\section{Introduction}

A new era has begun in coding theory with the discovery of graphical
codes -- low-density parity check codes (LDPC)
\cite{63Gal,95WLK,99Mac,03RU} and turbo codes\cite{93BGT}. These
codes are special, not only because they can virtually achieve the
error-free Shannon limit, but mainly because a family of
computationally efficient approximate decoding schemes is readily
available. This family includes iterative Belief Propagation (BP),
or simply message-passing, decoding \cite{63Gal,95WLK,99Mac} and
Linear Programming (LP) decoding \cite{05FWK}.

When operating at moderate noise values these decoding algorithms
show performance comparable to the ideal, but computationally not
feasible, Maximum Likelihood (ML) and Maximum-a-Posteriori
decodings. However sub-optimality of the approximate decodings
becomes a handicap at large SNR, in the so-called error-floor regime
\cite{03Ric}. An error-floor typically emerges due to the low-weight
fractional pseudo-codewords \cite{01FKKR,03KV,03Ric,05VK,Pascal}, or
if one uses a noise space description, due to the instantons
\cite{05SCCV} -- the most probable erroneous configurations of the
noise. Much effort has been invested in recent years on
understanding the pseudo-codewords/instantons and thus  the error
floor behavior of the graphical codes. Also, there were a few
attempts at decoding improvement. The Facet-Guessing (FG) algorithm
of \cite{06DW} and the Loop Erasure algorithm of \cite{06CCc}
constitute two recent advances in this direction.

Let us review these {\bf relevant prior results:}

\noindent $\bullet$ LP decoding can be considered as an asymptotic,
large SNR, version of the BP decoding \cite{03WJ,04VK} where the
latter is understood as the absolute minimum of the Bethe free
energy \cite{05YFW}. The LP decoding minimizes the linear functional
of beliefs, called the energy functional, under a set of non-strict
inequality and equality constraints \cite{05FWK}. (One can also
consider a small polytope formulation of the problem, where all
constraints are non-strict inequalities.)

\noindent $\bullet$ The error-floor of an LDPC code is typically due
to dangerous low-weight pseudo-codewords which are fractional, i.e.
which are not codewords \cite{03Ric}.

\noindent $\bullet$ The dangerous pseudo-codewords are rare and the
pseudo-codeword search algorithm \cite{06CS} is an efficient
heuristic for finding the troublemakers. This algorithm is based on
LP decoding.

\noindent $\bullet$ The Loop Calculus of \cite{06CCa,06CCb}
introduces the Loop Series, which is an explicit finite expression
for the MAP decoding partition function in terms of loop
contributions defined on the graphical representation of the
respective inference/decoding problem. Each loop contribution is
calculated explicitly from a solution of the BP equations and it is
represented as a product of terms along the loop. The LP limit of
the Loop Calculus is well defined.

\noindent $\bullet$ One experimentally verifies, and otherwise
conjectures, that all the dangerous pseudo-codewords are explained
in terms of a small number of critical loops \cite{06CCc}.
Typically, the respective partition function can be well
approximated in terms of the bare LP/BP term and one critical term
associated with a single connected critical loop. The bare term and
the critical terms are comparable while all other terms in the Loop
Series are much smaller. An efficient heuristic algorithm for
finding the critical loop has been constructed. It is based on
representing the loop contribution as a product of terms along the
loop, each smaller than or equal to unity in absolute value, and
pre-selecting elements of the critical loop to be larger than a
threshold close to unity.

\noindent $\bullet$ BP, corrected by accounting for the critical
loop, and its simplified LP version, coined the LP-erasure
\cite{06CCc}, are algorithms improving BP/LP. These algorithms,
applied when LP/BP fails, consist of modifying BP/LP along the
critical loop. Thus LP-erasure modifies log-likelihoods everywhere
along the critical loop (lowering log-likelihoods in absolute
value). It was shown experimentally (on the example of the test
$[155,64,20]$ code introduced in \cite{01TSF}) that the LP-erasure
is capable of correcting all the dangerous fractal pseudo-codewords,
previously found with the pseudo-codeword search algorithm
\cite{06CS,07CSa}.

\noindent $\bullet$ The Facet Guessing (FG) algorithm \cite{06DW} is
a graph local improvement of the LP algorithm. It applies if LP
decoding does not succeed. A non-active facet/inequality, i.e. the
one with the vertex/solution lying in the strict inequality domain,
is selected. The original LP problem is modified so that the
selected facet is enforced to be in its active (equality) state. The
number of non-active facets for a dangerous fractional
pseudo-codeword is typically small. (It is provably small for the
expander codes.) One constructs a full set of the single-facet
modified LP problems (where the number of problems is thus equal to
the number of active facets) or otherwise selects a random subset of
the LP problems. Running LP decoding for the set of the modified
problems, one chooses solution with the lowest energy functional and
calls it the outcome of the facet guessing algorithm. It was shown
experimentally in \cite{06DW} that the facet guessing algorithm
improves the bare LP decoding.

{\bf Results reported in this paper:}

\noindent $\bullet$ We introduce the Bit Guessing (BG) algorithm,
which is a simplified version of the Facet Guessing algorithm of
\cite{06DW}, enforcing single bits to be in $0$ or $1$ state. We
apply the algorithm to the set of pseudo-codewords found for the
bare LP decoding of the test $[155,64,20]$ code. It was found that
all the fractional dangerous pseudo-codewords are corrected by the
BG algorithm.

 \noindent $\bullet$ For each of the
dangerous pseudo-codewords, we identify the set of bits where local
BG leads to correct decoding and compare this set with the set of
bits forming a critical loop found via the critical loop search
algorithm of \cite{06CCc}.  The comparison shows very strong
correlations between the two sets: fixing a bit from the critical
loop almost always leads to correct decoding. This suggests that,
since the critical loop is relatively small, it is advantageous to
use the loop series and the critical loop analysis to pre-select the
set of single-bit corrected LP schemes in the BG algorithm. (One
will only need to consider fixing bits along the critical loop.)
Moreover, we consider the Loop Guided Guessing (LGG) algorithm built
on top of the bare LP with only one or two modified LP runs. The
modified LP scheme is constructed by adding to the bare LP scheme an
equality fixing the value of a randomly selected bit from the
critical loop.

 \noindent $\bullet$ We test the
LGG algorithm on the set of the LP-erroneous configurations of the
$[155,64,20]$ code for the Additive White Gaussian Noise (AWGN)
channel found in the finite Signal-to-Noise Ratios (SNR) Monte Carlo
simulations of \cite{07CSa}. We show that the LGG algorithm greatly
improves the bare LP algorithm and it also performs significantly
better than the LP-erasure algorithm of \cite{06CCc}.

\section{Error-Floor analysis of BP/LP decoding: Brief Review of Prior Results}

\subsection{Belief Propagation and Linear Programming}

We consider a generic linear code, described by its parity check
$N\times M$ sparse matrix, $\hat{H}$, representing $N$ bits and $M$
checks. The codewords are these configurations,
${\bm\sigma}=\{\sigma_i=0,1| i=1,\dots,N\}$, which satisfy all the
check constraints: $\forall \alpha=1,\dots,M$, $\sum_i H_{\alpha
i}\sigma_i=0$ (mod~$2$). A codeword sent to the channel is polluted
and the task of decoding becomes to restore the most probable
pre-image of the output sequence,  ${\bm x}=\{x_i\}$. The
probability for ${\bm\sigma}$ to be a pre-image of ${\bm x}$ is
\begin{eqnarray}
{\cal P}({\bm\sigma}|{\bm x}) &=& p({\bm\sigma}|{\bm x})Z^{-1},\quad
Z=\sum_{\bm\sigma}p({\bm\sigma}|{\bm x}),\label{Psx1}\\
p({\bm\sigma}|{\bm x}) &=& \prod_\alpha
\delta\biggl(\prod_{i\in\alpha}(-1)^{\sigma_i},1\biggr)
\exp\biggl(-\sum_ih_i\sigma_i\biggr), \label{Psx}
\end{eqnarray}
where one writes $i\in\alpha$ if $H_{\alpha i}=1$; $Z$ is the
normalization coefficient (so-called partition function); the
Kronecker symbol, $\delta(x,y)$, is unity if $x=y$ and it is zero
otherwise; and ${\bm h}$ is the vector of log-likelihoods dependent
on the output vector ${\bm x}$. In the case of the AWGN channel with
the {\rm SNR} ratio, $E_c/N_0=2s^2$, the bit transition probability
is $\sim \exp(-2s^2(x_i-\sigma_i)^2)$, and the log-likelihood
becomes $h_i=s^2(1-2x_i)$. The optimal block-MAP (Maximum
Likelihood) decoding maximizes ${\cal P}({\bm\sigma}|{\bm x})$ over
${\bm\sigma}$, $ \arg\max_{\bm\sigma}{\cal P}({\bm\sigma}|{\bm x})$
and symbol-MAP operates similarly, however in terms of the marginal
probability at a bit
$\arg\max_{\sigma_i}\sum_{{\bm\sigma}\setminus\sigma_i}{\cal
P}({\bm\sigma}|{\bm x})$. One can also think formally about ML in
terms of MAP,  i.e. in terms of summation over all possible
configurations of ${\bm\sigma}$ however with the weight and the
partition function in Eqs.~(\ref{Psx1},\ref{Psx}) transformed
according to
\begin{equation}
\ln( p({\bm\sigma}|{\bm x}))\to \rho\ln( p({\bm\sigma}|{\bm
x})),\quad \rho\to+\infty. \label{rho}
\end{equation}

