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\begin{document}
\vspace*{4cm}
\title{RECONCILING SUPERSYMMETRY AND THERMAL LEPTOGENESIS BY ENTROPY PRODUCTION}
\author{ J.~KERSTEN }
\address{II.\ Institute for Theoretical Physics, University of Hamburg,\\
Luruper Chaussee 149, 22761 Hamburg, Germany}
\maketitle\abstracts{
The entropy produced in the decays of super-weakly interacting particles
may help to reconcile thermal leptogenesis and Big Bang Nucleosynthesis
(BBN) in scenarios with gravitino dark matter, which is usually
difficult due to late decays of the next-to-lightest supersymmetric
particle (NLSP) spoiling BBN\@. We study this possibility for a general
neutralino NLSP\@. We discuss the constraints on the entropy-producing
particle, considering as an example the saxion from the axion multiplet.
We show that, in addition to enabling a solution of the strong CP
problem, it can indeed produce a suitable amount of entropy.
}
\section{The Gravitino Problem}
The tiny but non-zero neutrino masses, which constitute the first solid
evidence for physics beyond the Standard Model (SM), find a natural
explanation in the see-saw mechanism.%
\cite{Minkowski:1977sc,Yanagida:1980,Glashow:1979vf,Gell-Mann:1980vs,Mohapatra:1980ia}
In this setup the SM is extended by gauge-singlet neutrinos with very
large masses. The C- and CP-violating decay of these heavy neutrinos in
the early universe can provide the observed baryon asymmetry via
leptogenesis \cite{Fukugita:1986hr} as an almost free by-product.
The CP asymmetry of the decays
\begin{equation}
\epsilon =
\frac{\Gamma(\nuR \to \ell H) - \Gamma(\nuR \to \overline\ell \,
\overline{H})}{\Gamma(\nuR \to \ell H) + \Gamma(\nuR \to \overline\ell
\, \overline{H})}
\end{equation}
creates a lepton asymmetry, which is afterwards converted into a
baryon asymmetry
$\eta_\text{B} = \frac{n_\text{B}}{n_\gamma} \propto |\epsilon|$
by sphaleron processes.\cite{Kuzmin:1985mm}
We denote the lightest of the heavy neutrinos by $\nuR$ and its mass by
$\Mr$.
For hierarchical heavy neutrino masses and no fine-tuning, the CP
asymmetry is limited by \cite{Davidson:2002qv}
\begin{equation} \label{eq:EpsilonMax}
|\epsilon| < \frac{3}{16\pi} \frac{\Mr \sqrt{\Delta m^2_\text{atm}}}{v^2} \;.
\end{equation}
Using the mass squared difference $\Delta m^2_\text{atm}$
measured in atmospheric neutrino oscillations, the Higgs vacuum
expectation value $v$, the observed baryon asymmetry
$\eta_\text{B} \simeq 6 \cdot 10^{-10}$,
and other known quantities then leads to the lower limit
$\Mr \gtrsim 2 \cdot 10^9 \GeV$.\cite{Buchmuller:2002rq}
The only price to pay for all the baryons is a mechanism producing the heavy neutrinos in
the first place. In the simplest scenario, thermal leptogenesis,
the temperature is larger than $\Mr$, so the heavy neutrinos are
abundantly produced, since they
are in contact with the thermal bath via their Yukawa couplings.
Consequently, thermal leptogenesis requires
a sufficiently high reheating temperature after inflation,
\begin{equation}
\TR \gtrsim \Mr \gtrsim 2 \cdot 10^9 \GeV \;.
\end{equation}
This scenario does not address the biggest theoretical problem of the
SM, the hierarchy problem. This problem is elegantly solved by
supersymmetry (SUSY), which in turn offers a natural way to include
gravity in the form of supergravity.
