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\markboth{\quad \small \sl  L. Susskind \hfill}{\hfill\small
\sl Supersymmetry Breaking in the Anthropic Landscape \quad }
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\begin{document}

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\title{Supersymmetry Breaking\\[2mm] in the Anthropic
Landscape}

\author{L. Susskind}

\address{Department of Physics\\
Stanford University\\
Stanford, CA 94305-4060, USA}

\maketitle
%\bigskip\bigskip\bigskip\bigskip

%\centerline{\Large \bf {Supersymmetry Breaking in the Anthropic
%Landscape }}


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%\centerline{\it }
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%\centerline{\it L. Susskind  }
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%\centerline{Department of Physics} \centerline{Stanford
%University} \centerline{Stanford, CA 94305-4060}
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%\medskip

%\bigskip\bigskip
%\begin{abstract}
\medskip

\abstracts{%
In this paper I attempt to address a serious criticism of the
``Anthropic Landscape" and ``Discretuum" approach to cosmology,
leveled by Banks, Dine and Gorbatov. I argue that in this new and
unfamiliar setting, the gauge Hierarchy may not favor low energy
supersymmetry.
}%
%\end{abstract}

%\end{titlepage}
%\starttext \baselineskip=18pt \setcounter{footnote}{0}


\bigskip%\medskip
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{\sl 
I don't believe I ever met Ian Kogan but in many ways I felt he
was a friend. We shared both a struggle and a hobby. Ian's
passion was to understand the universe and like many others he
felt the key might be string theory \cite{afk}. The struggle to understand
was a defining feature of Kogan's work.

The Hobby? We both made an occasional foray into condensed matter
physics, again attempting to use lessons from string theory. His
last contribution was \cite{k}.
}%
\bigskip%\medskip

\setcounter{equation}{0}
\section{The Landscape }

Perhaps the struggle has entered a new phase. A new idea is
beginning to influence the way string theorists view the universe
and the laws that govern it. String theory it seems has a
Landscape with a
tremendous number of non-supersymmetric metastable vacuum states,
each with nonvanishing cosmological constant. The spectrum of
values of the cosmological constant will likely prove so dense
that it is called a discretuum. The number may be as high as
$10^{500}$. Given this, the question for string
theorists may not be--``What can be found in the universe?" but rather
"What can not be found?"

How does nature make use of this diversity of possibilities? Here
the ideas of Linde, Vilenkin and other cosmologists makes a
spectacular entrance--Eternal inflation fills the universe with what Alan Guth
calls pocket universes. Each pocket occupies a metastable valley
in the Landscape, with its own distinct laws of particle physics
and cosmological parameters. It is not surprising that the A-word,
the Anthropic principle is also finding a place in this new
framework.

If all of this is so, it is completely evident that the technical
notions of naturalness, which have played such a large role in
elementary particle physics have to be re-evaluated. For the
problem of  constant, this re-evaluation has already been done.
But it is entirely possible that other issues of naturalness will
also be affected. The obvious candidate is the gauge hierarchy.




\setcounter{equation}{0}
\section{The Banks Dine Gorbatov Argument }
Let's begin by reviewing a successful use of the anthropic
principle. In 1987  Steven Weinberg predicted that
if the anthropic principle was correct, the cosmological constant
should not be very much smaller than the bound provided by galaxy
formation \cite{sw}. The argument is straightforward: a much
smaller value would require fine tuning of the type which the
anthropic principle was supposed to eliminate. Since the anthropic
upper bound was only a couple of orders of magnitude above the
empirical upper bound, Weinberg argued that the anthropic
principle predicted that a \cc \ would soon be discovered. And it
was.

It is worth noting that there are also anthropic bounds on the
weak scale that might be strong enough to require a gauge
hierarchy like the one we actually observe. Increasing the Higgs
expectation value keeping everything else fixed would among other
things, increase the strength of gravity, making stars, galaxies
and the universe evolve through their life cycle much too fast.
Alternatively we could try to keep particle masses fixed by
decreasing Yukawa couplings. This would decrease the strength of
weak interactions and have many potentially disastrous effects on
the creation of heavy elements in stars. There would also be
effects on supernovaea, the primary agent for dispersing the
elements.