BP and LP decodings should be considered as computationally
efficient but suboptimal substitutions for MAP and ML. Both BP and
LP decodings can be conveniently derived from the so-called
Bethe-Free energy approach of \cite{05YFW} which is briefly reviewed
below. In this approach trial probability distributions, called
beliefs, are introduced both for bits and checks, $b_i$ and
$b_\alpha$, respectively. The set of bit-beliefs, $b_i(\sigma_i)$,
satisfy equality and inequality constraints that allow convenient
reformulation in terms of a bigger set of beliefs defined on checks,
$b_\alpha({\bm\sigma}_\alpha)$, where, ${\bm
\sigma}_\alpha=\{\sigma_i|i\in\alpha,\sum_i H_{\alpha
i}\sigma_i=0\mbox{ (mod~$2$)}\}$, is a local codeword associated
with the check $\alpha$. The equality constraints are of two types,
normalization constraints (beliefs, as probabilities, should sum to
one) and compatibility constraints:
\begin{equation}
\forall i,\ \forall\alpha\ni i:\
b_i(\sigma_i)=\sum\limits_{\sigma_\alpha\setminus\sigma_i}b_\alpha({\bm
\sigma}_\alpha),\
\sum\limits_{{\bm\sigma}_\alpha}b_\alpha({\bm\sigma}_\alpha)=1.\label{comp}
\end{equation}
Additionally, all the beliefs should be non-negative and smaller
than or equal to unity. The Bethe Free energy is defined as the
difference of the self-energy and the entropy, $F=E-S$:
\begin{eqnarray}
 && \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
 E\!=\sum_ih_i\sum_{\sigma_i}\!\sigma_ib_i(\sigma_i)\quad\mbox{and}\label{E_Bethe}\\
 && \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
  S\!\!=\!-\!\!\sum\limits_\alpha\! \sum_{{\bm \sigma}_\alpha}\!
  b_\alpha\!({\bm\sigma}_\alpha\!)
 \!\ln b_\alpha\!({\bm\sigma}_\alpha\!)\!
 +\!\!\sum\limits_i\!\sum\limits_{\sigma_i}
 (q_i\!-\!1)b_i(\sigma_i)\!\ln\! b_i(\sigma_i).
 \label{S_Bethe}
\end{eqnarray}
Optimal configurations of beliefs minimize the Bethe Free energy
subject to the equality constraints (\ref{comp}). Introducing the
constraints as the Lagrange multiplier terms to the effective
Lagrangian and looking for the extremum with respect to all possible
beliefs leads to
\begin{eqnarray}
 &&\!\!\!\!\!\!\!\!\!\! b_\alpha({\bm
 \sigma}_\alpha)=\frac{\exp\left(\sum_{i\in\alpha}(h_i/q_i+\eta_{\alpha
 i})(1-2\sigma_i)\right)}{\sum_{{\bm\sigma}_\alpha}\exp\left(\sum_{i\in\alpha}(h_i/q_i+\eta_{\alpha
 i})(1-2\sigma_i)\right)},\label{ba}\\
 &&\!\!\!\!\!\!\!\!\!\! b_i(\sigma_i)=\frac{\exp\left((\eta_{\alpha
 i}+\eta_{i\alpha})(1-2\sigma_i)\right)}{2\cosh\left(\eta_{i\alpha}+\eta_{\alpha
 i}\right)},\label{bi}
\end{eqnarray}
where the set of $\eta$ fields (which are Lagrange multipliers for
the compatibility constraints) satisfy
\begin{eqnarray}
 \eta_{\alpha i}=h_i+\sum_{\beta\ni i}^{\beta\neq \alpha}\eta_{i\alpha},\ \
 \eta_{i\alpha}=\tanh^{-1}\left(\prod_{j\in\alpha}^{j\neq
 i}\tanh\eta_{\alpha j}\right).
 \label{eta}
\end{eqnarray}
These are the BP equations for LDPC codes written in its standard
form. These equations are often described in the coding theory
literature as stationary point equations for the BP (also called the
sum product) algorithm and then $\eta$ variables are called
messages. The BP algorithm, initialized with $\eta_{i\alpha}=0$,
solves Eqs.~(\ref{eta}) iterating it sequentially from right to
left. Possible lack of the iterative algorithm convergence (to the
respective solution of the BP equation) is a particular concern, and
some relaxation methods were discussed to deal with this problem
\cite{06SCb}.

LP is a close relative of BP which does not have this unpleasant
problem with convergence. Originally, LP decoding was introduced as
a relaxation of ML decoding \cite{05FWK}. It can thus be restated as
$ \arg\min_{{\bm\sigma}\in {\cal
P}}\biggl(\sum_ih_i\sigma_i\biggr)$, where ${\cal P}$ is the
polytope spanned by all the codewords of the code. Looking for
${\bm\sigma}$ in terms of a linear combination of the codewords,
${\bm \sigma}_v$: ${\bm\sigma}=\sum_v\lambda_v{\bm \sigma}_v$, where
$\lambda_v\geq 0$ and $\sum_v\lambda_v=1$, one observes that the
block-MAP turns into a linear optimization problem. The LP-decoding
algorithm of \cite{05FWK} proposes to relax the polytope, expressing
${\bm\sigma}$ in terms of a linear combination of local codewords
associated with checks, ${\bm\sigma}_\alpha$. We will not give
details of this original formulation of LP here because we prefer an
equivalent formulation, elucidating the connection to BP decoding.
One finds that the BP decoding, understood as an algorithm searching
for a stationary point of the BP equations, turns into LP decoding
in the asymptotic limit of large {\rm SNR}. Indeed in this special
limit, the entropy terms in the Bethe free energy can be neglected
and the problem turns into minimization of a linear functional with
a set of linear constraints. The relation between BP and LP was
noticed  in \cite{03WJ,03KV} and it was also discussed in
\cite{04VK,06CCc}. Stated in terms of beliefs, LP decoding minimizes
the self-energy part (\ref{E_Bethe}) of the full Bethe Free energy
functional under the set of linear equality constraints (\ref{comp})
and also linear inequalities guaranteeing that all the beliefs are
non-negative and smaller than or equal to unity. This gives us a
full definition of the so-called large polytope LP decoding. One can
run it as is in terms of bit- and check- beliefs, however it may
also be useful to re-formulate the LP procedure solely in terms of
the bit beliefs.

\subsection{Error-floor. Instantons \& Pseudo-Codewords. Pseudo-Codeword Search Algorithm.}

The goal of decoding is to infer the original message from the
received output, ${\bm x}$. Assuming that coding and decoding are
fixed and aiming to characterize the performance of the scheme, one
studies the Frame-Error-Rate (FER)
 $\mbox{ FER} = \int d{\bm x} \,\, \chi_{\mbox{\scriptsize error}}({\bm
   x}) P({\bm x}|{\bm 0})$,

where $\chi_{\mbox{\scriptsize error}} = 1$ if an error is detected
and $\chi_{\mbox{\scriptsize error}} = 0$ otherwise. In a symmetric
channel, FER is invariant with respect to the original codeword,
thus the all-$0$ codeword can be assumed for the input. When {\rm
SNR} is large, FER, as an integral over output configurations, is
approximated by \cite{05SCCV}, $ \mbox{FER}\sim
\sum\limits_{\mbox{\scriptsize inst}} V_{\mbox{\scriptsize inst}}
\times P\left({\bm x}_{\mbox{\scriptsize inst}}|{\bm 0}\right)$,
where ${\bm x}_{\mbox{\scriptsize inst}}$ are the special instanton
configurations of the output maximizing $P({\bm x}|{\bm 0})$ under
the $\chi_{\mbox{\scriptsize error}} = 1$ condition, and
$V_{\mbox{\scriptsize inst}}$ combines combinatorial and
phase-volume factors. Generally, there are many instantons that are
all local maxima of $P({\bm x}|{\bm 0})$ in the noise space. For the
AWGN channel, the instanton estimate for FER at the high {\rm SNR},
$s \gg 1$, is $\sim \exp(-d_{\mbox{\scriptsize inst}} s^2/2)$. In
the instanton-amoeba numerical scheme, suggested in \cite{05SCCV},
instantons with the small effective distances, $d_{\mbox{\scriptsize
inst}}$, were found by a downhill simplex method also called
``amoeba'', with accurately tailored (for better convergence)
annealing. Instantons are closely related to the so-called
pseudo-codewords \cite{95WLK,01FKKR,03Ric,03KV}: decoding applied to
the instanton configuration results in the respective
pseudo-codeword. The effective distance, $d_{\mbox{\scriptsize
inst}}$, characterizing an instanton and its respective
pseudo-codeword, should be compared with the Hamming distance of the
code, $d_{\mbox{\scriptsize ML}}$. Instanton/pseudo-codewords with
$d<d_{\mbox{\scriptsize ML}}$ will completely screen contribution of
the respective codeword to the FER at $s\to\infty$.

In the case of LP decoding, one can actually develop a discrete
computational scheme, coined the Pseudo-Codeword-Search (PCS)
algorithm \cite{06CS}, which allows very efficient calculation of
the low weight pseudo-codewords and the respective instantons. It
was shown in \cite{06CS} that the PCS algorithm converges in a
relatively small number of iterations. The PCS algorithm, repeated
many times picking the initial noise configuration randomly,
generates a set of low-weight pseudo-codewords. Thus,  for the model
$[155,64,20]$ code studied in \cite{06CS,07CSa} some $\sim 200$
pseudo-codewords with the effective weight lower than the Hamming
distance of the code were found.

\subsection{Loop Calculus. Critical Loops. Loop Erasure Algorithm.}

Loop calculus is a technique which allows one to express explicitly
the partition function of the statistical inference problem
associated with Eq.~(\ref{Psx}) in terms of the so-called loop
series \cite{06CCa,06CCb}:
\begin{eqnarray}
 Z&=& Z_0\left(1+\sum\limits_{{\it
C}}r({\it C}) \right),\ \ r({\it C})=\prod\limits_{i,\alpha\in {\it
C}}\mu_\alpha \mu_{i}, \label{z_loop}\\
\mu_{i} &=& \frac{(1-m_i)^{q_i-1}\!+\!(-1)^{q_i}(1+m_i)^{q_i-1}}
 {2(1-m_i^2)^{q_i-1}},\ \ q_i\!=\!\sum\limits_{\alpha\in{\it C}}^{\alpha\ni i}\!1,
\nonumber\\
\mu_{\alpha}\!\!\!\!&=&\!\!\!\!\sum\limits_{\sigma_\alpha}\!b_\alpha(\sigma_\alpha)
 \!\!\prod\limits_{i\in{\it C}}^{i\in\alpha}\!(1\!-\!2\sigma_i\!-\!m_i),\;
m_i\!=\!\sum\limits_{\sigma_i}\! b_i(\sigma_i)(1\!-\!2\sigma_i),
\nonumber
\end{eqnarray}
where $b_\alpha({\bm\sigma}_\alpha)$ and $b_i(\sigma_i)$ are the
beliefs defined on checks and bits according to
Eqs.~(\ref{ba},\ref{bi}) and $Z_0=-\ln F=-\ln(E-S)$ with self-energy
and entropy expressed in terms of the beliefs according to
Eqs.~(\ref{E_Bethe},\ref{S_Bethe}).

The  loop series holds for the BP/MAP relation and it is also well
defined for the LP/ML relation, where transition from the former one
to the later one is according to Eq.~(\ref{rho}). Notice that in the
LP/ML version of the Loop Series $\mu_i$ can be singular, i.e.
$\to\pm\infty$ when $\rho\to\infty$ and $m_i\to\pm 1$, however the
resulting $r({\it C}$ is always finite,  because in this case the
corresponding $\mu_\alpha$ contribution approaches zero. Moreover,
the construction of the Loop Series is such that any individual
$r({\it C})$ is always smaller than unity in absolute value,  both
in the BP and LP cases.

If BP/LP performs well one expects that the loop corrections,
$r({\it C})$, are all significantly smaller than the bare unity.
Failure of the BP/LP decoding signals the importance of some loop
corrections. Even though the number of loops grows exponentially
with the size of the code, not all loops gives comparable
contributions to the loop series. Thus, an important conjecture of
\cite{06CCc} was that for the case of the low-weight
pseudo-codewords (i.e. for the values of log-likelihoods
corresponding to the instanton configuration decoded into the
pseudo-codewords) there exists a relatively simple loop contribution
(or a very few simple contributions), dominating corrections to bare
unity in the loop series. This conjecture was verified in
\cite{06CCc} for the example of the ($\sim 200$) instantons found
for the LP decoding of the $[155,64,20]$ code performing over the
Additive-White-Gaussian-Noise (AWGN) channel. It was demonstrated in
\cite{06CCc} that for each of the instantons, one can indeed
identify the corresponding critical loop, $\Gamma$, giving an
essential contribution to the loop series (\ref{z_loop}) comparable
to the bare LP contribution.