Within this theory, the large temperature in the early universe also
leads to a thermal production of gravitinos, the superpartners
of the graviton. Their relic density is approximately
\cite{Bolz:2000fu,Pradler:2006qh}
\begin{equation} \label{eq:Omega32}
\Omega_{3/2}^\text{tp} h^2
\simeq 0.11
\left( \frac{T_\text{R}}{2 \cdot 10^9 \GeV }\right)
\left( \frac{67 \GeV}{m_{3/2}}\right)
\left( \frac{M_{\widetilde g}}{10^3 \GeV}\right)^2 .
\end{equation}
Thus, the observed dark matter abundance
$\Omega_\text{DM} h^2 \simeq 0.11$ is compatible with the reheating
temperature required by thermal leptogenesis both for a gravitino
lightest superparticle (LSP) with a sufficiently large mass
$m_{3/2} \gtrsim 60\GeV$ and for a heavier non-LSP gravitino.
However, as it interacts only via gravity, a non-LSP gravitino has a
long lifetime between, very roughly, $10^{-2} \second$ and several
years. Consequently, it decays during or after Big Bang Nucleo\-synthesis
(BBN), releasing energetic decay products that destroy the light nuclei
produced by BBN\@.\cite{Falomkin:1984eu,Ellis:1984eq}
The observed primordial element abundances limit the gravitino density
and thus the reheating temperature. The result is
$\TR \ll 10^8\GeV$, unless $m_{3/2} \gg 1\TeV$.\cite{Kawasaki:2004qu}
So thermal leptogenesis is not possible for an unstable gravitino with a
mass similar to the other superparticle masses, as expected in most
scenarios of SUSY breaking.
Let us therefore concentrate on the case of a gravitino LSP with a mass
around $100\GeV$. For conserved $R$ parity, the gravitino is now stable
and does not cause any problems. However, the next-to-LSP (NLSP) can
only decay to the gravitino via gravity. Thus, it is long-lived and its
decay products threaten the success of BBN\@. If the NLSP relic density
is determined by the standard freeze-out mechanism, the resulting
changes of the primordial abundances are incompatible with observations
in the Minimal Supersymmetric Standard Model (MSSM) with TeV-scale SUSY,
with the exception of very small corners of the parameter space.
Consequently, the gravitino problem survives in the form of the NLSP
decay problem.
\section{Entropy Production}
We consider one of the many approaches to solve the gravitino problem,
the possibility that a large amount of entropy is produced after the
freeze-out of the NLSP, diluting its density by a factor $\Delta$.%
\cite{Buchmuller:2006tt,Pradler:2006hh,Kasuya:2007cy,Hasenkamp:2010if}
This reduces the impact of the NLSP decays on BBN, possibly making it
compatible with observations.
The entropy can stem from the decay of a non-relativistic particle
$\phi$. The energy density of such a particle only decreases as
$\rho_\phi \propto R^{-3}$, where $R$ is the scale factor of the
universe, while the energy density of radiation decreases faster,
$\rho_\text{rad} \propto R^{-4}$. Consequently, if $\phi$ is
sufficiently long-lived, $\rho_\phi$ will equal $\rho_\text{rad}$ at
some temperature $T_\phi^=$, and it will dominate the energy density of
the universe afterwards. Eventually, the particle decays into radiation
at a temperature $T_\phi^\text{dec}$, increasing the entropy per
comoving volume by a factor \cite{Scherrer:1984fd,Kolb:1990vq}
\begin{equation} \label{eq:Delta}
\Delta \simeq 0.75 \, \frac{T_\phi^=}{T_\phi^\text{dec}}
\end{equation}
and thus diluting all previously produced relic abundances by the same
factor.
We require radiation domination at the time of NLSP freeze-out, so that
the standard computation of its thermal relic density is valid. This
means that
$T_\phi^= < T_\text{NLSP}^\text{fo} \sim \frac{m_\text{NLSP}}{25}$.
The decay of $\phi$ has to happen before BBN to avoid changing the
primordial abundances, $T_\phi^\text{dec} > T_\text{BBN} \sim 4\MeV$.
