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KANAZAWA-02-14  \\
July, 2002
\end{flushright}
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\begin{center}
{\LARGE\bf $\mu$-term as the origin of baryon and lepton number asymmetry}\\
\vspace{1 cm}
{\Large  Daijiro Suematsu}
\footnote[1]{e-mail:suematsu@hep.s.kanazawa-u.ac.jp}
\vspace {0.7cm}\\
{\it Institute for Theoretical Physics, Kanazawa University,\\
        Kanazawa 920-1192, JAPAN}
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{\Large\bf Abstract}\\
%%%%%abstract
We study a possibility to combine an origin of the $\mu$-term and
the baryon and lepton number asymmetry. If we assume that the 
$\mu$-term is generated by
a vacuum expectation value of a singlet scalar field of the 
standard model gauge group at an 
intermediate scale, the decay of its oscillation caused by the 
deviation from the true vacuum can produce the Higgsino number asymmetry. 
This asymmetry can be distributed into the lepton and
baryon number asymmetry as far as it is produced at a scale where the
electroweak sphaleron interaction is in the thermal equilibrium.
Moreover, since the baryon number asymmetry is considered to be 
originated from the Higgsino number asymmetry, this scenario may 
explain such an important
feature of our universe that both the energy densities of the baryon and
the dark matter is the similar order. 
We also examine the compatibility between this scenario and the small
neutrino mass generation based on the bilinear $R$-parity violating
terms.
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%%%%%%%%%%%%%%%%%%%%text
In the present astroparticle physics it is one of the crucial problems
to clarify an origin of baryon number $(B)$ asymmetry in the 
universe \cite{bna}.
It has been suggested that the electroweak sphaleron in the standard 
model (SM) plays a very important role in that investigation
\cite{spha}. The baryon number asymmetry generated at high energy scales 
can be washed out by this effect and we need a suitable scenario to
escape this problem. 
Leptogenesis seems to be able to give an elegant scenario for the 
explanation of the baryon number asymmetry on the basis of the lepton number
$(L)$ asymmetry \cite{lept}.
Since the sphaleron interaction conserves $B-L$, a part of 
the lepton number asymmetry
can be converted into the baryon number asymmetry through that
interaction as far as $B-L\not= 0$.
Recent observations of the solar and atmospheric neutrinos suggest 
the existence of small neutrino masses and
this fact seems to indicate the existence of a lepton number violation 
at a certain energy scale. From this point of view the leptogenesis 
scenario seems to be very promising 
as an origin of the baryon number asymmetry.

Several leptogenesis scenarios have been proposed under the assumption
on the lepton number violating schemes \cite{lept,lept1}.
In the supersymmetric model one may consider
the lepton number violation by some condensates along a $D$-flat direction.
As such scenarios there are, for example, 
the decay of the Affleck-Dine condensate \cite{ad} 
in the $LH_2$ direction \cite{ad1} and
also the decay of the right-handed sneutrino condensate in the chaotic
inflation scenario \cite{sneu,sneu1}.
As an extension of the leptogenesis scenario, we may consider a
possibility in which an additional particle asymmetry is converted into
the baryon and lepton number asymmetry through the sphaleron interaction 
if there are extra global U(1) symmetries with the weak SU(2) anomaly
\cite{dark,iq,dr}.
In this paper we would like to propose such an example which can be 
related to an origin of the  $\mu$-term $\mu H_1H_2$.\footnote{In the following
argument we use the same notation for ordinary fields in the SM as the
one of chiral superfields and put a tilde on the character of the chiral
superfields for their superpartner fields.}
 
In the minimal supersymmetric standard model (MSSM)
the origin of a scale of the $\mu$-term is not known.
Since it is a supersymmetric mass term, we have no reason why it should
take a value of a weak scale $M_W$. 
To fix its scale at the weak scale is called the $\mu$-problem \cite{mu,mu1}.
Usually the $\mu$-term is considered to be spontaneously generated
through, for example,
a vacuum expectation value (VEV) of some SM singlet scalar field $S$
as $\mu=\lambda\langle S\rangle$. In such a case,
it is well-known that there is the Peccei-Quinn symmetry as an additonal 
global symmetry.
If this singlet field has an almost flat direction and 
stores a large oscillation energy due to the deviation from its true
vacuum value at a suitable
period of the expansion of the universe, the decay of this oscillation 
may induce the asymmetry
between the Higgsino $\tilde H_{1,2} $ number density and 
the anti-Higgsino $\tilde H_{1,2}^c$ number density through the coupling
$S\tilde H_1\tilde H_2$.

In the usual situation we cannot expect that this asymmetry
is related to the lepton and baryon number asymmetry.
However, if the electroweak sphaleron interaction is in the thermal
equilibrium and $\lambda SH_1H_2$ interaction is out-of-equilibrium, 
the Higgsino asymmetry may be related to both the lepton and
baryon number asymmetry.
Since the Higgsinos are the SU(2) doublets, they can
participate in the electroweak sphaleron interaction as far as they do
not decouple from it. This interaction can relate the Higgsino asymmetry 
to the lepton and baryon number asymmetry.\footnote{If we consider the
situation that the Higgsinos decouple from the electroweak sphaleron,
we need some interaction to connect the Higgsinos with the leptons in
order to transfer the Higgsino number asymmetry into the lepton number
asymmetry. }
The typical feature of this scenario is that the reheating 
temperature due to the thermalization of
the decay products from the oscillation of the condensate of $S$ is not 
so high compared with the weak scale $M_W$.
This is a result from the feature that the true VEV of $S$ is constrained
by the scale of $\mu$. 
So we need to study a viability of this scenario under such a
constraint.