The search for the critical loop suggested in \cite{06CCc} was
heuristic. One searches for a single-connected contribution
associated with a critical loop consisting of checks and bits with
each check connected to only two bits of the loop. According to
Eqs.~(\ref{z_loop}) this contribution to the loop series is the
product of all the triads, $\tilde{\mu}^{(bp)}$, along the loop,
\begin{eqnarray}
 r(\Gamma)\!=\!\prod_{\alpha\in\Gamma}\tilde{\mu}_\alpha,\quad
 \tilde{\mu}_{\alpha}\!=\!\frac{\mu_{\alpha}}
 {\sqrt{(1\!-\! m_i^2)(1\!-\! m_j^2}},
\label{tildemu}
\end{eqnarray}
where for any check $\alpha$ that belongs to $\Gamma$, $i,j$ is the
only pair of $\alpha$ bit neighbors that also belongs to $\Gamma$.
By construction, $|\tilde{\mu}_{\alpha;ij}|\leq 1$. We immediately
find that for the critical loop contribution to be exactly equal to
unity (where unity corresponds to the bare BP term), the critical
loop should consist of triads with all $\tilde{\mu}$ equal to unity
in absolute value. Even if degeneracy is not exact one still
anticipates the contributions from all the triads along the critical
loop to be reasonably large, as an emergence of a single triad with
small $\tilde{\mu}$ will make the entire product negligible in
comparison with the bare BP term. This consideration suggests that
an efficient way to find a single connected critical loop, $\Gamma$,
with large  $|r(\Gamma)|$ consists of, first, ignoring all the
triads with $|\tilde{\mu}|$ below a certain $O(1)$ threshold, say
$0.999$, and, second, checking if one can construct a single
connected loop out of the remaining triads. If no critical loop is
found, we lower the threshold until a leading critical loop emerges.

Applied to the set of instantons of the Tanner $[155,64,20]$ code
with the lowest effective distances this, triad-based search scheme
generates an $r(\Gamma)$ that is exactly unity in absolute value.
This is the special degenerate case in which the critical loop
contribution and the BP/LP contribution are equal to each other in
absolute value. Thus, only the sixth of the first dozen of
instantons has $r(\Gamma)\approx 0.82$ while all others yield
$r(\Gamma)=1$. To extend the triad-based search scheme to the
instantons with larger effective distance, one needs to decrease the
threshold. For the dangerous pseudo-codewords of the $[155,64,20]$
code this always resulted in the emergence of at least one single
connected loop with $r(\Gamma)\sim 1$.

Accounting for a single loop effect (when it is comparable to a bare
(BP) contribution) can be improved through the effective free energy
approach explained in \cite{06CCc}. This approach resulted in the
formulation of renormalized BP equations and the respective
Loop-corrected BP algorithm aimed at solving the renormalized
equations. Modification of the bare BP equations are  well localized
along the critical loop. This observation led to the suggestion an
LP counterpart of the loop-corrected BP, coined the  \underline{\bf
LP-erasure algorithm}:

\noindent $\bullet$ {\bf 1.} Run the LP algorithm. Terminate if LP
succeeds (i.e. a valid code word is found).

\noindent $\bullet$ {\bf 2.} If LP fails,  find the most relevant
loop $\Gamma$ that corresponds to the maximal amplitude $r(\Gamma)$.

\noindent $\bullet$ {\bf 3.} Modify the log-likelihoods
(factor-functions) along the loop $\Gamma$ introducing a shift
towards zero,  i.e. introduce a complete or partial erasure of the
log-likelihoods at the bits. Run LP with modified log-likelihoods.
Terminate if the modified LP succeeds.

\noindent $\bullet$ {\bf 4.} Return to {\bf Step 2} with an improved
selection principle for the critical loop.

This LP-erasure algorithm was tested in \cite{06CCc} on the
$[155,64,20]$ example. The results of the test are remarkable: all
$\sim 200$ low-weight instantons were actually corrected already
with the roughest version of the LP-erasure algorithm, corresponding
to the full erasure of the information (log-likelihoods) along the
critical loop.

\subsection{Facet Guessing Algorithm}

The Facet Guessing (FG) is an improvement of the LP decoder
suggested in \cite{06DW}. This algorithm applies when the bare LP
fails. Failure of LP means that some of the non-strict inequality
constraints in the LP formulation remain inactive for the LP
solution, i.e. the respective strict equalities are not satisfied.
Considering expander codes and the small-polytope version of the LP
decoding, the authors of \cite{06DW} proved that the set of active
constraints of any fractional pseudo-codeword is smaller by a
constant factor than the number of active constraints of any
codeword. This fact was exploited in \cite{06DW} to devise a
decoding algorithm that provably outperforms the LP decoder for
finite blocklengths. The FG algorithm proceeds by guessing the
facets of the polytope, i.e. enforcing the respective inactive
facets to be active with a new equality constraint, and resolving
the linear program on these facets. In its full version, the
algorithm thus consists of the set of modified LP algorithms. The
number of the modified schemes is equal to the number of inactive
facets in the fractional pseudo-codeword solution of the bare LP
algorithm. The configurational output of the FG algorithm is the
output of one modified LP from the set giving the lowest value of
the self-energy (optimization functional). The randomized version of
the FG algorithm consists of picking some fixed fraction of the
modified LP schemes at random from the full set, and then finding
the configuration minimizing the result on the subset. \cite{06DW}
also discussed experimental test of the theory done for couple of
codes, of which one is the $[155,64,20]$ code also considered in
this paper. It was experimentally demonstrated that the randomized
version of the FG algorithm improves the bare LP decoding.

\section{Breaking the critical loop}

We introduce a Bit Guessing (BG) procedure, which is a simplified
version of the FG algorithm.   The simplification comes with a
restriction imposed on the facet-activation (fixing) procedure.  In
BG, one only allows activation (fixing) of the inequalities
associated with bit beliefs, $b_i(\sigma_i)$, and not check beliefs,
$b_\alpha({\bm\sigma}_\alpha)$. Considering values of the
log-likelihoods resulting in a fractional LP pseudo-codeword, one
creates  a set of single bit corrected LP schemes, each different
from the bare LP schemes by only one extra equality condition,
enforcing the value of a bit to be $1$ or $0$. (If the value of the
marginal probability in the bare pseudo-codeword of LP is
fractional, we include two bit-modified LPs in the set,
corresponding to enforcing $0$ and $1$ values for the bit
respectively.  If the marginal probability of a bit is integer we
only includes one bit-modified LP in the set corresponding to fixing
the value of the bit to the integer opposite to the one observed in
the bare pseudo-codeword for the bit.) We run consequently all the
modified LP schemes,  forming the BG set, and choose the result with
the lowest energy functional as the outcome of the bit guessing
procedure.

The FG algorithm was tested on the set of dangerous fractional
pseudo-codewords described in \cite{06CS}. We found that all of the
pseudo-codewords were successfully corrected by FG! In other words,
the output of a corrected LP-scheme with the minimum self-energy is
the right codeword (the all-zero one in the simulations). One also
finds that there always exists a number of successful LP-corrected
schemes, each associated with a different pinned bit. This number
was actually relatively large for the fractional pseudo-codewords
with the lowest weight, $\sim 30-50$, but smaller values  were also
observed for some of the fractional pseudo-codewords with effective
distance from the dangerous range, $[16.4;20]$.

Next for any of the LP-dangerous,  but FG-correctable,
pseudo-codewords, one creates the list of ``successful" bits and
compares this list with the list of bits forming the respective
critical loop found in \cite{06CCc} with the thresholding of the
$\tilde{\mu}$ values. One finds that the set of bits forming the
critical loop forms a relatively small sub-set of the ``successful"
set. In other words,  fixing any bit of the critical loop describing
a dangerous fractional pseudo-codewords leads to correct decoding.

One draws a couple of useful conclusions from this simple
experiment. 1) One finds that the FG algorithm offers a very
successful strategy for decoding in accordance with the main claim
of \cite{06DW}. It corrects all the dangerous pseudo-codewords of
the model $[155,64,20]$ code. 2) The FG correction can be made with
the help of the critical loop procedure of \cite{06CCs}. Finding the
critical loop  helps to reduce the complexity of the operation
beacause it requires adding only one equality constraint to the bare
LP decoding by fixing the value of the marginal probability to zero
or one at any point of the critical loop.

These observations suggest the following decoding algorithm, coined
{\bf Loop Guided Guessing} (LGG):

\noindent $\bullet$ {\bf 1.} Run the LP algorithm. Terminate if LP
succeeds (i.e. a valid code word is found).

\noindent $\bullet$ {\bf 2.} If LP fails,  find the critical loop,
$\Gamma$, the one
 with maximal value of $|r(\Gamma)|$ in the loop series.

\noindent $\bullet$ {\bf 3.} Pick any bit along the critical loop at
random and form two corrected LP schemes,  different from the bare
LP schemes by only one extra equality condition, enforcing the value
of a bit to be $1$ or $0$ respectively.

\noindent $\bullet$ {\bf 4.} Run both LP-corrected schemes and
choose the output which corresponds to the smallest self-energy.
Terminate if the modified LP succeeds.

\noindent $\bullet$ {\bf 5.} Return to {\bf Step 3} selecting
another bit along the critical loop or to {\bf Step 2} for an
improved selection principle for the critical loop if the list of
all the bits along the previously selected loop is exhausted.

Notice that main advantage of the LGG algorithm, in comparison with
the Loop Erasure algorithm of \cite{06CCc}, is in the locality of
the bare algorithm (LP) modification. One finds that breaking a
loop, instead of modifying the algorithm along the loop, is
sufficient for successful decoding.

\begin{figure}
\begin{center}
\includegraphics[width=9.0cm]{ber.eps}
\end{center}
 \caption{ \label{fig1} Frame Error Rate vs SNR for the $[155,64,20]$ code and the AWGN channel.
 Diamonds represent data of the LP-decoding simulations from
 \cite{07CSa}. Stars stand for the LGG (Loop Guided Guessing)-decoding
 described in the text. Dashed and dashed-dotted lines show
 the MAP and BP/LP decodings $s\to\infty$ asymptotes,
 $\propto \exp(-20*s^2/2)$ and $\propto \exp(-16.407*s^2/2)$ respectively, where
 $20$ is the minimal (Hamming) distance of the code and
 the effective distance of the lowest weight LP pseudo-codeword found for the code is
 $\approx 16.407$.}
\end{figure}

We tested the performance of the LGG algorithm using Monte Carlo
(MC) simulations.  Our starting point was the set of configurations
whose bare LP failed in the MC-LP simulations for the $[155,64,20]$
code discussed in \cite{07CSa}. We apply the LGG algorithm to these
erroneous configurations and observed essential improvement.  Thus
for $s^2=2.4$, only one out of every ten LP-invalid configurations
is not correctable by LGG.   The results of the simulations are
shown in Fig.~(\ref{fig1}). ( Note that performance of the
LP-erasure algorithm of \cite{06CCc} was worse with only a few bare
LP failures corrected.)