This leads to the upper bound
\begin{equation}
\label{eq:d3}
\Delta \lesssim 0.75 \cdot 10^3 \left(\frac{m_\text{NLSP}}{100\GeV}\right).
\end{equation}
The amount of entropy production is also limited by leptogenesis, since
it dilutes the baryon asymmetry by a factor $\Delta$, too. According to
Eq.~\ref{eq:EpsilonMax}, this has to be compensated by increasing $\Mr$
by the same factor. However, for very large values of $\Mr$ the baryon
asymmetry is strongly reduced by washout processes.\cite{Buchmuller:2004nz}
This places an upper limit on $\Mr$ and thus on $\Delta$. We estimate
$\Delta \lesssim 10^3 \,\dots\, 10^4$, which roughly coincides with the
bound in Eq.~\ref{eq:d3} for NLSP masses around the electroweak scale.
Note that the increase of $\Mr$ raises the lower limit on the reheating
temperature by a factor $\Delta$ as well. Assuming $\TR \sim \Mr$, this
exactly compensates the dilution of the gravitino density, so we still
obtain the correct dark matter density.
As a concrete example, let us consider the
constraints from BBN on a neutralino NLSP for
a gravitino LSP mass of $100\GeV$ and a dilution factor $\Delta=10^3$.
Performing a scan over the low-energy
gaugino and higgsino mass parameters allowed by LEP and corresponding to
neutralino masses up to $2\TeV$, we arrive at the points shown in
Fig.~\ref{fig:DilutedNeutralino} for the case of a neutralino whose main
components are the bino and the higgsinos.
The horizontal axis of the plot is the neutralino lifetime.
The vertical axis is its relic density multiplied by the
hadronic branching ratio
and thus determines the energy released in the form of hadrons.
See \cite{Covi:2009bk} for details of the calculation of these quantities.
The curves in the figure are the bounds from BBN on hadronic energy
release.\cite{Jedamzik:2006xz}
All points above the uppermost line are definitely excluded, while those
between this line and the dashed line may be allowed. Everything below
the dashed line is definitely compatible with current observations.
\begin{figure}
\centering
\includegraphics[width=7cm,viewport=0 0 390 358,clip]{100GeVHadBinoHiggsinoD1000}
\caption{
Lifetime versus hadronic energy release of a bino-higgsino neutralino
compared with the hadronic BBN constraints \protect\cite{Jedamzik:2006xz}
for the case of a $100\GeV$ gravitino mass and a dilution factor
$\Delta=10^3$.
All points above the uppermost line are excluded, while those
between the curves should not be considered as strictly excluded.
The neutralino mass increases from right to left.
Its composition varies from bino at the top to higgsino at the
bottom, with the colors giving the dominant component
(from \protect\cite{Hasenkamp:2010if}).
\label{fig:DilutedNeutralino}}
\end{figure}
We see that even with considerable entropy production a large part of
the parameter space remains excluded. In particular, a neutralino with
dominant bino component is only possible for quite small lifetimes
correponding to masses above $1\TeV$.
However, unlike in the case without entropy production,
we do find allowed regions now.
There are states with comparable
bino and higgsino components and $m_\text{NLSP} \simeq 230 \GeV$
violating only the less conservative BBN bound. Neutralinos that are
mainly higgsino satisfy even these constraints, if they are
lighter than $250 \GeV$. They can be almost as light as the gravitino.
Thus, we have arrived at a scenario where thermal leptogenesis is
possible and the gravitino or NLSP decay problem is solved.
A change of $\Delta$ shifts all points vertically
by a corresponding factor. Therefore, it is straightforward to infer
the constraints for arbitrary $\Delta$ from the results shown here.
In~\cite{Hasenkamp:2010if} other possible neutralino
compositions and also the BBN constraints from electromagnetic energy
release have been discussed in detail. In particular, it turned out
that a neutralino with a large wino component is also possible.