There seems to be several interesting subjects related to this scenario.
One example is the explanation of the 
predicted ratio of the baryon energy density and the cold dark matter
energy density. 
This scenario may give a nice explanation for this problem if a
Higgsino is the lightest superparticle as discussed later.
The small mass generation for the neutrinos
may also have an intimate connection to this scenario, 
although the situation for this problem seems to be completely 
dependent on the model.
For example, one may consider a scenario based on the bilinear $R$-parity
violating terms $\epsilon_\alpha L_\alpha H_2$.
Various works suggest that the bilinear $R$-parity violating terms can
be successfully related to the neutrino masses \cite{rparity,hdprv,rparity1}.
However, if there are the bilinear $R$-parity violating terms and
Higgsino-neutrino mixing induced by these terms is in the thermal equilibrium, 
the produced lepton and baryon number asymmetry may be washed out.
Thus, it seems to be worthy to study what kind of neutrino mass generation
mechanisms are compatible with this scenario.

We consider a simplified model to clarify an essence of the scenario.
The model is defined by the superpotential
\begin{equation}
W_1=\lambda SH_1H_2 +{c_n\over M_{\rm pl}^{2n-3}}(S\bar S)^n,
\label{eqa}
\end{equation}
where $S$ and $\bar S$ are the SM singlet chiral superfields and 
$H_{1,2}$ are the ordinary doublet Higgs chiral superfields, respectively.
The Yukawa coupling $\lambda$ is, in general, complex when the VEV
of $S$ is taken to be real. 
We assume that $S$ and $\bar S$ are massless and then $n\ge 2$.
Here we also assume that there is an extra U(1) gauge symmetry 
in addition to the MSSM gauge structure and
$S$, $\bar S$ have an opposite charge of this symmetry each other.\footnote{An 
introduction of the extra U(1) gauge symmetry is also favored from a
view point to escape the tadpole problem and the domain wall problem,
which are expected to appear often in the models extended only by the
singlet chiral superfield \cite{rad}.}
Under this assumption there is a $D$-flat direction
$\vert S\vert=\vert\bar S\vert\equiv u$.
This direction is slightly lifted by the higher order term in the scalar 
potential coming
from the last term of $W_1$. Thus we expect $u=0$ before taking account of 
the supersymmetry breaking effect.
By introducing the soft supersymmetry breaking 
scalar mass terms $m_S^2\vert S\vert^2$ and $m_{\bar S}^2\vert\bar S\vert^2$
which can be expected to be negative by various
reasons \cite{rad,rad1},
a true vacuum value of $u$ is determined either by the pure radiative
symmetry breaking effect or by the effect of the nonrenormalizable term 
in eq.~(\ref{eqa}). Its value $u_0$ is determined by using
$m^2(\Lambda=u_0)=-{1\over 2}\beta_{m^2}$ where 
$\beta_{m^2}=\Lambda{dm^2\over d\Lambda}$ in the former case and 
is also roughly estimated as $u_0\sim (M_{\rm pl}^{2n-3}\vert
m_S\vert)^{1\over 2n-2}/c_n^2$ in the latter case \cite{rad1}.  
As a result, the $\mu$-term is generated from the first term in 
$W_1$ as $\mu=\lambda u_0$. 
As shown in \cite{rad1}, $u_0$ can take a value of an intermediate 
scale in a rather wide range and 
then $\lambda$ should be very small to realize a suitable value for
$\mu$ in that case.
Although we can consider it to be an effective coupling constant coming
from the nonrenormalizable term, for example,
we do not discuss a reason of this smallness here.\footnote{
Although the smallness of a bilinear $R$-parity violating parameter
$\epsilon_\alpha$ appeared in the latter discussion should 
be also explained similarly to the case of $\mu$
parameter, its origin of smallness can be considered not to be 
heavily relevant to the feature of the present scenario. 
Thus we will also treat it as an effective parameter.} 