For some of these individual configurations, we also run an
exhaustive BG algorithm (checking bit by bit if the respective
bit-corrected LP decodes correctly) and compare the resulting set of
``successful" bits with the set of bits forming the critical  loop.
Very much like the case of the respective dangerous pseudo-codewords
test, we found strong correlations between the two sets: bits of the
critical loop typically belong to the ``successful" set. This
justifies our decision to select the special bit along the critical
loop at random,  thus supporting the conjecture that a pin-point
bit-local correction of LP is sufficient for breaking the loop and
successful decoding.

Notice that configurations accessable at SNRs from Fig.~\ref{fig1}
via MC simulations  are typically those with relatively large
effective distances, $\sim 30-40$, while the Hamming distance of the
code is $20$ and the effective distance of the most dangerous
pseudo-codeword of the bare LP is $\approx 16.4$. We expect,
however, that majority of these configurations are from valleys of
the Bethe Free Energy functional with local minima correspondent to
effective distances from the $[16.4;20]$ range. See \cite{05SC} for
a related discussion of why the FER asymptote at moderate SNR shows
behavior controlled by pseudo-codewords with much smaller effective
distance than those representing a given SNR.

\section{Path forward}

Let us conclude listing some future problems/challenges:

\noindent $\bullet$ The LGG algorithm should be tested on longer
practically relevant codes.

\noindent $\bullet$ The LGG performance can be improved if a better
algorithm for finding the critical loop is implemented. An LDPC code
can be replaced by its MAP-equivalent dendro-counterpart
\cite{07CSa}. Then,  the problem of finding the single-connected
loop with the largest value of $|r(C)|$ is reduced to finding the
shortest path on the undirected graph with possibly negative
weights, however with a guarantee that all loop contributions over
the graph are positive. One may hope to develop an efficient graph
algorithm for solving this problem.

\noindent $\bullet$ Modern schemes of LDPC ensemble optimization
\cite{06AMU} are very successful in dealing with the water-fall
domain,   where performances of almost  all codes from the given
ensemble are identical.   It is known however that different codes
from the same ensemble show big performance variations in the
error-floor domain if a standard,  not yet optimized, decoding is
utilized. This problem is a serious handicap for the successful use
of random LDPC codes in the demanding high SNR regime.  We plan to
apply the decoding improvement strategy to the optimized LPDC
ensembles,  with the hope that the algorithm improvement may be
capable in lowering the error-floor for a majority of codes from the
ensemble.

The author acknowledge very useful, inspiring and fruitful
discussions with V. Chernyak, A. Dimakis, M. Stepanov and M.
Wainwright. This work was carried out under the auspices of the
National Nuclear Security Administration of the U.S. Department of
Energy at Los Alamos National Laboratory under Contract No.
DE-AC52-06NA25396.


\section{Introduction}
\label{sec:intro}
Supersymmetric extension of the standard model (SM) is one of the
most promising candidates for a new physics at the TeV scale.
In particular, the minimal supersymmetric standard model (MSSM)
is interesting from the viewpoint of its minimality.
The MSSM has several attractive aspects.
The MSSM realizes the unification of three gauge couplings
at the grand unified
theory (GUT) scale $M_{GUT} \sim 2 \times 10^{16}$ GeV.
Supersymmetry can stabilize the huge hierarchy
between the weak scale and the GUT/Planck scale.
Supersymmetric models with R-parity have a good candidate for dark matter,
that is, the lightest superparticle (LSP).

However, these attractive aspects are not perfectly
satisfying.
First of all, there is still a fine-tuning problem as follows.
By minimizing the Higgs scalar potential,
the $Z$ boson mass is obtained as
\begin{eqnarray}
\frac{1}{2}M_Z^2 &\sim& -\mu^2(M_Z)-m_{H_u}^2(M_Z),
\nonumber
\end{eqnarray}
where  $\mu$ is the SUSY mass of up- and down-sector Higgs fields
and $m_{H_u}$ is the soft SUSY breaking mass for
the up-sector Higgs field.
Thus, natural values of $|\mu^2|$ and $|m_{H_u}^2|$ would be
of $O(M_Z^2)$.
Otherwise, we need fine-tuning between $|\mu^2|$ and $|m_{H_u}^2|$
to cancel them and to lead to $M_Z$.
The soft mass $m_{H_u}$  receives a large radiative correction
between the weak scale and the cut-off scale $\Lambda$,
\begin{eqnarray}
\Delta m_{H_u}^2 &\sim&
-\frac{3y_t^2}{4\pi^2} m_{\tilde{t}}^2
\ln \frac{\Lambda}{m_{\tilde{t}}},
\nonumber
\end{eqnarray}
where $y_t$ is the top Yukawa coupling and
$m_{\tilde{t}}$ is the stop mass.
The cut-off scale may be the GUT scale or Planck scale,
and we may have $\Delta m_{H_u}^2 \sim -2 m_{\tilde{t}}^2 $
or $-3 m_{\tilde{t}}^2$.
On the other hand, the theoretical upper bound for
the lightest CP-even Higgs mass is obtained as \cite{Okada:1990gg}
\begin{eqnarray}
m_h^2 &\le&
M_Z^2  + \frac{3 m_t^4}{4 \pi^2 v^2}
\ln \frac{m_{\tilde{t}}^2}{m_t^2} + \cdots.
\nonumber
\end{eqnarray}
The experimental bound $m_h \ge 114.4$ GeV requires
$m_{\tilde{t}} \gtrsim 500$ GeV.
This value of $m_{\tilde{t}}$ leads to a quite large
correction $\Delta m_{H_u}^2$.
Hence, in order to realize $M_Z=91$ GeV,
we need a few percent fine-tuning
of the SUSY mass $\mu$ and the soft SUSY breaking mass
$m_{H_u}$ at the GUT scale, although these two masses are,
in general, independent parameters.
This fine-tuning problem is sometimes called as
the little-hierarchy problem between the weak scale and
a TeV scale ~\cite{Barbieri:1987fn}.

Several types of models have been proposed
to solve the little hierarchy problem.
Among them, the TeV scale mirage mediation
\cite{Choi:2005hd,Kitano:2005wc,Choi:2006xb} is
one of most interesting scenarios,
because the field content in the visible sector
is the same as one of the MSSM.
In the mirage mediation, the modulus
mediation and anomaly mediation \cite{anomaly} are comparable
\cite{Choi:2005ge,Choi:2005uz}, and such situation
can be realized in the KKLT-type
of moduli stabilization \cite{Kachru:2003aw}.
One of interesting aspects in the mirage mediation
is that the anomaly mediation effect and
renormalization group (RG) effects cancel each other
at the so-called mirage scale $M_{\rm mir}$.
That is, soft SUSY breaking terms at $M_{\rm mir}$
appear equivalent to the pure modulus mediation,
although there is no physical threshold at $M_{\rm mir}$.
Therefore, the model with $M_{\rm mir}=O({\rm TeV})$,
i.e., the TeV scale mirage model, is interesting
as a solution of the little hierarchy problem.
In the TeV scale mirage model, the superparticle
spectrum derived from the pure modulus mediation
appears at the TeV scale through the cancellation
between the anomaly mediation and RG effects.
In particular, the modulus mediation leading to
\begin{equation}
|m_{H_u}| \sim \mu \sim M_Z, \qquad m_{\tilde t}=O(1){\rm TeV}
\end{equation}
is interesting.
Indeed, concrete models realizing the above spectrum
have been studied in Ref.~\cite{Choi:2005hd,Choi:2006xb}.
In those models, gauge kinetic functions for
three MSSM vector multiplets are universal
and three gaugino masses are universal at $M_{\rm mir}$.
In addition, the universal gaugino mass is of $O(1)$ TeV.

One of interesting aspect in SUSY models is that
they have a good candidate for the dark matter as
the LSP.
In the above TeV scale mirage scenario,
the value of $\mu$ is of $O(M_Z)$
to avoid fine-tuning,
while the gaugino masses are universal around a TeV scale and
it is of $O(1)$ TeV.
Thus, the LSP is higgsino-like in the TeV scale mirage scenario.
In this case, the thermal relic density of the LSP is
much lower than cosmological observation \cite{Choi:2006im}.

Recently, the bottom-up analysis \cite{Abe:2007kf}
showed that the degeneracy between the wino and gluino masses
is most important to avoid fine-tuning in the Higgs sector,
but the bino mass can vary with keeping the same
degree of fine-tuning.
When the bino mass varies, several phenomenological aspects
would change.
The LSP is a mixture of the higgsino and bino,
and its thermal relic density would be totally different from
one of higgsino-like LSP.
Hence, in this paper we study the TeV scale partial
mirage unification, where the wino and gluino masses
are degenerate around $O(1)$ TeV, but the bino mass
is different.
We study phenomenological aspects of this scenario,
in particular the thermal relic density and direct detection possibility
for the neutralino LSP.\footnote{See
  Ref.\cite{naturaldm,Dermisek:2006ey}
for other studies on
a connection between naturalness of electroweak symmetry breaking and dark
matter phenomenology.}

Several authors have investigated phenomenological and
cosmological aspects of mirage mediation
\cite{endo05,falkowski05,yama,baer,kitano-lhc,Choi:2006im,kawagoe,cho}.
In particular, it has been noticed that moduli decay in the early
universe can produce so many gravitinos and neutralino LSPs that
successful Big Bang nucleosynthesis might be ruined and/or too large dark
matter abundance would be obtained \cite{yama}.
A possible way out of the cosmological moduli problem is to dilute the primordial moduli and
the subsequently produced gravitinos and LSPs, through some mechanism
such as the thermal inflation \cite{stewart}. In this work, we assume that
such a mechanism is realized and the neutralino dark matter is generated
through the conventional thermal production mechanism.

This paper is organized as follows.
In section 2, we study a concrete model leading to
the TeV scale partial mirage unification.
In section 3, we study phenomenological aspects of
our scenario, in particular, the thermal relic
density of the LSP.
Section 4 is devoted to conclusion and discussion.

\section{TeV scale partial mirage unification}

\subsection{Moduli stabilization
in the generalized KKLT scenario}

Indeed, our model is quite similar to the model
for the TeV scale mirage \cite{Choi:2006xb},
which is a generalization of the KKLT scenario for
moduli stabilization \cite{Kachru:2003aw,Abe:2005rx,Choi:2006bh}.

