(\Delta m^2(T))^2 \frac{\alpha_W^2 T^2}{(T^2+m_{\tilde{\nu}}^2)^4} \ \ ,
\ee
where $\alpha_W$ is the weak coupling constant and for non-vanishing
temperatures and a second order phase transition (see eq. \ref{relation})
\be
%\eta^2 (T_{out})=
%=(1-\frac{T_{out}^2}{T_C^2})= 
\frac{\Delta m (T)}{\Delta m (T=0)}= 
\frac{\Delta m^2 (T)}{\Delta m^2 (T=0)}=
\frac{\langle v (T)\rangle^2}{\langle v (T=0) \rangle^2}=
\left( 1 - \frac{T^2}{T_C^2} \right) \ .
\ee
%
%However, it has
%been suggested in \cite{cdm} that the lightest sneutrino mass state
%could be the cold component of the dark matter in the universe under
%the condition, that the mass-splitting of the lighter sneutrino mass
%states is not less than around $10GeV$. \\
%
The full zero temperature cross section can be found in \cite{phen}. 
%and 
%depends on the gaugino mixing 
%matrices which themselves vary considerably with $\langle v(T) \rangle$ 
%since some of the off diagonal entries in the gaugino mass 
%matrices are proportional to the gauge boson masses, but we will
%assume that the relevant mixing matrix element is not much smaller
%than one.

\begin{figure}[t]
\hspace*{3mm}
\epsfxsize130mm
\epsfbox{ressplit.eps}
\vspace{-0.5cm}
\caption{The limits on the zero temperature light sneutrino mass-splitting
splitting $\Delta m (T=0)$ in case {\bf I)} (see text) in dependence of 
the critical temperature $T_C$ from eq. (\ref{boundeq}) (the regions 
above the curves are excluded) 
for $k=0.3$ (left) and $k=0.01$ (right) and for different values of 
the sneutrino mass: $m_{\xi^l}=70 GeV$ (solid curve),
$m_{\xi^l}=150 GeV$ (dotted curve), $m_{\xi^l}=300 GeV$
(short-dashed curve), $m_{\xi^l}=500 GeV$ (long-dashed
curve), $m_{\xi^l}=1TeV$ (dot-dashed curve).} 
%For the case $m_{\tilde{\nu}}=70 GeV$ the limits
%become weaker for higher values of $T_C$. 
%The results stemming from
%eq. (\ref{boundeq}) have been scaled by a factor 
%$\langle v (T=0) \rangle^2 / \langle v (T=T_{out}) \rangle^2$ in order
%to compare the results with low energy constraints on $\Delta m$.}
\label{figresult}
\end{figure} 


%In order to compare the limits from baryogenesis with the low-energy 
%constraints one has to take into account the suppression due to the 
%non-vanishing ratio $T_{out}/T_C$.
%since $\Delta m^2 \sim \langle v (T) \rangle^2$. 
%Since%
%
%the bounds originating from eq. (\ref{boundeq}) have to be scaled
%up by a factor $\langle v (T=0) \rangle^2/\langle v (T=T_{out})\rangle^2$.
%it is possible to compare the bound from baryogenesis {\it e.g.} with 
%the bounds on $\Delta m^2$ which can be derived
%from the upper limits on neutrino masses. 
%Taking $\eta \approx 0.2$
%the constraint (\ref{finresult}) becomes for $\sigma^0=0.1pb$
%$(10pb,10^{-3}pb)$
%\be \label{finresult0}
%\Delta m^2 (T=0)_{baryo} < 2.5 GeV^2 \; (0.25 GeV^2,\; 25GeV^2)
%\ee
%Bigger values of $\kappa$ increase $\eta$ and result in slightly more 
%stringent bounds on $\Delta m^2$. \\
The results, which hold for any generation, are plotted in figure 
\ref{figresult} for several values for the sneutrino mass and $k$
in dependence of $T_C$ which has been varied freely between $50GeV$
and $250GeV$. For example for $T_C \approx 150 GeV$ the limits range 
from $\Delta m \lsim {\cal O}(few \ 100 MeV)$ to 
$\Delta m \lsim {\cal O}(few \ GeV)$ for the chosen sneutrino masses.
The limit on the mass-splitting becomes less stringent
for smaller values of $T_C$ (since the temperature range 
(\ref{range}) becomes smaller) and for larger values of 
$m_{\tilde{\nu}}$. For small values of $T_C$ and large values of 
$m_{\tilde{\nu}}$ the mass-splitting is basically not constrained.
%Furthermore, for too small values of $T_C$ it is likely that all
%superpartners (but the LSP in the case of conserved $R$-parity) have 
%already decayed away so that light sneutrinos do not have the potential 
%to erase $n_B$. 