When a Hubble parameter $H$ satisfies a condition $H\sim M_S $ where
$M_S$ is a physical mass of $S$ and $O(\vert m_S\vert)$,
$u$ begins to oscillate around $u_0$ starting from its initial value $u=0$. 
This oscillation energy density scales as $R^{-3}$ like the energy
density of the matter fields as the universe expands with the scale
parameter $R$, whereas 
the radiation energy density decreases as $R^{-4}$. 
Thus this oscillation energy is expected to dominate 
the energy density of the universe at a certain period before its decay.
The decay of this oscillation into the Higgsinos
$\tilde H_{1,2}$ occurs due to the first term in $W_1$ if 
$m_{\tilde H_1}+m_{\tilde H_2} < M_S$ is satisfied. 
The decay width $\Gamma_u$ of the condensate $u$ 
is estimated as $\Gamma_u\sim {\vert\lambda\vert^2\over 4\pi}M_S $
and then the reheating temperature due to this decay will be
\begin{equation}
T_{\rm RH}^u\simeq 0.25\sqrt{M_{\rm pl}\Gamma_u}\simeq
 0.07~{\vert\mu\vert\over u_0}\sqrt{M_{\rm pl}M_S},
\label{eqb}
\end{equation}
where we used the fact that $\mu$ is generated as $\mu=\lambda u_0$ in
this scenario.
As far as there is no additional entropy production after the decay of the
oscillation of $u$, the constraint from the gravitino production 
requires $T_{\rm RH}^u~{^<_\sim}~10^9$~GeV \cite{grav}.
On the other hand, if we impose that the electroweak sphaleron interaction 
can be in the thermal 
equilibrium \cite{lept}, $T_{\rm RH}^u~{^>_\sim}~10^2$~GeV should also 
be satisfied.
Since the naturalness requires $\mu$ and $M_S$ to be of the order of 
weak scale $M_W$, 
these conditions for $T_{\rm RH}^u$ give a constraint such as $10^3~{\rm
GeV}~{^<_\sim}~u_0~{^<_\sim}~10^{10}$~GeV on the value of $u_0$.
Its realization is related to, for example, the contents of the extra matter 
fields which couple with $S$, $\bar S$ and also the power 
$n$ of the nonrenormalizable term in $W_1$. These are crucial to fix a 
magnitude of $u_0$ \cite{rad1}.
Since the oscillation of the condensate $u$ is non-thermal, 
the decay of the condensate 
$u$ is considered to be out-of-equilibrium.
Thus, if there are $CP$ violating complex phases and also
$\Gamma_u<H_{T\simeq M_S}$ is satisfied, the asymmetry between 
Higgsino $\tilde H_{1,2}$ number density and the anti-Higgsino 
$\tilde H_{1,2}^c$ number density is expected to be produced through 
this decay process.
This restricts the region of $u_0$ obtained above, furthermore, into 
\begin{equation}
10^8~{\rm GeV}~{^<_\sim}~u_0~{^<_\sim}~10^{10}~{\rm GeV},
\end{equation}
which corresponds to $10^2~{\rm GeV}~{^<_\sim}~T^u_{\rm
RH}~{^<_\sim}~10^4~{\rm GeV}$. 
%%%%
\input epsf 
\begin{figure}[tb]
\begin{center}
\epsfxsize=8.4cm
\leavevmode
\epsfbox{asym.eps}\\
%\vspace*{-0.3cm}
{\footnotesize Fig. 1~~\  One loop diagrams contributing to 
$S\rightarrow\tilde H_1\tilde H_2$.}
\end{center}
\end{figure}
%%%%%

In order to estimate this asymmetry, we need to define the $CP$ 
structure of the model. The model is naturally expected to have 
the soft supersymmetry breaking parameters corresponding to each term in 
the superpotential $W_1$ such as
\begin{equation}
{\cal L}_{\rm soft}= A_\lambda\lambda SH_1H_2 +\cdots,
\label{eqc}
\end{equation}
where $A_\lambda$ is assumed to be $O(M_W)$ and complex.
The Higgsino asymmetry $\varepsilon$ can be considered to be 
dominantly produced by 
the interference terms between a tree diagram 
of $S\rightarrow\tilde H_1\tilde H_2$ 
and one-loop diagrams with a scalar trilinear $A_\lambda$ vertex as
shown in Fig.~1.
This asymmetry $\varepsilon$ can be roughly estimated as
\begin{eqnarray}
\varepsilon&\equiv&{\Gamma_u(S\rightarrow\tilde H_1\tilde H_2)
-\Gamma_u(S^c\rightarrow\tilde H_1^c\tilde H_2^c)\over 
\Gamma_u(S\rightarrow\tilde H_1\tilde H_2)
+\Gamma_u(S^c\rightarrow\tilde H_1^c\tilde H_2^c)}\nonumber \\
&\simeq&{\vert A_\lambda\vert \over 16\pi}
\left({g_2^2\over M_2}+{g_1^2\over M_1}\right)\sin\delta_A
+ {\vert\lambda\vert^2\over 32\pi}
{\vert A_\lambda^2\mu\vert\over M_{\tilde S}^3}\sin(\delta_A+2\delta_\lambda),
\label{eqd}
\end{eqnarray}
where $M_{1,2}$ and $M_{\tilde S}$ are the masses of the gauginos 
$\lambda_{1,2}$ and the superpartner of $S$, respectively. 
By imposing $u_0$ to be real, we find that $\delta_A+\delta_\lambda=0$
is satisfied where $\delta_A\equiv {\rm arg}~A_\lambda$ and 
$\delta_\lambda\equiv {\rm arg}~\lambda$.
Since the coupling constant $\lambda$ is very small in this scenario
as mentioned before, $\varepsilon$ is approximated by the first term. 
The energy density $\rho_S$ stored in the $u$ oscillation and 
the total energy density $\rho_{\rm tot}$ of
the universe are written as
$\rho_{\rm tot}=sT_{\rm RH}$ and $\rho_S=n_SM_S$, respectively.
In these relations $s$ is the
entropy density and $n_S$ is the number density of $S$.
Thus, by using these,
the total asymmetry of $\tilde H_2$ can be estimated as
\begin{equation} 
{n_{\tilde H_2}-n_{\tilde H_2^c}\over s}
\simeq \varepsilon{\rho_S\over\rho_{\rm tot}}{T_{\rm RH}^u\over M_S}
\simeq 3\times 10^4\left({\vert\mu\vert\over u_0}\right)\sin\delta_A, 
\label{eqe}
\end{equation}
where we assume the energy density of 
the universe is dominated by the
oscillation of $u$ at this period and $\rho_{\rm tot}\sim \rho_S$, 
for simplicity. We also assume $M_{1,2}\sim \vert A_\lambda\vert$ in the last 
similarity relation.