We consider the IIB string model with the dilaton $S$,
a single K\"ahler modulus $T$ and complex
structure moduli $Z_\alpha$.
First, we assume that the dilaton $S$ and complex structure moduli
$Z_\alpha$ are stabilized by the flux-induced superpotential
$W_{\rm flux}(S,Z_\alpha)$ \cite{Giddings:2001yu}, that is, they have
heavy masses of $O(M_P)$, where $M_P$ is the Planck scale.
At this stage, the K\"ahler modulus $T$ is not stabilized.
To stabilize $T$, we introduce a $T$-dependent non-perturbative effect
in the superpotential.
In the original KKLT model, a simple term is considered as
\begin{equation}
W_{\rm np} = A e^{-aT},
\end{equation}
where $A = O(M_P^3)$ and $a$ is a constant.
Here and hereafter we use the unit, where $M_P=1$.
Such non-perturbative effect can be generated by a gaugino
condensation of the hidden gauge sector on D7 branes,
whose gauge kinetic function is proportional to $T$.
In general, the gauge kinetic function is a linear
combination of $S$ and $T$ as,
\begin{equation}
f_a=k_a T + \ell_a S,
\end{equation}
e.g. on magnetized D-branes, where
$k_a$ and $\ell_a$ are rational numbers \cite{Lust:2004cx}.
The gaugino condensation in the hidden gauge sector may generate
a non-perturbative term
like $W_{\rm np} \sim e^{-8 \pi^2 (k_h T + \ell_h S)}$.
Thus, we consider the superpotential \cite{Abe:2005rx}
\begin{equation}
W = \langle W_{\rm flux}\rangle - A_h e^{-8 \pi^2 (k_h T + \ell_h S)},
\end{equation}
where $A_h=O(1)$ and $8 \pi^2 k_h =O(10)$.
In the second term of the right hand side, the dilaton $S$ is
replaced by its vacuum expectation value (VEV) $S_0$,
because $S$ is assumed to be stabilized with a mass of $O(M_P)$
by the flux-induced superpotential $W_{\rm flux}$.\footnote{
We can replace $S$ by its VEV only when the  flux-induced
superpotential $W_{\rm flux}$ includes its supersymmetric mass,
which is heavier than the mass of $T$ and the gravitino mass.
Otherwise, such analysis is not
valid \cite{Choi:2004sx,deAlwis:2005tf}.}
With this superpotential and the K\"ahler potential,
\begin{equation}
K_0=-3\ln (T + T^*),
\end{equation}
we can write the scalar potential,
\begin{equation}
V_F = e^{K_0}\left[ K_0^{T T^*}|D_TW|^2 -3|W|^2 \right],
\end{equation}
where
\begin{equation}
D_TW \equiv (\partial_T K_0 )W + \partial_T W.
\end{equation}
The K\"ahler modulus $T$ is stabilized at the SUSY point $D_TW =0$,
where we can estimate $\langle W \rangle \approx \langle W_{\rm
  flux}\rangle$ because $8 \pi^2 k_h =O(10)$.
At this SUSY point,
the vacuum energy is negative,
\begin{equation}
V_F = -3m^2_{3/2},
\end{equation}
where $m_{3/2}$ denotes the gravitino mass, $m_{3/2}=e^{K_0/2}W$.
The modulus $T$ has a mass of $O(8 \pi^2 m_{3/2})$, which is
much larger than the gravitino mass.

To obtain a de Sitter (Minkowski) vacuum, we add the uplifting
potential,
\begin{equation}
V_{\rm lift} = e^{2K_0/3}{\cal P}_{\rm lift}.
\end{equation}
Such potential can be generated by putting anti D3-brane
at a tip of warp throat  \cite{Kachru:2003aw},
and the warp factor leads to a suppressed value of
${\cal P}_{\rm  lift}$ \cite{Giddings:2001yu},
\begin{equation}
{\cal P}_{\rm lift} \sim e^{-32\pi^2K{\rm Re}(S_0)/3M},
\end{equation}
where $K$ and $M$ are integer-valued NS and R 3-form fluxes.
We tune our parameters to realize almost vanishing
vacuum energy, i.e. $V_F+ V_{\rm lift} \approx 0$.
Since $V_F \approx -3 e^{K_0}|W_{\rm flux}|^2$,  the above fine-tuning
requires $|W_{\rm flux}|^2 \sim {\cal P}_{\rm lift} \sim
e^{-32\pi^2K{\rm Re}(S_0)/3M}$.
Hence, we can parameterize $W_{\rm flux}$ as
\begin{equation}
W_{\rm flux} = A_0 e^{-8 \pi^2 \ell_0 S_0},
\end{equation}
where $\ell_0$ is a rational number and $A_0 =O(1)$.
We consider the low-energy SUSY breaking, i.e.
$m_{3/2} = O(10)$ TeV.
That requires
\begin{equation}
8 \pi^2 \ell_0 {\rm Re}(S_0) \simeq \ln (M_P/m_{3/2}) \sim
4 \pi^2.
\end{equation}
At the minimum of $V_F + V_{\rm lift}$, the values of
$T$ and the F-term  $F^T$ are obtained as
\begin{eqnarray}
k_hT &\simeq& (\ell_0 -\ell_h) S_0, \\
\frac{F^T}{T + T^*} &\simeq& \frac{\ell_0}{\ell_0 - \ell_h}
\frac{m_{3/2}}{\ln (M_P/m_{3/2})}.
\end{eqnarray}
When $\ell_0/(\ell_0 - \ell_h) =O(1)$,
$F^T/(T + T^*)$ is smaller than $m_{3/2}$ by a factor
of $O(4 \pi^2)$.
That implies that the modulus mediation due to $F^T$
and the anomaly mediation are comparable in this scenario
\cite{Choi:2005ge,Choi:2005uz}.

We have put anti D3-brane at the tip of warp throat.
Instead of that, a similar uplifting can be realized by
adding a spontaneous SUSY breaking sector, i.e.
the F-term uplifting \cite{fuplifting,fuplifting-2}.

\subsection{Soft SUSY breaking terms in the visible sector}

Now, we consider soft SUSY breaking terms in
the visible sector.
We assume that the compactification scale is close to the
GUT scale.
Thus, the following initial values are obtained at
the GUT scale.

First, we study the gaugino masses of the visible gauge sector,
i.e. $SU(3) \times SU(2) \times U(1)_Y$.
Here, these gauge groups are denoted by $G_a$, ($a=1,2,3$), i.e.
$G_1=U(1)_Y$, $G_2=SU(2)$ and $G_3=SU(3)$.
Suppose that the SU(3) and SU(2) gauge kinetic functions
are given as
\begin{equation}
f_v = T + \ell S,
\label{f-visible}
\end{equation}
 where $\ell$ is a rational number.
The gauge coupling unification in the MSSM, $g^{-2}_{\rm GUT} \simeq
2$, requires ${\rm Re}(T) + \ell {\rm Re}(S_0) \simeq 2$.
The modulus-mediated contributes to the gluino and wino masses
are obtained as
\begin{equation}
M_0 = F^T \partial_T \ln ({\rm Re}(f_v))=
\frac{F^T}{T + T^*}\left(
\frac{\ell_0 - \ell_h}{\ell_0 - \ell_h + k_h  \ell} \right).
\end{equation}
Since $F^T/(T + T^*)=O(m_{3/2}/(4\pi^2))$, the contributions
due to the anomaly mediation are comparable.
Thus, just below $M_{GUT}$ the gluino mass $M_3$ and wino mass $M_2$
are obtained as
\begin{equation}
M_a = M_0 + \frac{b_a}{16 \pi^2}g_{GUT}^2m_{3/2},
\end{equation}
with $b_a=1,-3$ for $a=2,3$.
Then, at the energy scale $Q$, these gaugino masses are
given as
\begin{equation}
M_a(Q)=M_0\left[ 1 - \frac{1}{8 \pi^2}b_a g_a^2(Q) \ln \left(
M_{\rm mir}/Q \right) \right] ,
\end{equation}
where the so-called mirage scale $M_{\rm mir}$ is defined as
\begin{equation}
M_{\rm mir} = \frac{M_{GUT}}{(M_P/m_{3/2})^{\alpha/2}},
\end{equation}
with
\begin{equation}
\alpha = \frac{m_{3/2}}{M_0 \ln (M_P/m_{3/2})} =
\frac{\ell_0 - \ell_h + k_h \ell}{\ell_0}.
\end{equation}
When $\alpha =2$, we have $M_{\rm mir} \sim 1$ TeV, that is,
the gluino and wino masses are unified
around 1 TeV.
Note that there is no physical threshold around $M_{\rm mir}$.
Here we consider the model with $\alpha =2$.

If we consider the same gauge kinetic function for the $U(1)_Y$
group as Eq.~(\ref{f-visible}), the bino mass is also unified
at $M_{\rm mir}$.
However, the degeneracy between the bino and gluino masses
is less important to reduce the degree of fine-tuning
in the Higgs sector, although the degeneracy between
the wino and gluino masses are important \cite{Abe:2007kf}.
Hence, we consider a generic case for the gauge kinetic
function of the $U(1)_Y$ group as
\begin{equation}
f_Y = k_Y T + \ell_Y S,
\end{equation}
in the $U(1)_Y$ charge normalization, which can be embedded
into the $SU(5)$ GUT.
We assume that  $k_Y {\rm Re}(T) + \ell_Y {\rm Re}(S) \simeq 2$,
because of the gauge coupling unification, $g^{-2}_{GUT} \simeq 2$.
Then, the modulus-mediated contribution to the bino mass is
obtained as $k_YM_0$.
The bino mass also has a contribution due to the anomaly
mediation, and at $M_{GUT}$ the bino mass is obtained as
\begin{equation}
M_1 = k_Y M_0 + \frac{b_1}{16 \pi^2}g^2_{GUT}m_{3/2},
\end{equation}
where $b_1=33/5$.
Obviously the bino mass $M_1$
is not degenerate at $M_{\rm mir}$ unless $k_Y =1$.