{\bf II)} In the case that all gauginos and sleptons but the
sneutrinos have already decayed away at temperatures above $T_C$
the processes depleting $L$ are $2\leftrightarrow2$ scatterings 
of sneutrinos into neutrinos and $L$-violating decays of non-LSP 
sneutrinos into the LSP sneutrino $\xi^h \ra \xi^{LSP}\nu\nu$ mediated
by neutralinos. 

The cross-section for gaugino-mediated $L$-violating scattering of 
sneutrinos into neutrinos can be approximated by (compare with 
\cite{cdm})
\be \label{sneuinitial}
\langle \sigma |v| \rangle \approx 
\frac{(\Delta m^2 (T))^2}{T^4} 
\left( \frac{\alpha_Y M_1}{T^2+M_1^2} +
       \frac{\alpha_W M_2}{T^2+M_2^2} \right)^2
\ee
where $\alpha_Y$ is the coupling associated with $U(1)_Y$.  
%Only pair annihilation will be considered in what follows, since
This cross-section depends sensitively on the relation $M_2/M_1$.
In figure \ref{ressneuinitial} the bounds on the $\Delta m$ are
shown for several values of $M_1$ and for the cases of $M_2/M_1$
considered in \cite{cdm,newpap} which have been shown to be compatible
with the sneutrino CDM hypothesis as long as $M_1 \gsim 200 GeV$
(see next section).

\begin{figure}[t]
\epsfxsize130mm
\hspace*{0.3cm}
\epsfbox{sneuinitial.eps}
\vspace{-0.8cm}
\caption{The limits on $\Delta m(T=0)$ from eqs. 
(\ref{sneuinitial}),(\ref{boundeq}) in case {\bf II)} (see text)
for several values of the
ratio $M_2/M_1$ in dependence on the critical temperature and
for a depletion factor $k=1/3$. The regions above the lines are 
excluded. The limits correspond to 
the parameters $M_1=300 GeV$ (solid line), $M_1=500 GeV$ (dotted line)
and $M_1=1 TeV$ (dashed line).}
\label{ressneuinitial}
\end{figure}

The resulting limits are more stringent than the bounds in {\bf I)}
if $M_1$ and $M_2$ have the same sign. This is simply due to the
propagator structure of the two different scattering processes. 
Even for very large gaugino masses of order ${\cal O}(TeV)$ the
mass-splitting $\Delta m (T=0)$ is only of order ${\cal O}(few \ GeV)$
for plausible values of $T_C$. This statement is also valid in the limit
$M_2 \ra 0$. On the other hand, if $M_1$ and 
$M_2$ are of opposite sign there is a partial cancellation of the
contributions and the resulting limits on $\Delta m (T=0)$ become
weaker. In particular for $T_C \approx 200 GeV$ for 
$M_2/M_1=-3\alpha_W/5\alpha_Y$ ($-\alpha_W/\alpha_Y$) a mass difference
$\Delta m \sim {\cal O}(few \ GeV)$ is allowed if $M_1 \approx 500 GeV$
($300 GeV$).