Now we examine whether this asymmetry can remain as a nonzero value and
be partially converted into the $B$ and $L$ asymmetry under the situation 
that the electroweak sphaleron interaction is in the thermal equilibrium.
For this study, it is convenient to consider the detailed balance of
various interactions and solve
the chemical equilibrium equations \cite{lb,bp}. The particle-antiparticle
number asymmetry $n_f$ can be approximately related to the corresponding 
chemical potential $\mu_f$, in the case of $\mu_f \ll T$, such as
\begin{equation}
\delta n_f\equiv n_{f} -n_{f^c}=\left\{ \begin{array}{ll}
\displaystyle
{g_f\over 6}T^2\mu_f & (f~:~{\rm fermion}),\\
\displaystyle
{g_f\over 3}T^2\mu_f & (f~:~{\rm boson}), \\ \end{array} \right. 
\label{eqee}
\end{equation}
where $g_f$ is a number of relevant degrees of freedom of $f$.
By using the chemical potential $\mu_f$,
we can write the detailed valance equations after the reheating.
If the SU(2) and SU(3) sphaleron interactions are in the thermal
equilibrium, we have the conditions such as
\begin{eqnarray}
&&\sum_{i=1}^N\left(3\mu_{Q_i}+\mu_{L_i}\right)+\mu_{\tilde{H_1}}
+\mu_{\tilde{H_2}}+ 4\mu_{\tilde W}=0, \nonumber \\  
&&\sum_{i=1}^N\left(2\mu_{Q_i}-\mu_{\bar U_i}-\mu_{\bar D_i}\right)
+6\mu_{\tilde g}=0,
\label{eqee1}
\end{eqnarray}
where $N$ is the number of generation and we will put it as $N=3$ 
finally.
The cancellation of the total hypercharge or the electric charge 
of plasma in the universe requires
\begin{eqnarray}
&&\sum_{i=1}^N\left(\mu_{Q_i}+2\mu_{\bar U_i}-\mu_{\bar D_i}-\mu_{L_i}-
\mu_{\bar E_i}\right)+\mu_{\tilde{H_2}}-\mu_{\tilde{H_1}} \nonumber \\
&&\hspace{2cm}+2\sum_{i=1}^N(\mu_{\tilde{Q_i}}+2\mu_{\tilde{\bar U_i}}-
\mu_{\tilde{\bar D_i}}-\mu_{\tilde{L_i}}-\mu_{\tilde{\bar E_i}})
+2\left(\mu_{H_2}-\mu_{H_1} \right)=0.
\end{eqnarray}
Yukawa interactions of the quarks and leptons in the thermal 
equilibrium impose the conditions
such as
\begin{equation} 
\mu_{Q_i}-\mu_{\bar U_j}+\mu_{H_2}=0, \quad
\mu_{Q_i}-\mu_{\bar D_j}+\mu_{H_1}=0, \quad
\mu_{L_i}-\mu_{\bar E_j}+\mu_{H_1}=0,
\end{equation}
where we assume that the right-handed neutrinos decouple or
do not exist. Thus there are no conditions for the Yukawa couplings
for the neutrinos.
There are also the conditions for the gauge interactions in the
thermal equilibrium, which are summarized as
\begin{equation}
\mu_{\tilde{Q_i}}=\mu_{\tilde g}+\mu_{Q_i}=\mu_{\tilde W}+\mu_{Q_i}
=\mu_{\tilde B}+\mu_{Q_i}.
\label{eqee2}
\end{equation}
There are similar equations to eq.~(\ref{eqee2}) 
for leptons $L_i$, Higgs $H_{1,2}$ and other
fields $\bar U_i, \bar D_i, \bar E_i $ which have the SM gauge interactions.
It should be noted that a detailed balance equation for $\lambda S\tilde 
H_1\tilde H_2$ is not imposed since we assume that it is out-of-equilibrium.
Flavor mixings of quarks and leptons due to the Yukawa couplings
allow us to consider the flavor independent chemical potential such as
$\mu_{Q}=\mu_{Q_i}$ and $\mu_{L}=\mu_{L_i}$.
Thus, in these equations we find that there are two independent
parameters, which can be taken as $\mu_{\tilde H_2}$ and $\mu_{\tilde g}$.