Banks, Dine and Gorbatov (BDG) \cite{bdg} have recently argued
that similar logic can be applied to proton stability and that the
exceptionally long life of the proton falsifies the \ap . The
context of the \bdg \ argument is the ``Landscape" or
``Discretuum" hypothesis \cite{bp, kklt, ls, ad}. They argue
(correctly I think) that the anthropic bound on the proton
lifetime is about $10^{17}$ years. Therefore  the vast majority of
{\it anthropically acceptable} landscape sites have proton
lifetimes many orders of magnitude shorter than the experimental
lower limit $~10^{32}$ years.

One way out of the \bdg \ argument is to note that in the
non-supersymmetric standard model, there is no mechanism for
proton decay. If the scale signaling the breakdown of the
standard model is high enough, there is no problem. Thus BDG
begin by first arguing that the vast majority of \aa \ vacua
have a very low supersymmetry breaking scale. Their argument goes
as follows:

Without supersymmetry both the \cc \ and the Higgs mass scale,
$\mu$ must be fine tuned. The combined fine tuning is about one
part in $10^{150}$. If the \sbs \ is called $M$ then the natural
scale for radiative corrections to the \cc \ is of order $M^4$. If
the actual \cc \ is $\lambda$ then the likelihood of radiative
corrections canceling and leaving the small value $\lambda$ is of
order $\lambda / M^4$. Thus making $M$ as small as possible will
yield the largest number of \aa \ vacua. Similarly they argue that
since the Higgs mass is quadratically sensitive to $M$ the actual
measure of fine tuning is
\be
P(M,\mu,\lambda) =\lambda \mu ^2/ M^6.
\label{fintun}
\ee
 For $M $ of order the Planck mass we get $10^{-150}$ but for $M$ of order the
weak scale, the fine tuning is only of order $10^{-60}$. There is
therefore a very strong statistical bias toward low energy \sb .

But low energy supersymmetry breaking is not without its problems.
Many of the things which the standard model solved so
neatly--proton stability, neutral strangeness changing currents
and the like-- are neither automatic nor especially natural in
supersymmetric extensions.  In particular, dimension 5 baryon
violating operators are dangerous. Without some special
non-generic symmetry, the proton lifetime in supersymmetric
theories is several orders of magnitude too small. I might point
out that dimension 4 operators can also occur in generic
supersymmetry models. These lead to extremely short lifetimes
unless R-parity forbids them. But unlike the dimension 5
operators, the dimension 4 operators are so bad that they can be
ruled out anthropically.

The \bdg \ criticism is quite serious in my opinion, and needs to be
addressed. In this note I will argue that \bdg \ overlooked one
important statistical factor in their analysis which, when
included, may change the conclusion. Here is the way I think the
argument should go:

What we want to compute is the conditional probability that the supersymmetry
breaking scale is $M$, given that the \cc \ and weak scale are in
the anthropic range. This is not the function $P$ in \ref{fintun}.
Rather the function $P$ is the conditional probability that the \cc \ and weak
scale are in the allowed range, given that the \sbs \ is $M$. The
two distributions are related by Baye's theorem. Define the
probability that the \cc \ and weak
scale are in the allowed range, given that the \sbs \ is $M$ to
be
$$
P(\lambda,\mu|M)
$$
and the probability that the supersymmetry
breaking scale is $M$, given that the \cc \ and weak scale are in
the anthropic range to be
$$
P(M|\lambda,\mu)\,.
$$
Also define the unconditional probabilities for given \sbs \ and
for the values $\lambda,\mu$ to be
$$
P(M), \ \ P(\lambda,\mu)\,.
$$
Then Baye's theorem relates these probabilities
\be
P(M|\lambda,\mu)= P(\lambda,\mu|M) P(M)/P(\lambda,\mu)\,.
\label{bayes}
\ee

Thus in comparing the likelihood of different values of $M$ the
factor $1/M^6$ should be multiplied by $P(M)$. The factor in the
denominator of \ref{bayes} can be ignored since it is independent
of $M$. The question is, what is the value of the unconditional
probability $P(M)$?

I believe it is likely that $P(M)$ tends to zero as a power of $M$
as $M$ tends to zero. The reason is that the conditions for
supersymmetry generally pick out a subspace of the landscape whose
dimensionality may be a good deal lower than the number of
available dimensions including those that parameterize
supersymmetry breaking. In this case $P(M)$ could overwhelm the
factor $1/M^6$. To illustrate the point I will concoct an
illustrative example within a
 framework similar to that of \cite{kklt}. The \sb
\ mechanism and the discrete fine tunings of the \cc \ and Higgs
scale take place in different sectors of the theory. In particular
the \sb \ sector is located at the infrared end of a warped
compactification while the tunings of the \cc \ and $\mu$ are done
through the choice of fluxes in the ultraviolet part of the
compactification manifold. The only modification of \cite{kklt} is to suppose
that there are $n$ throats instead of just one.