%Substituting the sneutrino mass by the gaugino mass the
%$L$-violating scattering cross-section can be approximated by eq. 
%(\ref{twotwocross}) and the limits on the mass-splitting $\Delta
%m(T=0)$ are the same as the ones depicted in figure \ref{figresult}. 
The decays of non-LSP sneutrinos into the LSP sneutrino have been
discussed in detail in \cite{newpap}. Neglecting gaugino mixing
the zero-temperature decay width of the $L$-violating mode to 
leading order in the parameter $\Delta m^2$ is
\be
\Gamma_{\Lbar}\approx \frac{\alpha_W^2}{16 \pi}
    \frac{(\Delta m^2(T))^5}{(m_{\chi}^2-m_+^2)^4 \sqrt{m_+^2}} 
    \sim (\Delta m^2 (T=0))^5\left(1-\frac{T^2}{T_C^2}\right)^5
\ee   
where $m_+^2=m_{\xi^{LSP}}^2+m_{\xi^h}^2$. This expression does not
result into a constraint when inserted into eq. (\ref{evolution}) due
to its dependence on the fifth power of the suppression factor 
$(1-T^2/T_C^2)$: {\it e.g.} for $T_C \lsim 250 GeV$ the constraint
on the mass-splitting is $\Delta m \lsim {\cal O}(100 GeV)$ and for
smaller values of $T_C$ becomes even less stringent. 

%\section{Comparison with low-energy constraints and implications for
%sneutrino Cold Dark Matter}

\vspace{0.5cm}
\noindent
{\bf V \hspace{0.2cm} Comparison with low-energy constraints and 
implications for sneu-\-
\hspace*{0.8cm} trino Cold Dark Matter}\\ 


In \cite{theorem,susyseesaw}
it has been shown that $L$ violating sneutrinos give rise to loops
containing neutralinos and (light) sneutrinos which contribute to the 
(Majorana) mass of the neutrino. In \cite{sndb} a scan over a wide range
of the SUSY parameter has been carried out and ``average'' upper
limits 
\be \label{neutrinobound}
\Delta m (e) < 8 MeV \ \ , \ \ 
\Delta m (\mu) < 6 GeV
%\Delta m^2 (\tau) < 4 \cdot 10^4 GeV^2.
\ee
have been deduced (for neutrino mass limits $m(e)<15eV$ and 
$m(\mu)<170keV)$ while the third generation mass-splitting remains 
virtually unconstrained. The current bound on the effective Majorana 
neutrino mass $\langle m_{ee} \rangle = \sum_i' U_{ei}^2 m_i < 0.36 eV$
(90$\%$ $c$.$l$.) from the 
Heidelberg-Moscow experiment \cite{0nexp} yields in the case
$U_{ei} \approx \delta_{ei}$ the relation 
$\Delta m (e) < 13  keV$ for an average sneutrino mass of $100 GeV$. 
Direct contributions to $\0n$ have been considered in \cite{sndb} and 
the resulting limits on $\Delta m (e)$ are less stringent. 
Therefore the limits from baryogenesis are considerably less severe
than the ones from neutrino masses in the first generation case for
plausible ranges of $T_C$ and superpartner masses. 
For the second generation the limit is comparable or slightly 
more stringent for values of $T_C$ not much smaller than $150GeV$
and superpartner masses not bigger than $500 GeV$.
In the third generation case for plausible values of
$T_C$ baryogenesis yields more stringent limits than neutrino
masses do. 
%For the model eq. (\ref{massmatrix}) the result shown in 
%figure \ref{figresult} corresponds to the case that 
%$B_N \gg m_Z$. For $B_N \approx m_Z$ the mass-splitting $\Delta m^2$
%is about a factor $10^{-3}$ smaller, {\it c.f.} the discussion in 
%\cite{susyseesaw}.
% While the bounds (\ref{neutrinobound}) are
%compatible with a substantial mass-splitting of the light sneutrino mass
%states in the second and third generation case the generic bound 
%(\ref{finresult0}) rules out any sizable light sneutrino mass-splitting.