We have not taken account of the equilibrium conditions for the soft 
supersymmetry breaking terms in the above equations yet. Here 
we should remind the fact that the soft supersymmetry breaking terms 
are in the thermal equilibrium when $T~{^<_\sim}~10^7$~GeV is satisfied 
\cite{iq} and it is the case in the present scenario.
As its result, we find that $\mu_{\tilde g}=0$ is satisfied.
If we define the $B$ and $L$ in such a way as $\delta n_B\equiv BT^2/6$ and 
$\delta n_L\equiv LT^2/6$ by using eq.(\ref{eqee}),
they can be written by solving eqs.~(\ref{eqee1}) $\sim$ (\ref{eqee2})
with the chemical potential of Higgs field $H_2$ as
\begin{equation}
B=-3(2N+1)\mu_{\tilde H_2}, \qquad L={3\over 4}(14N+9)\mu_{\tilde H_2}.
\label{eqeee}
\end{equation}
This shows that all of $B$, $L$ and $B-L$ take nonzero values as far as 
$\mu_{\tilde H_2}\not= 0$.
If we use eqs.~(\ref{eqe}) and (\ref{eqee}), we can rewrite 
the above $B$ and $L$ in the
following way,
\begin{equation} 
B=-{3(2N+1)\over 2}D, \qquad  
L={3(14N+9)\over 8}D,
\end{equation}
where $D$ is defined as $\delta n_{\tilde H_2}\equiv DT^2/6$.
Thus the expected baryon asymmetry in this scenario is estimated as
\begin{eqnarray}
Y_B\equiv {\delta n_B\over s} ={n_{\tilde H_2}-n_{\tilde H_2^c}\over s}~
{B\over D}\kappa 
\simeq -4.5\times 10^4~(2N+1)
\left({\vert\mu\vert\over u_0}\right)\kappa\sin\delta_A,
\label{eqe4}
\end{eqnarray}
where $\kappa\le 1$ is introduced to take account of the washout effect and we 
discuss it later.
From this result, we find that this scenario can produce the presently
observed baryon number asymmetry $Y_B=(0.6~-~1)\times 10^{-10}$
as far as $u_0\sim 10^{15}\vert\mu\vert\kappa\sin\delta_A$~GeV is satisfied.
Since $\vert\mu\vert$ is considered to be $O(M_W)$, 
$10^{-10}~{^<_\sim}~\kappa\sin\delta_A~{^<_\sim}~10^{-8}$ seems to be 
required to satisfy the constraint obtained from the condition on
the reheating temperature 
$T_{\rm RH}^u$. The most severe constraint on the magnitude of $\delta_A$ 
comes from the electric dipole moment of a neutron and an electron 
depending on the masses of the superpartners. 
In any case, it is well-known that this constraint is satisfied as far as 
$\delta_A~{^<_\sim}~10^{-2}$. We have no contradiction on this point.
The problem is that an unnaturally small value of $\delta_A$ seems 
to be required for the realization of the appropriate baryon number asymmetry.
To escape this situation we need some kind of washout effects $\kappa \ll 1$.

Leaving this problem to the later discussion, we would like to 
stress a nice feature of this scenario 
with respect to the dark matter abundance here.
If the Higgsinos $\tilde H_{1,2}$ are the dominant components of the lightest
superparicle which seems to be a natural possibility, 
they can be the cold dark matter candidates. Their abundance
is also determined by the chemical potential $\mu_{\tilde H_2}$ as
\begin{equation}
{\Omega_B \over \Omega_{\rm CDM}}\simeq
{m_P \over m_{\tilde H_{1,2}}}{B\over D}\simeq 
0.01~(2N+1) \left({100~{\rm GeV} \over m_{\tilde H_{1,2}}}\right),
\label{eqe5}
\end{equation}
where $\Omega_i$ is the ratio of the energy density $\rho_i$ to the
critical density of the universe. 
It is noticeable that this relation may explain 
the observationally predicted value of $\Omega_B/\Omega_{\rm CDM}$ such as
$0.01~{^<_\sim}~\Omega_B/\Omega_{\rm CDM}~{^<_\sim}~0.2$,
which is an important aspect of our universe.

It is an interesting and also important issue to consider the compatibility 
of this scenario for the baryon and lepton number asymmetry 
with the small neutrino mass generation.
Here we consider the typical two schemes for the neutrino mass generation.
One is the usual seesaw scheme characterized by the superpotential
\begin{equation}
W_2=y_N^{\alpha\beta}L_\alpha\bar N_\beta H_2 
+ M_{\alpha\beta}\bar N_\alpha N_\beta. 
\end{equation}
In this type of model the leptogenesis is usually considered on the basis of 
the out-of-equilibrium decay of the right-handed neutrino or 
the decay of sneutrino condensate. 
However, as far as we consider the spontaneous $\mu$-term generation
along the almost flat direction of $S$, 
the baryon number asymmetry produced by this usual leptogenesis 
cannot be the dominant one.
The reheating temperature due to the thermalization of the oscillation 
energy of $S$ is not high enough compared with the right-handed neutrino 
masses and then the asymmetry produced through the usual scenario is
washed out by this entropy production.
Thus the baryon number asymmetry produced by the present scenario 
can be the one which exists in the present universe,
if we can prepare a suitable dilution mechanism in the range of $10^{-8}\sim
10^{-6}$ for $\delta_A\sim 10^{-2}$.
Another entropy production below the weak scale is required to
satisfy this condition. On the other hand,
the nucleosynthesis requires that this reheating temperature $T_N$
should satisfy $T_N~{^>_\sim}~1$~MeV at least. The dilution effect
$\kappa$ due to this reheating will be written as $\kappa\sim
T_N/T_{\rm RH}^u$ and then estimated at most as 
$10^{-7}~{^<_\sim}~\kappa~{^<_\sim}~10^{-5}$.
This shows that the present scenario can work in the ordinary seesaw
scheme if we can present this reheating scenario.