Next, we consider soft SUSY breaking scalar masses as well as
A-terms.
Such SUSY breaking terms are determined by the kinetic term
of chiral superfield $\Phi^i$,
\begin{equation}
\int d^4 \theta C C^* e^{-K_0/3}Z_i \Phi^{i*}\Phi^i,
\end{equation}
where $Z_i$ is the K\"ahler metric of the matter field $\Phi^i$.
Here, $C$ denotes the chiral compensator superfield, i.e.
$C=C_0 + F^C \theta^2$, and its F-component is obtained as
$F^C/C_0=m^*_{3/2}$ in our model.
Then, the modulus-mediated contributions to A-terms
and soft scalar masses are obtained as
\begin{eqnarray}
\tilde A_{ijk} &=& a_{ijk}M_0 = F^T \partial_T \ln (e^{K_0}Z_iZ_jZ_k), \\
\tilde m^2_i &=& c_iM^2_0 = - |F^T|^2 \partial_T \partial_{\bar T}
\ln(e^{-K_0/3}Z_i),
\end{eqnarray}
where we have assumed that holomorphic Yukawa couplings are
independent of the modulus $T$.
Here we take the following form,
\begin{equation}
e^{-K_0/3}Z_i=(T + T^*)^{n_i},
\end{equation}
where $n_i$ is a rational number.
Then, $a_{ijk}$ and $c_i$ are obtained as
\begin{eqnarray}
a_{ijk} &=& (n_i+n_j+n_k)\left(
\frac{\ell_0 - \ell_h + k_h \ell_h}{\ell_0 - \ell_h} \right), \\
c_i &=& n_i \left(
\frac{\ell_0 - \ell_h + k_h \ell_h}{\ell_0 - \ell_h} \right)^2.
\end{eqnarray}
A-terms and soft scalar masses have contributions due to
the anomaly mediation.
Thus, these values at $M_{GUT}$ are given as
\begin{eqnarray}
A_{ijk} &=& \tilde A_{ijk} - \frac{1}{16\pi^2}(\gamma_i +
\gamma_j + \gamma_k)m_{3/2}, \\
m^2_i &=& \tilde m^2_i-\frac{1}{32\pi^2}\frac{d\gamma_i}{d \ln Q}
m_{3/2}^2 + \left[ \sum_{jk}\frac{1}{4}|y_{ijk}|^2\tilde A_{ijk} -
\sum_a g^2_a C^a_2(\Phi^i)r_aM_0 \right] m_{3/2},
\end{eqnarray}
with $r_{2,3}=1$ and $r_1=k_Y$, where $\gamma_i$ denotes the anomalous
dimension of $\Phi^i$ and $y_{ijk}$ is Yukawa couplings.
In addition, $C^a_2(\Phi^i)$ denotes quadratic Casimir
of the field $\Phi^i$ under the gauge group $G_a$.
We include RG effects to obtain A-terms and soft scalar masses
at the energy scale $Q$.
If $k_Y=1$ and the following relations
\begin{equation}
a_{ijk} = c_i+c_j+c_k=1,
\end{equation}
are satisfied for large Yukawa couplings $y_{ijk}$,
RG effects and the anomaly mediation contributions cancel
each other at $M_{\rm mir}$.
Then, we have $A_{ijk}(M_{\rm mir})=\tilde A_{ijk}$ and
$m^2_i(M_{\rm mir}) = \tilde m^2_i$.
Even for $k_Y \neq 1$, this spectrum is realized approximately
because RG effects due to $U(1)_Y$ are not important except
for right-handed slepton masses unless $|k_Y| \geq O(1)$.
We take
\begin{equation}
c_{H_u}=0, \qquad c_{t_L}=c_{t_R}=\frac{1}{2}.
\end{equation}
Then we can realize the little hierarchy between
$m_{H_u}=O(M^2_0/(4\pi^2))$
and $m_{\tilde t}^2=M_0^2/2$.
We consider $M_0 =O(1)$ TeV and a moderate value of $\tan \beta$,
e.g. $\tan \beta=10$.
Then, we neglect all of Yukawa couplings except the top Yukawa
coupling.
For the down-sector Higgs field, we take $c_{H_d}=1/2$.

With the above assignment of $a_{ijk}$ and $c_i$,
we have a smaller higgsino mass $\mu=100 - 200$ GeV.
Thus, a small value of $|k_Y|$, $|k_Y| < 1$, is interesting
because in such a case the LSP would be a mixture between
the higgsino and bino.
If $k_Y$ and the bino mass are quite small,
right-handed slepton masses for $c_{\ell}=1/2$
may become tachyonic at the weak scale.
Thus, we take $c_{\ell}=1$ for both left-handed and right-handed
slepton masses.
At any rate, slepton masses are irrelevant to the fine-tuning
problem of the Higgs sector.

Alternatively, in order to increase slepton masses
we could consider the scenario with
an extra (anomalous) $U(1)$ gauge group.
We assume that such $U(1)$ sector is separated away from the
SUSY breaking anti D3 brane, and $U(1)$ is broken
at a certain scale, e.g. $M_{GUT}$.
Such breaking induces another source of soft scalar masses, which
are proportional to $U(1)$ charge $q_i$ of the fields $\Phi^i$,
\begin{equation}
m^2_{i(D)}= q_i D.
\end{equation}
The size of $D$ is model-dependent.\footnote{See
Ref.~\cite{Choi:2006bh,Brummer:2006dg} for the D-term
contributions in the KKLT scenario
and Ref.~\cite{Kawamura:1996wn,Higaki:2003ig} for the D-term
in dilaton-moduli mediation of heterotic string and type I string
theory.}
This type of contribution could also increase
slepton masses.

The size of sleton masses are important for analysis
on the thermal relic density of the LSP as shown in the next section.
Hence, in the following section we consider two cases for slepton
masses, 1) the case that slepton masses are determined from
$c_{\ell}=1$ and 2) the case that slepton masses vary.
The latter case can be realized by the D-term contributions.

\section{Neutralino Dark Matter}

In this section, we consider neutralino dark matter phenomenology.
Recent WMAP and other observations imply that the cold dark matter abundance is \cite{wmap3}
\begin{eqnarray}
0.085 < \Omega_{\rm DM} h^2 <0.119~ (95\%~ \rm CL),
\label{wmap}
\end{eqnarray}
where $h \simeq 0.7$ is the scaled Hubble constant.
We assume that the neutralino LSPs were in thermal equilibrium when the temperature
of the Universe is larger than the LSP mass $m_\chi$. As the temperature drops below $m_\chi$,
the number density of the LSP is exponentially suppressed. At some point neutralino LSP
annihilation rate becomes smaller than the Hubble expansion rate.
Then the neutralino LSPs fall out of equilibrium and the LSP number density in a comoving volume
remains constant \cite{jungman}.
We also assume that the neutralino LSP constitutes
all the cold dark matter in the Universe at the current epoch.

\begin{figure}[ht!]
\vskip 0.6cm
\begin{center}
\epsfig{figure=mass.ps,width=7cm,height=7cm,angle=270}\hskip 0.5cm
\epsfig{figure=oh2.ps,width=7cm,height=7cm,angle=270}
\end{center}
\vskip -0.4cm
\caption{\it (a) sparticle masses at the weak scale and (b) thermal relic density $\Omega_\chi h^2$
of neutralino LSP, as a function of $M_1 (M_{\rm GUT})$.}
\label{fig:oh2}
\end{figure}

In the TeV scale mirage mediation, in which all three gaugino masses are unified at TeV scale,
it turns out that the neutralino LSP is higgsino-like \cite{Choi:2006im}.
This is because the gluino mass $M_3$ is smaller than bino mass $M_1$ (and wino mass $M_2$)
at higher energy scales. (Notice that $M_3: M_2: M_1 \simeq (1-0.3\alpha) g_3^2:
(1+0.1\alpha) g_2^2: (1+ 0.66\alpha) g_1^2$, and $\alpha \sim 2$ for
the TeV scale mirage mediation.)
Such a small $M_3$ gives a small stop mass squared and in turn
leads to a small $|m_{H_u}^2|$ and thus $|\mu|$ at the weak scale,
compared to bino mass $M_1$ and wino mass $M_2$.

For higgsino-like LSP, the lighter chargino $\chi_1^\pm$ and the two light neutralinos
$\chi_1^0$, $\chi_2^0$ are nearly degenerate. In this case
dominant annihilation processes for the neutralinos and chargino
are neutralino pair annihilation into gauge bosons, and
the neutralino-neutralino and neutralino-chargino coannihilations
into fermion pair \cite{coanil}.
These annihilation processes are so effective that the thermal relic density of
the neutralino LSP is too small unless neutralino LSP is rather heavy
($m_{\chi_1^0} \sim |\mu| \sim 1$ TeV).
Therefore, the higgsino LSP with $|\mu| \sim O(M_Z)$, which avoids fine-tuning
in the Higgs sector, cannot provide the correct amount of thermal
relic density in Eq.(\ref{wmap}).

On the other hand, the TeV scale partial mirage unification scenario can provide
not only a solution of the little hierarchy problem  but also
a right amount of thermal relic density.
As a specific numerical example, we choose a parameter set;
\begin{eqnarray}
\alpha=2,\  M_0=800~ {\rm GeV},\
c_{H_u}=0,\ c_{H_d}=1/2,\  c_q=1/2,\ c_l=1,\ \rm tan\beta=10,
\label{param1}
\end{eqnarray}
while varying the bino mass at the GUT scale within some range, 100 GeV $< M_1 (M_{\rm GUT}) <$ 2 TeV.
Figures $\ref{fig:oh2}$ show (a) sparticle masses at the weak scale and
(b) thermal relic density $\Omega_\chi h^2$ of the neutralino LSP, as a function of $M_1 (M_{\rm GUT})$.
One can notice that $\mu$ values at weak scale remain small $i.e,$
$130 \lesssim \mu \lesssim 160$ GeV so that there is no little hierarchy problem in this case.
Our model leads to the CP even Higgs mass, $m_h \sim 116$ GeV.

For a large $M_1 (M_{\rm GUT})$ value, $\mu$ is much smaller than
$M_1$ at the weak scale implying higgsino-like
LSP. It leads to a very small relic density $\Omega_\chi h^2 \sim O(10^{-3})$.
The bino mass at the weak scale decreases as $M_1 (M_{\rm GUT})$ decreases,
and becomes similar to $\mu$ value at the weak scale when $M_1 (M_{\rm GUT}) \sim 350$ GeV.
In the bino-higgsino mixed region of LSP,
the relic density $\Omega_\chi h^2$ increases rapidly
as $M_1 (M_{\rm GUT})$ decreases, due to the enhanced bino-component of neutralino LSP.
When $M_1 (M_{\rm GUT}) \sim 300$ GeV, $\Omega_\chi h^2 \simeq 0.1$, thus providing
a right amount of relic density which is consistent with the WMAP bound
on the dark matter density.

As $M_1 (M_{\rm GUT})$ further decreases, the neutralino LSP becomes bino-like and the relic density
$\Omega_\chi h^2$ gets too large and increases until $M_1 (M_{\rm GUT}) \sim 200$ GeV.
Below this point, an interesting annihilation channel for neutralino LSP is open.
For the region around $M_1 (M_{\rm GUT}) \simeq 160$ GeV, the mass of the neutralino LSP is equal to
the half of the light CP even higgs mass, $i.e, m_{\chi_1^0} \sim m_h /2$.
In this case the neutralino pair annihilation through s-channel higgs exchange becomes
very efficient so that the relic density $\Omega_\chi h^2$ is reduced to a very small values $O(10^{-3})$,
passing acceptable ones $O(10^{-1})$. When $m_{\chi_1^0} \sim m_Z /2$, $Z$ resonance effect is dominant
for reducing the relic density.

\begin{figure}[ht!]
\vskip 0.6cm
\begin{center}
\epsfig{figure=msl-mass.ps,width=7cm,height=7cm,angle=270}\hskip 0.5cm
\epsfig{figure=msl-oh2.ps,width=7cm,height=7cm,angle=270}
\end{center}
\vskip -0.4cm
\caption{\it (a) slepton masses at the weak scale and (b) thermal relic density $\Omega_\chi h^2$
of neutralino LSP, as a function of $m_{slepton}$(GUT).}
\label{fig:slepton}
\end{figure}

As we discussed in the last section, for the scenario with an extra (anomalous) $U(1)$ gauge group,
additional $D$-term contributions to soft terms would make slepton mass a free parameter in practice.
In order to see possible effects of the $D$-term contributions, we fix $M_1 (M_{\rm GUT})=200$ GeV
with the parameter set $(\ref{param1})$ while
varying $m_{\rm sl} (M_{\rm GUT})$, the slepton mass at the GUT scale.
Notice that in this case, the neutralino LSP is bino-like and $m_{\chi_1^0} \simeq 75$ GeV.
Figures 2 show (a) slepton masses at weak scale and (b) the thermal relic density
$\Omega_\chi h^2$, as a function of $m_{\rm sl} (M_{\rm GUT})$.
When the slepton mass is large ($m_{e_R} \simeq 500$ GeV),
the relic density is quite large ($\Omega_\chi h^2 \simeq 0.6$), as expected for the bino-like LSP
with rather heavy sparticle mass spectrum.
As $m_{\rm sl}(M_{\rm GUT})$ decreases, however, slepton masses at weak scale decrease.
Accordingly, the relic density $\Omega_\chi h^2$ decreases and gets close to
the WMAP bound (\ref{wmap}) when $m_{e_R} \sim 150$ GeV.
It is known that in this case, the LSP relic density is mainly determined
from neutralino pair annihilation into lepton pair through $t$-channel exchange of SU(2) singlet
sleptons \cite{dn}.