\section{Candidates for the Entropy Producer}
Let us next discuss candidates for the field $\phi$ producing the
entropy. A list of general requirements is shown in Tab.~\ref{tab:req}.
Most of them are already clear from the discussion in the previous
section. Requirement vii is that the presence of $\phi$ be compatible
with gravitino dark matter. This would be violated, for example, if the
gravitino could decay into $\phi$ with a lifetime shorter than the age
of the universe $t_0$.
The last requirement concerns other particles that have to be introduced
together with $\phi$, such as its superpartners.
They must not violate ii or vii, must not produce many NLSPs or gravitinos
in their decays (v, vi) and must not introduce new problems on their own.
In fact, the requirements in the table either have to be fulfilled or are
generically fulfilled in any scenario containing long-lived particles.
As a consequence, the solution of the generic problems of long-lived
particles may automatically lead to the desired entropy production.
\begin{table}[b]
\caption{List of requirements for our scenario of entropy produced by
$\phi$ to dilute the NLSP (from \protect\cite{Hasenkamp:2010if}).
\label{tab:req}}
\vspace{0.4cm}
\centering
\begin{tabular}{|ccc|}
\hline
No. & Requirement & Reason or Comment
\\ \hline
i & $T^\text{dec}_\phi < T^\text{fo}_{\text{NLSP}}$ &
dilute $\Omega_{\text{NLSP}}$
\\
ii & $T^\text{dec}_\phi > T_{\text{BBN}}$ & do not spoil BBN
\\[1mm]
iii & $\frac{\rho_\phi}{\rho_\text{rad}}(T^\text{dec}_\phi) > 1$ &
needed for $\Delta \gg 1$
\\
iv & $\frac{\rho_\phi}{\rho_\text{rad}}(T^\text{fo}_\text{NLSP}) < 1$ & for standard NLSP freeze-out
\\[1mm]
v & Br$(\phi \to \text{NLSP}) \simeq 0$ & avoid NLSP decay problem
\\
vi & Br$(\phi \to \text{gravitino}) \simeq 0$ & avoid gravitino overproduction
\\[1mm]
vii & e.g., $\tau_{3/2} \gg t_0$ & compatibility with gravitino dark matter
\\[1mm]
viii & ii and v--vii & for by-products; no new problems
\\ \hline
\end{tabular}
\end{table}
One potential candidate for the entropy producer exists if the strong CP
problem is solved by the Peccei-Quinn
mechanism.\cite{Peccei:1977hh,Peccei:1977ur} This mechanism involves
the axion supermultiplet containing two real scalars, the axion and
the saxion $\phi_\text{sax}$, as well as their superpartner, the axino
$\widetilde a$.
Their interactions with the MSSM particles are suppressed by the
Peccei-Quinn scale $f_a \gtrsim 6 \cdot 10^8 \GeV$, which makes them
long-lived.
In particular, the saxion is a suitable candidate to produce
entropy, since it has even $R$ parity and therefore can decay into SM
particles without producing superparticles.
If its dominant decay mode is into a pair of gluons, the decay
temperature is~\cite{Lyth:1993zw}
\begin{equation} \label{eq:tsaxdec}
T_\text{sax}^\text{dec} \simeq
53 \MeV \left(\frac{10^{12}\GeV}{f_a}\right)
\left(\frac{m_\text{sax}}{1\TeV}\right)^\frac{3}{2} .
\end{equation}
Thus, a decay shortly before BBN is possible.
If the saxion is produced in thermal equilibrium, its density starts to
dominate at
\begin{equation} \label{eq:tsaxeq}
T_\text{sax}^= \simeq 1.6 \GeV \left(\frac{m_\text{sax}}{1\TeV} \right) .
\end{equation}
Together with Eq.~\ref{eq:Delta}, this yields
\begin{equation}
\Delta \lesssim 55 \left( \frac{f_a}{10^{12}\GeV} \right)^\frac{2}{3} .