Another scheme for the small neutrino mass may be characterized by 
a bilinear $R$-parity violating superpotential\footnote{In the following 
discussion we will assume $\epsilon_\alpha=\epsilon$, for simplicity.
The conclusion does not depend on this assumption.}
\begin{equation}
W_3=\epsilon_\alpha L_\alpha H_2.
\end{equation} 
In the model with $W_3$ we can consider the small neutrino mass generation 
on the basis of the neutralino-neutrino mixing.
In fact, in the MSSM extended by $W_3$ the first term in $W_1$ 
collaborates with $W_3$ to induce a mixing mass matrix between neutrinos 
and neutralinos such as 
\begin{eqnarray}
&&{\cal M}=\left(\begin{array}{cc}
0 & M_m \\  M_m  & M_{\cal N} \\
\end{array}\right), \quad M_m=\left(
\begin{array}{cccc}
\sqrt 2g_2\langle\tilde\nu_e\rangle &\sqrt 2g_1\langle\tilde\nu_e\rangle 
& 0 & \epsilon_e \\
\sqrt 2g_2\langle\tilde\nu_\mu\rangle 
&\sqrt 2g_1\langle\tilde\nu_\mu\rangle & 0 & \epsilon_\mu \\
\sqrt 2g_2\langle\tilde\nu_\tau\rangle 
&\sqrt 2g_1\langle\tilde\nu_\tau\rangle & 0 &\epsilon_\tau \\
\end{array}\right), \nonumber  \\
&&M_{\cal N}=\left(\begin{array}{cccc}
M_2 & 0 & {1\over\sqrt 2}g_2v_1 & -{1\over\sqrt 2}g_2v_2\\ 
0  & M_1 & -{1\over\sqrt 2}g_1v_1 & {1\over\sqrt 2}g_1v_2 \\
{1\over\sqrt 2}g_2v_1 &-{1\over\sqrt 2}g_1v_1  & 0 & \mu \\ 
-{1\over\sqrt 2}g_2v_2 &{1\over\sqrt 2}g_1v_2  & \mu & 0 \\
\end{array}\right),
\label{eqf}
\end{eqnarray}
where we write ${\cal M}$ by using the basis 
$(L_\alpha,~-i\lambda_2^3,~-i\lambda_1,~\tilde H_1,~\tilde H_2^c )$
and $M_{\cal N}$ corresponds to the usual neutralino mass matrix.
Nonzero sneutrino VEVs $\langle\tilde\nu_\alpha\rangle$ are
expected by the minimization of the scalar potential which contains a
soft supersymmetry breaking term 
$B_{\epsilon_\alpha}\epsilon_\alpha L_\alpha H_2$ for $W_3$.
We can estimate it as $\langle\tilde\nu_\alpha\rangle\sim O(\epsilon_\alpha)$.
If we assume that both the absolute values of 
$\epsilon_\alpha$ and $\langle\tilde\nu_\alpha\rangle$ 
are much smaller than the ones of $\mu$ and $ M_{1,2}$ which 
are considered to be $O(M_W)$,
the small neutrino masses are generated by the weak scale seesaw 
mechanism \cite{rparity,hdprv}.
It is easily checked that ${\cal M}$ has two zero and 
five nonzero eigenvalues. The four nonzero eigenvalues correspond to
the ones of the neutralinos and $O(M_W)$. 
On the other hand, as discussed in \cite{hdprv}, 
the tree-level mass eigenvalue is characterized 
by the quantities
$\Lambda_\alpha\equiv\epsilon_\alpha\langle 
H_1\rangle+\mu\langle\tilde\nu_\alpha\rangle$
in the effective neutrino mass matrix. 
Using this $\Lambda_\alpha$, the smallest one of the nonzero eigenvalues is
written as $(M_1g_1^2+M_2g_2^2)\vert \vec{\Lambda}\vert^2/4 {\rm
det}(M_{\cal N})$. The remaining massless states become massive by
taking account of the radiative corrections.
By taking account of the constraints for the mass and mixing 
required by the explanation of the 
anomalies of the solar neutrinos and the atmospheric neutrinos \cite{sol,atm},
it is shown that $\epsilon$ should satisfy a 
condition such as $\vert\epsilon\vert/\vert\mu\vert
\sim O(10^{-4\sim -3})$ 
by using the numerical analysis \cite{hdprv}.\footnote{The 
author would like to thank Dr.~M.~Hirsch for informing this point to him.}
If we assume that two massless states at the tree level are 
proportional to $\nu_e\pm \nu_\mu$, the eigenstate with a small 
non-zero eigenvalue is expected to be represented 
as $\nu_\tau+O(\vert\epsilon\vert/\vert\mu\vert){\cal N}$ 
where ${\cal N}$ is a neutralino which is dominated by $\tilde H_{1,2}$.

The model with the bilinear $R$-parity violating term can be extended by
introducing the generation dependent extra U(1)$_X$ gauge symmetry at the
TeV region \cite{rparity1}. In this case, the neutrino mass degeneracy
can be resolved at the tree-level and then the mass and mixing of
neutrinos can be directly related to the $\epsilon_\alpha$.
The largest mass eigenvalue can be written as 
$O(\vert\epsilon\vert^2/\vert\mu\vert)$,
where the gaugino masses are assumed to satisfy $M_{1,2,X}\sim \vert\mu\vert$.
By imposing the condition for the $\nu_\tau$ mass required from the
explanation of the atmospheric neutrino anomaly, we can obtain a
condition $\vert\epsilon\vert/\vert\mu\vert 
\sim O(10^{-6})$ for a reasonable
value of $\mu$. The value of $\vert\epsilon\vert$ can be smaller than 
the one in the previous case by the order two or three magnitude, although  
the mixing between $\nu_\tau$ and $\tilde H_{1,2}$ is similarly given by 
$O(\vert\epsilon\vert/\vert\mu\vert)$.