In \cite{kklt} the mechanism for supersymmetry breaking is one or
more antibranes placed at the end of the throat. The \sbs \ is the
combined mass of the branes. Thus
\be
M = N M_D\,,
\ee
where $M_D$ is the antibrane mass and N is the number of
antibranes.

If there are $n$ throats a given total number of antibranes can be
distributed among the throats in many ways. For example $n_1$
antibranes can be placed in the first throat, $n_2$ in the second
throat and so on. The number of ways of partitioning the mass $M$
among $n$ throats is of order $M^{n-1}$. Thus if the number of
throats exceeds six, the distribution favors high energy \sb .

Which kind of throat structure--many or one-- is favored by
statistics of the landscape? This may be a delicate question. With
no condition of supersymmetry breaking, one throat may dominate.
But the statistics may be tilted toward many throats just because
there are more ways to break supersymmetry if there are many
throats.



If nothing else, I hope this note shows that the measure on the
landscape is a very subtle issue that demands better quantitative
methods of the kind being developed by Douglas and collaborators
\cite{ad}. The whole idea of fine-tuning has to be re-evaluated
and redefined in this new unfamiliar framework.

\vspace{-4mm}

\section*{Acknowledgments}
I would like to thank Tom Banks, Michael Dine,  Ben Frievogel,
Savas Dimopoulos Steve Shenker, Scott Thomas, Shamit Kachru and
Michael Douglas for very useful discussions.

\begin{thebibliography}{999}

\bibitem{afk}S.~Abel, K.~Freese and I.~Kogan,
{\em Hagedorn inflation: Open strings on branes can drive inflation},
hep-th/0303046.

\bibitem{k} A. Gorsky, I. Kogan and C. Korthels-Altes,
%    Dualities in Quantum Hall System and Noncommutative Chern-Simons Theory, 
 JHEP {\bf 0201}, 002 (2002)%\\{}
[hep-th/0111013].



\bibitem{bdg} T.~Banks, M.~Dine and  E.~Gorbatov, %Tom Banks, Michael Dine, Elie Gorbatov,
{\em Is There A String Theory Landscape},\\{} hep-th/0309170.

\bibitem{sw} S.~Weinberg, %Steven Weinberg, 
% ANTHROPIC BOUND ON THE COSMOLOGICAL CONSTANT,
Phys. Rev. Lett. {\bf 59}, 2607 (1987).


\bibitem{bp} R.~Bousso and  J.~Polchinski,% Raphael Bousso, Joseph Polchinski,
%Quantization of Four-form Fluxes and Dynamical Neutralization of
%the Cosmological Constant, 
JHEP {\bf 0006}, 006 (2000) [hep-th/0004134]. 

\bibitem{kklt} 
S.~Kachru, R.~Kallosh, A.~Linde and S.~P.~Trivedi,
%``De Sitter vacua in string theory,''
Phys.\ Rev.\ D {\bf 68}, 046005 (2003)
[hep-th/0301240].
%%CITATION = HEP-TH 0301240;%%

\bibitem{ls} L.~Susskind, % Leonard Susskind, 
{\em The Anthropic Landscape of String 
Theory}, hep-th/0302219.


\bibitem{ad} M.~R.~Douglas,% Michael R. Douglas, 
%The statistics of string/M vacua, hep-th/0303194,  
JHEP {\bf 0305}, 046 (2003) [hep-th/0303194];\\
S.~Ashok and M.~R.~Douglas,
%``Counting flux vacua,''
JHEP {\bf 0401}, 060 (2004)
[hep-th/0307049];\\
%%CITATION = HEP-TH 0307049;%%
%Sujay Ashok, Michael R. Douglas, Counting Flux Vacua,
%hep-th/0307049
M.~R.~Douglas, B.~Shiffman and S.~Zelditch,
% Michael R. Douglas, Bernard Shiffman, Steve Zelditch,
{\em  Critical points and supersymmetric vacua}, math.CV/0402326


\end{thebibliography}

\end{document}