\end{equation}
This dilution factor is much smaller than the value $\Delta = 10^3$
considered previously and in fact inconsistent with gravitino dark
matter, since saxions enter thermal equilibrium only if
$\TR \gtrsim f_a$. Besides, the decays of axinos would produce a
disastrous amount of NLPSs for $f_a \gtrsim 10^{10} \GeV$.
We have to conclude that the thermally produced saxion is not suited to
produce a sufficient amount of entropy. While it satisfies all the
requirements of Tab.~\ref{tab:req} (if we choose $f_a \lesssim 10^{10}\GeV$
to fulfill viii), the resulting dilution factor is simply too small.
This can be traced back
to two conflicting requirements: on the one hand sufficient saxion
production requires sufficiently strong couplings (small $f_a$), while on the other
hand sufficiently late decay requires weak couplings (large $f_a$),
where later decay corresponds to more entropy production. In the
considered case, the allowed parameter ranges fail to overlap.
Using simple estimates we can generalize this negative conclusion to a
generic thermally produced particle.\cite{Hasenkamp:2010if}
Fortunately, we do not have to rely on thermal production of saxions.
It can be abundantly produced in coherent oscillations about its potential
minimum, if the saxion field is displaced from this minimum during
inflation. In this case, Eq.~\ref{eq:tsaxeq} changes to
\cite{Kawasaki:2007mk}
\begin{equation} \label{eq:rhoosc}
T_\text{sax}^= \simeq
6.4 \GeV \left(\frac{m_\text{sax}}{1 \TeV}\right)^{\frac{1}{2}}
\left(\frac{f_a}{10^{14} \GeV}\right)^2
\left(\frac{\phi_\text{sax}^\text{i}}{f_a}\right)^2 ,
\end{equation}
where $\phi_\text{sax}^\text{i}$ denotes the initial amplitude of the
oscillations. Now production and decay are decoupled, so we are able to
choose parameter values that yield a large dilution factor
saturating the upper bound of Eq.~\ref{eq:d3}.
For example, this is the case for $m_\text{sax} \sim 10\GeV$,
$m_{\widetilde a} \sim 1\TeV$, $f_a \sim 10^{10}\GeV$, and
$\phi_\text{sax}^\text{i} \sim 10^4 f_a$.
\section{Conclusions}
We have considered the early universe in a scenario where a relatively
heavy gravitino is the LSP and forms the dark matter, enabling a
reheating temperature large enough for thermal leptogenesis. In order
to prevent late NLSP decays from ruining the success of BBN, we have
required a dilution of the NLSP relic density by a factor
$\Delta \sim 10^3$. This dilution can be caused by the entropy from the
decay of a long-lived non-relativistic particle. A diluted neutralino
NLSP can be compatible with BBN, if it has a large higgsino or wino
component.
We have discussed the general requirements for the entropy-producing
particle. Afterwards, we have studied the saxion from the axion
supermultiplet as a specific example. We have found that the saxion
will not have the desired effects if it is produced only thermally.
However, non-thermal production in coherent oscillations overcomes this
problem and allows the saxion to produce a large amount of entropy.
Thus, we may conclude that we have arrived at a scenario with a
completely consistent cosmology. Thermal leptogenesis produces the
correct baryon asymmetry, the density of the gravitino dark matter is
compatible with the observed value, and BBN works as successfully as in
the Standard Model. In addition, the strong CP problem is solved by the
Peccei-Quinn mechanism.
\section*{Acknowledgments}
I'd like to thank Jasper Hasenkamp for the collaboration
on \cite{Hasenkamp:2010if}, on which this talk was based, as well as the
organizers of the Rencontres de Moriond for financial support.
This work was also supported by the German Science Foundation (DFG) via the
Junior Research Group ``SUSY Phenomenology'' within the Collaborative
Research Centre 676 ``Particles, Strings and the Early Universe''.
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