There is another constraint on the value of 
$\vert\epsilon\vert/\vert\mu\vert$. 
If we assume the existence of the bilinear $R$-parity violating 
terms $\epsilon_\alpha L_\alpha H_2$ 
to generate the neutrino mass, it can crucially affect
the abundance of the baryon and lepton number asymmetry
since these terms violate the lepton mumber. 
In fact, when these terms are in the thermal equilibrium
and then $\mu_L+\mu_{\tilde H_2}=0$ is satisfied, 
the chemical detailed balance equations result in $\mu_{\tilde H_2}=0$ 
and we cannot have the nonzero $B$ and $L$ as it can 
be seen from eq.~(\ref{eqeee}).
Thus, we must assume that this term should not be in the thermal 
equilibrium at the period when the electroweak sphaleron is in 
the thermal equilibrium.
To estimate roughly what a kind of constraint comes out from this condition, 
it is convenient to redefine the chiral superfield $H_1$ as 
$H_1^\prime\equiv {\epsilon_\alpha\over \mu}L_\alpha +H_1$.
By this manipulation, $\epsilon_\alpha L_\alpha H_2$ disappears from the
superpotential but there appear the new $R$-parity violating terms
\begin{equation}
W_{\rm RPV}=
-y_e^{\alpha\beta}{\epsilon_\gamma\over\mu}L_\alpha\bar E_\beta L_\gamma
-y_d^{\alpha\beta}{\epsilon_\gamma\over\mu}Q_\alpha\bar D_\beta L_\gamma.
\label{eqff}
\end{equation}
We need to require that these interactions do not completely wash out
the baryon and lepton number asymmetries.
The stringent constraints on these terms are derived by requiring 
that the lepton number violating scattering processes with both 
the Yukawa coupling $y_{e,d}$ in eq.~(\ref{eqff}) and the 
gauge interaction as two vertices are 
not in the thermal equilibrium. 
By estimating these processes, we can find a condition such as \cite{dr,bp}
\begin{equation}
\left\vert~ y_{e,d}^{\alpha\beta}~{\epsilon_\gamma\over\mu}~ 
\right\vert~{^<_\sim}~10^{-7}. 
\label{eqfff}
\end{equation} 
Since these Yukawa coupling constants $y_{e,d}$ are constrained by the
masses of quarks and leptons, eq.~(\ref{eqfff}) gives us a condition on
$\vert\epsilon_\alpha\vert/\vert\mu\vert$.
If we assume 
$\langle H_1^\prime\rangle=100$~GeV, we find
$\vert\epsilon_\alpha\vert/\vert\mu\vert~{^<_\sim}~10^{-6}$. 
\footnote{
The thermal corrections on the Higgsino masses may be expected to loose
this bound. In the present scenario, however, the reheating 
temperature is not much higher than the supersymmetry breaking scale. 
Then such corrections is not expected to be so large as to make this bound 
change substantially.}  
Since this seems to impose the severe constraint on the neutrino 
mass generation scheme in the present scenario, the more detailed
study on this point is necessary.

As discussed before, we need rather strong dilution effect
$\kappa \ll 1$ in eq.~(\ref{eqe4}) to obtain an appropriate baryon
number asymmetry
without the extreme fine tuning of $\delta_A$.
Since the larger value of $\vert\epsilon_\alpha\vert/\vert\mu\vert$ makes 
the lepton number violating process effective, the estimation of 
the washout factor $\kappa$ is crucial to determine the unwashed value
of the lepton and baryon number asymmetry due to the interactions 
in eq.(\ref{eqff}) with the values of 
$\vert\epsilon_\alpha\vert/\vert\mu\vert$ which are 
consistent with the neutrino mass generation.
The washout factor $\kappa$ can be estimated by the analysis 
of the Boltzmann equation for the particle 
number density which includes the effects of the scattering 
$\psi_L\psi_a\rightarrow\psi_i\psi_j$. 
It can be written as \cite{bna,bp}
\begin{equation}
{d N_L\over dx}=-{\Gamma_A\over H(m)}{x\over N_L^{\rm EQ}}\left(N_LN_a-
N_L^{\rm EQ}N_a^{\rm EQ}\right),
\end{equation}
where $N_f$ stands for the number density per comoving volume $N_f\equiv
n_f/s$ and $N_f^{\rm EQ}$ represents its value at the thermal equilibrium.
We introduce a dimensionless parameter $x=m/T$ for a certain mass scale 
$m$. $H(m)$ is the Hubble constant at $T=m$.
This can be deformed into the equation for the lepton number asymmetry
$Y_L\equiv N_L- N_{L^c}$.  From that equation we can obtain the 
washout factor $\kappa$ in such a way as
\begin{equation}
\kappa= \exp\left(-\int_1^{x_0} dx{x\Gamma_A\over H(m)}
{Y_a^{\rm EQ}\over Y_L^{\rm EQ}}\right),
\label{eqf4}
\end{equation}
where $x_0$ should be taken as a value at which the electroweak sphaleron
interaction becomes the out-of-equilibrium. 
If we assume that $m$ is the supersymmetry breaking scale $O(1)$~TeV
which seems to be a reasonable choice in the present scenario,
$x_0$ should be fixed as 10. 