For $m_{\rm sl} (M_{\rm GUT}) \lesssim 115$ GeV,
the neutralino LSP and the lighter stau are almost degenerate.
Then LSP-stau coannihilation \cite{ellis}
becomes very effective to reduce thermal relic density of the neutralino LSP.
>From the Fig. 2(b), one can notice that
the thermal relic density $\Omega_\chi h^2$ reaches
the WMAP range and then drops quickly below 0.01
in the small $m_{\rm sl} (M_{\rm GUT})$ region.

\begin{figure}[ht!]
\vskip 0.6cm
\begin{center}
\epsfig{figure=sd.ps,width=7cm,height=7cm,angle=270}\hskip 0.5cm
\epsfig{figure=bsg.ps,width=7cm,height=7cm,angle=270}
\end{center}
\vskip -0.4cm
\caption{\it (a) spin-independent cross section of neutralino and proton
and (b) $b\rightarrow s\gamma$ branching ratio, as a function of $M_1(M_{\rm GUT})$.}
\label{fig:scattering}
\end{figure}

Our model has also an interesting aspect for the direct detection search of
neutralino dark matter. For the spin-independent cross section of neutralino-proton
scattering, the contributions from $t$-channel CP even Higgs exchanges are usually dominant \cite{jungman}.
The cross section $\sigma_{SI}$ would enhance if neutralino LSP is a mixed state of gaugino and higgsino,
due to the nature of neutralino-neutralino-Higgs couplings.
Figure \ref{fig:scattering}(a) shows the spin-independent
scattering cross section as a function of $M_1(M_{\rm GUT})$ with the parameters (\ref{param1}).
The $\sigma_{SI}$ is quite small ($\sim 5\times 10^{-9}\ \rm pb)$ for large $M_1(M_{\rm GUT})$ region $i.e$,
higgsino-like LSP case. There is, however, about one order of
magnitude increase of $\sigma_{SI}$ in
the bino-higgsino mixed region. For $M_1(M_{\rm GUT}) \simeq 300$ GeV, which provides
a right amount of thermal relic density $\Omega_\chi h^2 \simeq 0.1$,
the spin-independent scattering cross section is $\sigma_{SI} \simeq 4\times 10^{-8}\ \rm pb$ with
$m_{\tilde\chi_1^0} \simeq 105$ GeV. This cross section value is quite close to the current limit from
XENON experiment \cite{xenon}, $i.e.,\ 8.8\times 10^{-8}$ pb for a WIMP mass of 100 GeV.
Therefore, our model would be explored in the near future experiments on the direct searches.

Before closing this section, we comment on experimental constraints.
Our benchmark point (\ref{param1}) satisfy the experimental bounds on particle masses such as
$m_{\chi^+} > 104$ GeV for chargino and $m_h > 114$ GeV for light Higgs boson.
Figure \ref{fig:scattering}(b) shows $b\rightarrow s\gamma$ branching ratio $BR(b\rightarrow s\gamma)$
for the parameter choice (\ref{param1}), as a function of $M_1(M_{\rm GUT})$.
The NLO calculation for $BR(b\rightarrow s\gamma)$ gives about $3\times 10^{-4}$ for our
parameter choice, insensitive to $M_1(M_{\rm GUT})$.
These predictions are rather smaller than the current world average of experimental values \cite{bsg},
$BR(b\rightarrow s\gamma)^{exp}=(3.55\pm 0.26)\times 10^{-4}$,
due to large contribution from chargino-stop loop which adds destructively to
Standard Model contribution for our choice on the sign of $\mu\ (> 0)$.
Considering theoretical and experimental uncertainties, it turns out that the calculated branching ratio is
consistent with the measured one within $2\sigma$ range.

\section{Conclusions}

We have studied the TeV scale partial mirage unification scenario,
where the gluino and wino masses are degenerate, but the bino mass
is not degenerate.
We have shown an example leading to such a spectrum.
This spectrum has phenomenologically interesting aspects.
First, there is no fine tuning problem because of
the degeneracy of the gluino and wino masses, that is,
our model leads to $130 \lesssim \mu \lesssim 160$ GeV.
The LSP is the mixture of the bino and higgsino.
In the TeV scale partial mirage unification,
a right amount of thermal relic density of neutralino LSP
can be obtained through various channels for neutralino annihilations.
A mixed bino-higgsino LSP, which is available through adjusting the bino mass at the GUT scale,
may lead to an appropriate neutralino annihilation rate into gauge bosons
and so the right amount of the relic density.
The neutralino pair annihilation via s-channel higgs exchange
play an important role for obtaining the suitable relic density, when $m_\chi \sim m_h/2$
in bino-like LSP region. Furthermore, if the slepton mass can vary independently,
LSP annihilations through $t$-channel SU(2) singlet slepton exchange or LSP-stau coannihilation
can make the thermal relic density satisfy the WMAP bound on dark matter density.
The TeV scale partial mirage unification scenario also provides a sizable
spin-independent scattering cross section between neutralino dark matter and nucleon,
which can be explored in near future experiments, when the neutralino dark matter is
a mixture of bino and higgsino.

\subsection*{Acknowledgement}

H.~A.\/ and T.~K.\/ are supported in part by the Grand-in-Aid for
Scientific Research \#182496, \#17540251, respectively.
T.~K.\/ is also supported in part by the Grant-in-Aid for the 21st Century
COE ``The Center for Diversity and Universality in Physics'' from the
Ministry of Education, Culture, Sports, Science and Technology of Japan.
This work was supported by the KRF Grant KRF-2005-210-C000006 funded by the Korean Government and
the Grant No. R01-2005-000-10404-0 from the Basic Research Program of the Korea
Science \& Engineering Foundation (Y.G.K. and Y.S.).


\section{Introduction}\label{lbl:intro}

Correlation between identical pions emitted from a thermalized source with
similar momenta is due to the bosonic nature of the pions. The phenomenon is
often referred to as Bose-Einstein Correlation (BEC). The correlation functions carry
important information about the space-time extent and the dynamics of the particle
emitting source. R.~Hanbury Brown and R.~Q.~Twiss used the bosonic correlation of particles
to measure the angular diameter of distant stars~\cite{HBT}. BEC, or the HBT effect has
become an important tool in high energy physics. In most heavy ion experiments there
is enough statistics to measure the two- and three-particle correlation function
of charged pions. We define the correlation function $C(k_1,k_2)$ as
\bl{eq:Cdef} C(k_1,k_2)=\frac{N_2(k_1,k_2)}{N_1(k_1)N_1(k_2)} .
\ee Here $k_1$ and $k_2$ are the four-momenta of the emitted
particles, $N_1(k)$ is the single particle momentum distribution,
$N_2(k_1,k_2)$ is the two-particle momentum distribution. To raise
the statistics, one integrates over some components of $k_1$,
$k_2$. From the experimental point of view the adopted definition
is \bl{eq:Cqdef} C(q)=\frac{N_{corr}(q)}{N_{mix}(q)} , \ee the
number of pairs with $q$ relative momentum from the same event
divided by the number from mixed events. This is measured directly
in experiments, and it must not to be confused with the ideal
Bose-Einstein correlation function without any final state
effects, $C_0(q)$. In our notation $q=\sqrt{-(k_1-k_2)^2}$ is the
invariant relative momentum of the pair, which is equal to
$2k_{12}$ in the pairs c.m.s. frame. The definition of $k_{12}$
is: $k_{12}=\rec{2}\left|\v{k}_1-\v{k}_2\right|$.

In the approximate case when multi-particle correlation effects
are negligible, the BEC has good theoretical description even in
the case when final state interactions (FSI) play important role.
The two-particle correlation function of the pions can be calculated from the
source function as follows \cite{Koonin}: \bl{eq:PPGinteq}
C(q)-1=\rec{2}\int\ud^3\v{r}\z{\left|\Phi_C\z{q,\v{r}}\right|^2-1}S\z{\v{r}}
, \ee where $\Phi_C(q,\v{r})$ is the symmetrized final state
two-particle outgoing wave function. In case of
Coulomb-interaction, this has an analytic form~\cite{Messiah,Alt}. With
the imaging method (by inverting this integral equation) one can
determine the $S(\v{r})$ source function. If we assume spherical
symmetry, then $S(\v{r})$ depends only on $r:=|\v{r}|$.

\section{PHENIX Source Imaging}\label{lbl:img}
Emission source functions have been extracted from correlation
functions constructed from charged pions produced at mid-rapidity
in Au+Au collisions at $\sqrt{s_\mathrm{NN}}=200\
\mathrm{GeV}$ \cite{PPG052}. A model-independent imaging technique
developed by Brown and Danielewicz \cite{ImgSrc} has been carried out (See
Fig.~\ref{fig:ppg052_1}). The source parameters extracted from
low \ensuremath{k_T}~PHENIX data give the indication of a long tail, which
differs from that of a Gaussian distribution.

\begin{figure}[htb]
  \begin{tabular}{cc}
    \begin{minipage}{50mm}
      \includegraphics[width=60mm]{ppg052_fig1.ps}
    \end{minipage}
    &
    \begin{minipage}{75mm}
      \caption[]{\it (a) Correlation function, C(q), (full circles) for
      \ensuremath{\pi^\mathrm{+}}\pip{} and \ensuremath{\pi^\mathrm{-}}\pim pairs (\ensuremath{200\ \MeV<\kT<360\ \MeV}{}) produced in the central Au+Au collisions of
      0\%-20\% centrality. Open squares represent the restored
      correlation from the imaging technique. The dotted line represents
      the result of direct correlation fitting. The open triangles
      represent the 1D angle-averaged correlation of the 3D correlation
      function. (b) and (c) show the 1D source function $S(r)$ and
      $4\pi{}r^2S(r)$, respectively, from imaging (open squares),
      Spheroid fit to the correlation function (full circles), and
      angle-averaging of the 3D-Gaussian source function (open
      triangles). Systematic errors are less than the size of the data
      points.} \label{fig:ppg052_1}
    \end{minipage}
  \end{tabular}
\end{figure}

The aim of this talk was to present our investigations carried out
with THERMINATOR and HRC Monte Carlo simulatons, that concluded in
possible kinematic explanations of the long-range components of
the radial source function seen in PHENIX.

