A New Strong Interaction Sector as the origin for the Dark Energy and Dark Matter
P.Q. Hung
A new strong interaction sector is proposed as a possible origin for the Dark Energy and Dark matter. It is described by an unbroken gauge group $SU(2)_Z$ which grows strong at a scale $\Lambda_Z \sim 10^{-3}\,eV$ provided its coupling is of the order of the electroweak coupling at some high energy scale (either $\sim 10^{16}\,GeV$ or $\sim O(TeV)$ . The present vacuum energy which is consistent with a cosmological constant is related to $\Lambda_Z$ and one of the $SU(2)_Z$ ``matter fields'' provides a candidate for the Dark Matter in the form of a weakly interacting massive particle (WIMP) with an annihilation cross section naturally of the order of the electroweak one at the time of its decoupling.
hep-ph0504060.tex
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\begin{document}
\title{A New Strong Interaction Sector as the origin
for the Dark Energy and Dark Matter}
\author{P.Q. Hung}
\email[]{pqh@virginia.edu}
\affiliation{Dept. of Physics, University of Virginia, \\
382 McCormick Road, P. O. Box 400714, Charlottesville, Virginia 22904-4714, 
USA}
\date{\today}
\begin{abstract}
A new strong interaction sector is proposed as a possible origin
for the Dark Energy and Dark matter. It is described by an unbroken
gauge group $SU(2)_Z$ which grows strong at a scale $\Lambda_Z
\sim 10^{-3}\,eV$ provided its coupling is of the order of the
electroweak coupling at some high energy scale (either
$\sim 10^{16}\,GeV$ or $\sim O(TeV)$ . The present vacuum 
energy which is consistent with
a cosmological constant is related to $\Lambda_Z$ and one
of the $SU(2)_Z$ ``matter fields'' provides a candidate for the Dark Matter
in the form of a weakly interacting massive particle (WIMP)
with an annihilation cross section naturally of the order of the
electroweak one at the time of its decoupling.
\end{abstract}
% insert suggested PACS numbers in braces on next line
\pacs{}
\maketitle

The nature of the dark matter and dark energy 
(responsible for an accelerating
universe \cite{acceleration}) is one of the deepest problem in 
contemporary cosmology \cite{FT}.
With the equation of state relating the pressure to the
energy density being $p=w\rho$, the latest CTIO Lensing 
Survey \cite{CTIO} performed in conjunction
with CMB and Type Ia supernovae data up to redshift $z \sim 0.4$
found a value of $w$ which is consistent with $-1$. This,
in turn, is consistent with the scenario in which the present
total energy density is dominated by the vacuum energy or
cosmological constant.
%There exists several models of dark energy, most prominently
%among which is the quintessence model \cite{quintessence}. 
%(String theory landscape \cite{landscape} is the latest
%idea which is still developing.) If
%one writes the equation of state for the dark energy as $p=w\rho$, 
%it is found that various measurements favor $-1.2<w<-0.8$.
%The scenario of a cosmological constant gives $w=-1$, a
%value which is perfectly consistent with observation.
%In fact, the latest CTIO Lensing Survey \cite{CTIO} performed in conjunction
%with CMB and Type Ia supernovae data supports the cosmological
%constant scenario.
%In other words, the vacuum energy, a.k.a cosmological constant, is
%one (if not the most preferred) candidate for the dark energy.
However, it is well-known that this vacuum energy is 
surprisingly tiny, namely $\rho_{V} 
\approx (10^{-3}\,eV)^4$. We shall take the point of view that
this could represent a {\em new scale of physics} on the
same footing as other known scales ($\Lambda_{EW} \sim
250\,GeV$, $\Lambda_{QCD} \sim 200\,MeV$) rather than 
one in which the associated vacuum energies 
{\em so finely cancel} down to $(10^{-3}\,eV)^4$. (We assume
that the electroweak and QCD vacuum energies are set to zero
after the associated phase transitions are completed.)
%What is then
%the dynamics that can give rise to this new scale? 
The purpose
of this paper is to propose a new interaction which grows
{\em strong} at $\Lambda_{Z} \sim 10^{-3}\,eV$ with interesting
implications for the Dark Energy and Dark Matter.