The constraints on the sneutrino mass-splitting are especially
interesting for the hypothesis of sneutrino CDM. In \cite{cdm} it
has been shown that a sneutrino with mass of around $70 GeV$ may
be a viable CDM candidate if the mass-splitting exceeds about
$5GeV$ (so that $Z^0$-mediated $s$-channel coannihilation of the
LSP sneutrino with its heavier partner is sufficiently suppressed)
and $M_1$ is bigger than about $200 GeV$ (so that $L$-conserving
LSP-sneutrino pair annihilation is sufficiently suppressed). Therefore the 
low energy limits eq. (\ref{neutrinobound}) imply that the light 
third generation sneutrino is a viable CDM candidate. 
On the other hand, in case {\bf I)} discussed in the last chapter 
the results displayed in figure \ref{figresult} (solid line) imply 
that sneutrino CDM is firmly ruled out for plausible values of the
critical temperature $T_C \gsim 150 GeV$. 

The same conclusion is valid in scenario {\bf II)} for plausible
values of $T_C$ if $M_1$ and $M_2$ bear the same sign as long as
the gauginos are not exceptionally heavy ($M_1 \gsim 1TeV$). 
However, in case {\bf II)} the bound on the sneutrino mass-splitting 
could become sufficiently large to allow for sneutrino CDM if $M_1$ and
$M_2$ are of opposite sign. For plausible values of $T_C \gsim 150 GeV$
the ratio $M_2/M_1=-3\alpha_W/5\alpha_Y$ ($M_2/M_1=-\alpha_W/\alpha_Y$) allows
for a sneutrino mass-splitting being large enough to be compatible with
the sneutrino CDM hypothesis as long as $M_1$ is bigger than about 
$500 GeV$ ($300 GeV$).
%CDM as long as $M_{1,2}\gsim 500 GeV$ for $T_C \sim 150 GeV$ and
%correspondingly larger for increasing values of $T_C$. 
In particular, this statement is compatible with the lower limit 
on $M_1$ from the requirement that the sneutrino relic abundance 
should be sufficiently high to account for the CDM. 
%However, in view of figure \ref{figresult} (solid line) this
%possibility is definitely ruled out for plausible values of the
%critical temperature $T_C \approx 150 GeV$ if the mass of at least
%one of the gauginos or charged sleptons is not much bigger than
%$T_C$. Assuming
     