\section{Hadron Rescattering}\label{lbl:hrc}
It has been shown \cite{Humanic:2002iw,Humanic:2006ve} that
calculations based on a hadronic rescattering model agree
reasonably well with experimental results on single particle
spectra, elliptic flow, HBT radii in Au+Au collisions at RHIC
energies. The model we use here to compare PHENIX data with is Tom
Humanic's Hadronic Rescattering Calculations. The model deals with the 8
most abundant resonances and simulates their decays and rescattering.

Simulated source of charged pions is discussed in three groups,
determined by time of creation. The {\it core} consists
of the primordial particles
and the decay products of the short lived resonances. The secondary
pion sources of long lived resonances are referred to as the {\it
halo}, while the decay products of the $\omega(782)$ meson make up
a separate class (see Table \ref{tab:pisrc}).

\begin{table}[h!]
\vspace*{-12pt}
\caption[]{Components of charged pion sources in the HRC simulation as grouped in this discussion.}
\label{tab:pisrc}
\vspace*{-14pt}
\begin{center}
\begin{tabular}{|l|l|l|}
\hlin
Group & Lifetime & Components in HRC \\
\hlin
\hline
core & $<20~\mathrm{fm/c}$ & primordial $\pi^\pm$, $\rho, \Delta, \mathrm{K}^\star$ \\
\hline
omega & $\approx 23~\mathrm{fm/c}$ & $\omega(782)$ \\
\hline
halo & $>25~\mathrm{fm/c}$ & $\Lambda, \Phi, \eta, \eta\prime$ \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}[htb]
\begin{center}
  \includegraphics[width=55mm]{HRC_0020_020036.eps}
  \includegraphics[width=55mm]{HRC_0020_048060.eps}
  \\
  \includegraphics[width=55mm]{HRC_5090_020040.eps}
  \caption[]{\it Pion pair source measured in PHENIX and HRC simulation for central
    ($0-20\%$) \ensuremath{200\ \MeV<\kT<360\ \MeV}~collisions (top left); high \ensuremath{k_T}~(\ensuremath{480\ \MeV<\kT<600\ \MeV}) from the 20\% most
    central Au+Au midy collisions (top right); low kT (\ensuremath{200\ \MeV<\kT<360\ \MeV}) from
    peripheral collisions ($40-90\%$) (bottom center).
    Solid dots represent data on each figure.
    Black crosses are simulation, while color online symbols stand for different
    components of the simulated source as explained on the legend.
    The Gauss fit to $S(r<15~\mathrm{fm})$ is shown as a continuous line.}
\label{fig:hrc}
\end{center}
\end{figure}

On Fig.\ \ref{fig:hrc}, note the excellent agreement between
PHENIX data and HRC simulation in the central, low $\mathrm{k_T}$
case. The agreement is also acceptable in the full range for the
peripheral low $\mathrm{k_T}$ case. However, HRC fails to describe
the $\mathrm{k_T}$ dependence of the source.

Further investigations have shown that the $\omega(782)$ does not
account for the heavy tail in the simulation. Core-core pairs
themselves have a power-law tail in the \ensuremath{S(\ensuremath{r_{12}})\;} distribution. In the
HRC code, rescattering goes on until it self-quenches. The mean
free path increases strongly as the system expands, and
rescattering in such a time dependent mean free path system
corresponds to an anomalous diffusion. Anomalous diffusion is known
to lead to power-law tailed distribution, as stated by a generalization
of the Central Limit Theorem \cite{Metzler_Klafter,0702032}.

\section{Single Freezeout with THERMINATOR}

THERMINATOR---A Thermal Heavy Ion Generator \cite{Therm}, based on
the Cracow Single Freezeout Model \cite{Cracow} was used for the
following simulations. In this model, freezeout of the partonic matter occurs
on a single space-time surface, governed by universal thermodynamic
parameters. Particle distribution is determined by the
thermodynamical equilibrium. Particles propagate freely and the
decay of 385 resonances is fully implemented. In opposition to the HRC simulation,
rescattering effects are neglected here.

\subsection{PHENIX source simulated}

THERMINATOR simulation provides a flat rapidity distribution.
Several checks were done to verify that a realistic distribution
and $p_\mathrm{T}$ spectrum of particles is generated.

Centrality can not be directly set in THERMINATOR, but through the
parameters $\ensuremath{\rho_{max}}$ and $\tau$, where $\ensuremath{\rho_{max}}$ stands for
a radius of a boost invariant axially symmetric cylinder, while
$\tau$ denotes the proper time of a simultaneous freezeout and hadronization
\cite{Cent}. Although fitted parameters for the spectra of all four RHIC
experiments including PHENIX were published, PHENIX \ensuremath{p_T}\ spectra is not well
described by THERMINATOR, being primarily developed for, and tuned to STAR.
Thus we used the parameters corresponding to STAR spectrum fits.
This indicates that THERMINATOR might be able to show much better
agreement with data with further tuning. The central data region
$0-20\%$ has been reproduced using simulated events from the three
most central regimes in proportion to their span. The $40-90\%$
regime was substituted with equal number of events simulated in the regimes
of $50-60\%$ and $70-80\%$.

A separate set of events were simulated for each centrality bin of
5\% for central, and 10\% for peripheral collisions. Then these
sets were concatenated to make up the wider regimes of PHENIX
data. THERMINATOR was able to reproduce the centrality dependence
of the data. It is also capable to describe the \ensuremath{k_T}{} dependence
to a much higher accuracy than HRC does, although there is a
significant discrepancy at the higher \ensuremath{k_T}{} region. (See Fig.\
\ref{fig:th}).

\begin{figure}[htb]
\begin{center}
  \includegraphics[width=55mm]{ThSr_fig1b.eps}
  \includegraphics[width=55mm]{ThSr_0020_048060.eps}
  \\
  \includegraphics[width=55mm]{ThSr_5090_020040.eps}
  \caption[]{\it Pion pair source measured in PHENIX and THERMINATOR simulation for central
    ($0-20\%$) \ensuremath{200\ \MeV<\kT<360\ \MeV} collisions (top left); high \ensuremath{k_T}{} (\ensuremath{480\ \MeV<\kT<600\ \MeV}) from the 20\% most
    central Au+Au midy collisions (top right); low kT (\ensuremath{200\ \MeV<\kT<360\ \MeV}) from
    peripheral collisions ($40-90\%$) (bottom center).
    Solid dots represent data on each figure. Black crosses are simulation, while
    color online symbols stand for different
    components of the simulated source as explained on the legend.}
\label{fig:th}
\end{center}
\end{figure}

It must be noted that there is a significant excess in the
simulated \ensuremath{S(\ensuremath{r_{12}})\;}{} at small $r$ values. Our studies have shown that
neither the primordial nor the resonance particles are responsible
for it alone, but the $r\rightarrow{}0$ peak vanishes if all the
core particles and the short lived ($\Gamma\geq{}150\ensuremath{\mathrm{MeV}}$)
resonances are excluded from the source (see Fig.\
\ref{fig:th_r_studies}). This can be explained with the freezeout
that occurs in a single surface in this model, and the lack of
rescattering.

\begin{figure}[htb]
\begin{center}
  \includegraphics[width=55mm]{ThSr_0510_020036_.eps}
  \includegraphics[width=55mm]{ThSr_0510_020036_noprimord.eps}
  \\
  \includegraphics[width=55mm]{ThSr_0510_020036_noresoance.eps}
  \includegraphics[width=55mm]{ThSr_0510_020036_nocore-lt150.eps}
  \caption[]{\it $r\rightarrow{}0$ behaviour of THERMINATOR simulated \ensuremath{\pi^\mathrm{+}}\pip{} and \ensuremath{\pi^\mathrm{-}}\pim{} \ensuremath{S(\ensuremath{r_{12}})\;}{} for central
    low \ensuremath{k_T} collisions. Plots from top left to bottom right show all pion pairs, all pairs from resonance,
    all primordial pairs, and all pairs coming from the short lived resonances
    $\Gamma\geq{}150\ensuremath{\mathrm{MeV}}$ respectively. Solid dots represent data on each figure.
    Black crosses are simulation, while the color online symbols stand for different
    components of the simulated source as explained on the legend.}
\label{fig:th_r_studies}
\end{center}
\end{figure}

Further simulations indicate that rescattering and resonance decay create
very different sources for different particles ($\pi^\pm, \mathrm{K}^\pm$ and
$\mathrm{p}^\pm$).

\vspace{5mm}
\noindent
A study of direction-dependent PHENIX pion-pair source functions
using THERMINATOR Blast Wave and Single Freezeout and models has
been carried out by P. Chung et al.\ \cite{wwnd_pd} and concluded
that these models, although they do well in the "sideways" and
"long" directions, fail in the "outward" direction
(in the terms of the Bertsch-Pratt parametrization \cite{Pratt}).

\subsection{Discussion of results}
THERMINATOR is a simple model that shows a high power in the
description of particle spectra of the large heavy ion
experiments. Although there is no rescattering implemented in this
model, it reproduces the PHENIX image with the observed heavy
power-law tail for the lower \ensuremath{k_T}~events with a higher accuracy
than HRC---although it still slightly overpredicts the tail for the high
\ensuremath{k_T}~regime, and fails for $r\rightarrow{}0$ because of its
inherent limitations.

The relatively good agreement including the reproduction of the long-range tail
can be understood by taking into account that the high number of
resonances provide us with an almost-continuum distribution of
lifetimes, increasing virtually to infinity. This results in an approximate
power-law-like final distribution \cite{Bialas:1992ca} just as anomalous diffusion
does \cite{Metzler_Klafter}. However, THERMINATOR fails to describe correctly the
intercept $\lambda$ and the HBT radii $R_{side}, R_{long}$ and $R_{out}$
\cite{Warsaw_picorr}.
So it is questionable that resonance decays alone were sufficient
to describe the experimentally seen structure of source as Bialas had proposed.

\section{Conclusions}\label{lbl:concl}
Source images of PHENIX data have been studied with two fundamentally different,
simple models. Both were able to reproduce the observed 1D images
to a certain level, but had their limits. The different kinematic
mechanisms give similar shapes, that are determined by the same
underlying mathematical principle (a generalization of the Central Limit Theorem),
and so that any of them can be a potential explanation of the measured heavy power-law
tails.

\begin{enumerate}
\item HRC simulations describe low \ensuremath{k_T}~PHENIX image, but do
not have the right \ensuremath{k_{\mathrm{t}}}~dependence, as for higher \ensuremath{k_T}~values, the
simulated \Sr12~still has the large tail, but not the PHENIX data.
The reason for the heavy tail is rescattering, change of mean free
path and thus anomalous diffusion.

\item THERMINATOR simulations provide a reasonable reproduction of the \ensuremath{k_T}~and centrality
dependent behaviour of the 1D PHENIX image. The heavy tail in those simulations can be
explained by the large number of resonances in that model. This produces
an effect that is similar to the anomalous diffusion.

\item However, THERMINATOR fails to describe the direction-dependently
parametrized PHENIX source functions, suggesting that resonance decays
alone are not a sufficient explanation of the measured structure of the source.

\item Measurement of $S(r)$ images for $\pi^\pm, \mathrm{K}^\pm$ and
$\mathrm{p}^\pm$ could distinguish between rescattering and resonance decay effects.

\end{enumerate}