In the calculation of $\kappa$, the scattering processes such as $LQ
\rightarrow \lambda_3\bar D$ and its supersymmetric 
correspondence become the dominant contribution to the reaction rate 
$\Gamma_A$ in eq.~(\ref{eqf4}).
For these tree diagrams, it can be, respectively, represented as
\begin{equation}
\Gamma_A\sim\alpha_g\alpha_{y_{e,d}}\left(\epsilon\over\mu\right)^2
{T^5\over (T^2 +m^2)^2},
\qquad
\Gamma_A\sim\alpha_g\alpha_{y_{e,d}}\left(\epsilon\over\mu\right)^2T,
\end{equation}
where we use $\alpha_g=g^2/4\pi$ and so on.
For simplicity, we put the masses of the superpartners $m$. 
Using these $\Gamma_A$'s and practicing the calculation of
eq.~(\ref{eqf4}), we find that $\kappa$ takes the values 
shown in Table~1 for various values of $\vert\epsilon\vert/\vert\mu\vert$. 
We also find that the value of $\kappa$ becomes constant at 
$x_0~{^>_\sim}~10$ for these values of $\vert\epsilon\vert/\vert\mu\vert$. 
It means that these lepton number violating
processes also become out-of-equilibrium at $T\sim 100$~GeV.
For $\vert\epsilon\vert/\vert\mu\vert <10^{-5}$ the value 
of $\kappa$ is less than $10^{-11}$ and we cannot obtain the sufficient 
baryon number asymmetry.
\begin{figure}[tb]
\begin{center}
\begin{tabular}{cc|cc}
$\vert\epsilon\vert/\vert\mu\vert$ & $\kappa$&~~ 
$\vert\epsilon\vert/\vert\mu\vert$ 
& $\kappa$\\ \hline
$10^{-6}$ & 0.85 & $10^{-5.2}$ & $1.5\times 10^{-3}$ \\
$10^{-5.5}$ & 0.2 & $10^{-5.1}$ & $3.5\times 10^{-5}$  \\ 
$10^{-5.3}$ & $1.7\times 10^{-2}$ & $10^{-5}$ & $8.6\times 10^{-8}$ \\ \hline
\end{tabular}
\vspace*{3mm}

{\footnotesize Table~1}
\end{center}
\end{figure}
This estimation of $\kappa$ shows that 
the magnitude of the $R$-parity violating coupling is severely
constrained by the washout effect and as a result 
the neutrino mass generation scheme based on the bilinear $R$-parity
violation seems to be strictly restricted in the present scenario. 
On the other hand, this washout effect may make it possible to produce 
an appropriate baryon number asymmetry for a certain range of 
$\vert\epsilon\vert/\vert\mu\vert$ without 
any fine tuning of the value of CP phases.
Unfortunately, in this case, the $R$-parity violating term makes the
Higgsinos $\tilde H_{1,2}$ unstable and then the explanation of
$\Omega_B/\Omega_{\rm CDM}$ based on  eq.~(\ref{eqe5}) 
cannot be applicable. 
  
We have left some important problems undiscussed.
One of them is whether we can have the models which cause the symmetry 
breaking for $S$ at a required scale.
In ref.~\cite{rad1} this kind of study has been done extensively 
for the similar models in the different context and they found that
the symmetry breaking scale required here could be successfully
realized. It is necessary to construct such a concrete model
in which the present scenario for the baryogenesis is applicable.
In the present study we have not discussed the relation between this
scenario and the inflation of the universe, either. It is necessary 
to investigate the possibility to embed this scenario into a 
suitable inflation model to examine the viability of this scenario. 
These subjects will be discussed in other places. 

In conclusion we have proposed a lepto- and baryogenesis scenario 
which is intimately related to the origin of the $\mu$-term and the small
neutrino masses. 
If the $\mu$-term is assumed to be originated from an intermediate
scale VEV of the SM singlet field $S$, the deviation of this condensate 
from the true vacuum value induces the coherent oscillation and its
decay may generate the asymmetry between the Higgsinos number density and the
anti-Higgsinos number density.  
Since the Higgsinos are the SU(2) doublets, they contribute to the electroweak
sphaleron interaction. If the sphaleron interaction is in the thermal
equilibrium, this asymmetry is distributed into the lepton and baryon
number asymmetry. 
Since the reheating temperature realized by the decay of the $S$
condensate is high enough for sphaleron interaction to be in the thermal 
equilibrium, the conversion of the Higgsino number asymmetry into the baryon 
asymmetry can be substantially proceeded.
Since the produced abundance is too large, we need some kind of washout
effect. As such an example, we may be able to use the bilinear 
$R$-parity violating term which can induce the small neutrino masses.
The produced Higgsino number asymmetry may also explain the energy 
ratio between the baryon and the cold dark matter since their origins
are the same in this scenario.
This possibility seems to be very interesting 
and deserve for further study since it can be intimately 
related to various problems in particle and astroparticle physics.

\vspace*{10mm}
\noindent
This work is supported in part by a Grant-in-Aid for Scientific 
Research (C) from Japan Society for Promotion of Science
(No.~14540251) and also by a Grant-in-Aid for Scientific 
Research on Priority Areas (A) from The Ministry of Education, Science,
Sports and Culture (No.~14039205).

\vspace*{5mm}
\newpage
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\end{document}