%Furthermore, $L$-violating transitions of the type 
%$\tilde{\nu}^{non-LSP}\ra \tilde{\nu}^{LSP} + \ $leptons which have been
%considered in \cite{newpap,cdm} have the potential to erase $B$ as well, 
%even in the case that all gauginos have already decayed away (conserving $L$) 
%before the phase transition took place. In the latter case 
%the constraints in figure \ref{figresult} describing the contribution
%of scatterings only of course do not apply. The question under which
%circumstances such decays are compatible with the sneutrino CDM hypothesis 
%will be discussed elsewhere. 
%
%However, the impact of generational mixing possibly softens the
%limits on the mass-splitting. In the three generation framework
%the mass matrix (\ref{massmatrix}) is a $12 \times 12$ matrix. 
%To illustrate the effect of generational mixing consider the
%gaugino-lepton-slepton vertex
%$V_{i1}^* \ovl{\chi}_i l_{L,j}^{weak} \tilde{\nu}_{L,j}^{weak*}$. 
%The transformation into the mass basis can be expressed 
%in terms of unitary rotations ($w$ ``weak'', $m$ ``mass'') 
%\be \label{rotation}
%l_{L,i}^w = U^{l_L}_{ij} l_{L,j}^m,\;
%\tilde{\nu}_{L,i}^w = U^{\tilde{\nu}}_{ij} \tilde{\nu}_j^m, \;%\nn \\ 
%M_{kl} = U^{l_L}_{jk} U^{\tilde{\nu}*}_{jl},\; %\nn \\
%V_{i1}^* \ovl{\chi}_i l_{L,j}^w \tilde{\nu}_{L,j}^{w *} \ra
%V_{i1}^* M_{kl} \ovl{\chi}_i l_{L,k}^m \tilde{\nu}_l^m. \nn
%\ee
%The constraints from the neutrino masses now read
%\be
%\sum_k M^2_{ik} \Delta m^2 (k) < 0.03 GeV^2,\; 
%340 GeV^2,\; 4 \cdot 10^4 GeV^2\nn
%\ee
%for $i=1,2,3$ respectively and 
%similarly for (\ref{finresult}). The dependence on the rotation 
%matrices possibly renders the limits on the mass-splitting less  
%stringent.\\
%However, since the rotation matrices are unknown the values
%(1) will be used in the following for the sake of definiteness, but
%one should keep in mind that they might be less stringent.
%
%As a consequence $L$ violating processes 
%will not be detectable at present or future collider
%facilities as such processes require a substantially higher mass
%splitting.
%For example, the process $l^{\pm} l^{\pm} \ra \chi^{\pm} \chi^{\pm}$,
%where $l$ is an electron or a muon, to be detectable at the NLC or a
%muon-collider requires $\Delta m^2$ to be not much smaller 
%than $200 GeV^2$ for $\sigma^0=1pb$ \cite{phen}. On the other hand
%for $\Delta m^2 = 25 GeV^2$ sneutrino oscillations could still be
%observable, while for $\Delta m^2$ substantially lower the total
%decay width of the sneutrino states has to be exceptionally small
%to yield a observable signal, cf. the discussion in 
%\cite{phen,susyseesaw}. 
%
%The lightest sneutrino being a candidate for CDM requires a mass 
%splitting not much smaller than $\Delta m \approx 5GeV$ \cite{cdm},
%so that the constraints (\ref{neutrinobound}) allow that the 
%lighter of the third generation sneutrinos accounts for the CDM
%(and in this case is the lightest SUSY particle). However,
%according to $\Delta m^2=2 m \Delta m$ (m being the average of the
%light sneutrino masses) the constraint $\Delta m^2 < 25GeV^2$ restricts 
%$\Delta m$ to be less than about $0.1GeV$ for $m\approx 100GeV$. 
%Therefore sneutrino CDM is excluded not only in the case that the 
%interaction states are the mass states, but also in the case that the 
%sneutrino sector exhibits $L$ violating properties.\\
%
%n particular, in view of this bound the lightest sneutrino mass state 
%as to be discarded as the dominant component of the dark matter
%n the universe since in this case the mass-splitting has to be of
%order $\Delta m^2 \approx {\cal O}(10^2 GeV^2)$, {\it c.f.} the
%discussion in \cite{cdm}. \\

%\section{Summary}

%\newpage

\vspace{0.5cm}
\noindent
{\bf VI \hspace{0.2cm} Summary}\\

In conclusion, in scenarios with $L$ violation in the light sneutrino 
sector it has to be made sure that sneutrino-induced interactions which
take place in the early universe at temperatures below the electroweak 
symmetry breaking scale do not erase the BAU generated at some higher 
temperature. This may happen if the electroweak phase transition is
second or weakly first order, so that sphaleron-induced processes are
still operative at temperatures below the critical temperature $T_C$.
The $L$-violating interactions considered here result in  
constraints on the mass-splitting of the light sneutrino mass states 
which are less stringent than the ones which can be derived from the 
contribution of $L$-violating sneutrinos to neutrino masses for the
first generation. For the second generation the limits are more
stringent for plausible values of $T_C \gsim 150 GeV$ and for 
superpartner masses smaller than $500 GeV$ except for the case of
an almost complete cancellation of the bino and wino contributions
for a sneutrino LSP of mass ${\cal O}(100GeV)$. 
%comparableor more stringent for the 
%second generation but more stringent than the constraints from third 
%generation neutrino masses. 

In particular, the constraint on the third generation mass-splitting 
is of interest for the sneutrino Cold Dark Matter hypothesis. Low energy
limits on the
sneutrino mass-splitting suggest that the lighter of the third
generation sneutrino states with mass of around $70 GeV$ may serve 
as a CDM candidate if $M_1 \gsim 200 GeV$. This hypothesis is not
compatible with the observed Baryon asymmetry if the gauginos or further
sleptons other than the sneutrino are still present in the plasma
at temperatures below $T_C$ or if the masses associated to the bino
and wino bear the same sign. On the other hand, a scenario with
opposite sign gaugino mass parameters and both gauginos sufficiently 
heavy $M_1,M_2 \gsim 500 GeV$ (and slepton masses
bigger than $T_C$) is compatible with the sneutrino CDM hypothesis.\\   

%In particular the smallness of the third generation 
%sneutrino mass-splitting seems to firmly exclude the lightest
%sneutrino accounting for the dominant part of the Cold Dark Matter 
%in the universe except for the case that all gauginos decayed away
%at temperatures above the electroweak phase transition. 
%\\

\centerline{\bf Acknowledgments}

\vspace{0.5cm}
We thank D. Semikoz and U. Sarkar for helpful discussions. St. K. would 
like to thank the Institute for Nuclear Research in Moscow for hospitality
and the Deutscher Akademischer Austauschdienst for financial support. 




\begin{thebibliography}{99}

\bibitem{kamiokande} Y. Fukuda {\it et al.}, the Super-Kamiokande
                     Collaboration, Phys. Lett. B 467 (1999) 185.

\bibitem{oscover} S.M. Bilenky, C. Giunti and W. Grimus, Prog. Part.
                  Nucl. Phys. 43 (1999) 1 and .

\bibitem{klaptrento} H.V. Klapdor-Kleingrothaus in Proc. Int. Conf. on
                     Lepton- and Baryon-Number Non-Conservation, Trento, 
                     Italy, April 20-25, 1998, IOP, Bristol (1999);
                     H.V. Klapdor-Kleingrothaus, Int. J. Mod. Phys.
                     A 13 (1998) 3953.

\bibitem{theorem} M. Hirsch, H.V. Klapdor-Kleingrothaus and S. Kovalenko,
                  Phys. Lett. B 398 (1997) 311.

\bibitem{haberkane} H.E. Haber and G.L. Kane, Phys. Rep. 117 (1985) 76.

\bibitem{susyseesaw} Y. Grossmann and H.E. Haber, Phys. Rev. Lett. 78
                     (1997) 3438.

\bibitem{baugen} B.A. Campbell, S. Davidson and K.A. Olive, Nucl. Phys.
                 B 399 (1993) 111; 
                 M. Pl\"umacher, Nucl. Phys. B 530 (1998) 207;
                 L. Covi, E. Roulet and F. Vissani, 
                 Phys. Lett. B 384 (1996) 169.

\bibitem{phen} M. Hirsch, H.V. Klapdor-Kleingrothaus, St. Kolb and 
               S. Kovalenko, Phys. Rev. D 57 (1998) 2020.

\bibitem{sndb} M. Hirsch, H.V. Klapdor-Kleingrothaus and S. Kovalenko,
               Phys. Rev. D 57 (1998) 1947.

\bibitem{cdm} L.J. Hall, T. Moroi and H. Murayama, Phys. Lett. B 424
              (1998) 305. 

\bibitem{sph} V.A. Kuzmin, V.A. Rubakov and M. Shaposhnikov, 
              Phys. Lett. B 155 (1985) 36.

\bibitem{kotu} E.W. Kolb and M.S. Turner, The Early Universe
               (Addison-Wesley, Redwood City, CA, 1990).

\bibitem{utpal} U. Sarkar, Phys. Lett. B 390 (1997) 97.

\bibitem{valuetc} K. Kajantie, M. Laine, K. Rummukainen and 
                  M. Shaposhnikov, Nucl. Phys. B 466 (1996) 189.

\bibitem{sphrate} V.A. Rubakov and M.E. Shaposhnikov,
                  Usp. Fiz. Nauk 166 (1996) 493, 
                  Phys. Usp. 39 (1996) 461; 
                  A. Riotto, 

\bibitem{moore} G.D. Moore, Phys. Rev. D 59 (1999) 014503.

\bibitem{0nexp} L. Baudis {\it et al.}, Heidelberg-Moscow Collab.,
                Phys. Rev. Lett. 83 (1999) 41.

\bibitem{newpap} St. Kolb, M. Hirsch, H.V. Klapdor-Kleingrothaus
                 and O. Panella, Phys. Lett. B 478 (2000) 262.

\end{thebibliography}    

\end{document} 