%even compared with the smallest known
%vacuum energy density of order $(93\, MeV)^4$ which arises from 
%chiral symmetry breaking in QCD. The fact
%that even this vacuum energy is many orders of magnitude larger
%than the ``observed'' one is very perplexing.
%constitutes what is now called
%the Old Cosmological
%Constant problem. 
%The {\em old problem} of trying to understand
%why the Cosmological Constant should be exactly equal to {\em zero}
%has now been transformed into the problem of why it
%should be so small. It is fair to say that it is completely
%unclear which one is relatively ``easier'' to tackle.
%From a purely theoretical prejudice point of view, one
%might suppose that it is more natural to search for
%a physical/symmetry principle whereby the exact cancellation of
%the vacuum energy occurs than to construct a mechanism
%in which a cancellation is so fine-tuned as
%to leave a {\em tiny} remnant that is the present dark energy.
%This fine tuning is made worse by the presence of at least
%two kinds of vacuum energy related to the electroweak and 
%quark chiral symmetry breakings.

%Might it be that $\rho_{V} \approx (10^{-3}\,eV)^4$ is actually
%associated with a new scale in physics related to
%some new symmetry breaking, giving rise to the
%aforementioned vacuum energy? Might it be that this
%vacuum energy is associated with a false vacuum that we
%are {\em now} living in? Might it be that we are entering
%an inflationary era whose scale is approximately
%$10^{-3}\,eV$?

%There has been some renewed interest in trying to find a
%dynamical reason for the cosmological constant to be {\em
%exactly equal to zero}\cite{cosmo}. In the absence of a convincing
%argument for this to be so, the usual
%practice is to add by hand a constant vacuum energy in such
%a way as to {\em exactly cancel} the one produced when the phase
%transition from the false to true vacuum occurs. The vacuum energy
%then ``relaxes'' to zero. Briefly speaking, one writes an
%effective potential as $V_{eff}(\phi)= U(\phi) + V(0)$ with
%$\phi$ being a generic scalar field and $V(0)$ being the vacuum
%energy. The potential $U(\phi)$ satisfies $U(\phi=0)=0$ and
%$U(<\phi> =v) = -V(0)$. (One simple example is $V_{eff} =
%\lambda(\phi^2 - v^2)^2 /4$.)
%Despite
%being an ad-hoc procedure, it has nevertheless deep implications,
%one of which being a driving mechanism for the 
%generic Inflationary model when the universe
%was still in the false vacuum and that the
%total energy density was dominated by that
%{\em ad-hoc} term $V(0)$ mentioned previously. 
%Although a compelling model for
%the Inflationary Universe is yet to be found, it is generally
%believed that, if it is associated with a first-order phase
%transition, the scale associated with the vacuum energy is
%probably around some Grand Unified Theory (GUT) scale of
%the order of $10^{16}\,GeV$. During the inflationary stage,
%the energy density of the universe is dominated by the vacuum
%energy density $\sim (10^{16}\,GeV)^4$. The scale of
%the vacuum energy for the Inflationary Scenario is thus
%well motivated, lying near the Planck scale.

%It is tempting to make a similar assumption 
%concerning the origin of the
%dark energy, namely there exists a first-order phase transition whose
%effective vacuum energy density is of the order $(10^{-3}\,eV)^4$ during
%the transition and vanishes {\em when it is completed}. However,
%there are several crucial questions
%that need to be addressed. First, where does 
%this vacuum energy come from? Second, why is it so small?
%The answers to these two questions require a little more
%than just writing down a ad-hoc quasi-flat potential with the
%required vacuum energy. 
%What we need here are some concrete
%dynamical reasons which lead to the scale $\sim 10^{-3}\,eV$.
%Can attach a 
%{\em dynamical meaning} to the scale $\sim 10^{-3}\,eV$?
%After all, we already
%know of the existence of two physical scales which are connected to
%known interactions: the electroweak and the QCD scales. Is it then
%reasonable to expect that the scale $\sim 10^{-3}\,eV$ might actually be 
%related to some {\em extra, as-yet-undiscovered
%new interaction}? If that is so, what might it be and what
%are its implications? Last but not least, what would the
%Once this is done, the next step would be a study of the
%dynamics of the phase transition be? 


%We will offer a simple model where the scale
%$\sim 10^{-3}\,eV$ arises dynamically through the renormalization
%group evolution of the gauge coupling of 
The model we would like to propose is based on a vector-like 
unbroken $SU(2)_Z$ gauge group whose coupling starts out
with
%(with
%$SU(2)_Z$ matter field transforming as a triplet) 
%from
a value assumed to be close to that of the SM $SU(2)_L$
at some high energy scale and 
%evolves to a value which
becomes large at $\sim 10^{-3}\,eV$. 
%, triggering
%the dynamical chiral symmetry breaking and/or confinement of
%the triplet fermions. 
Here the subscript $Z$ refers to an
ancient greek word {\em zophos} which means {\em darkness}.
Hence, this new interaction will be referred to
as Quantum Zophodynamics or QZD.
The  $SU(2)_Z$ matter fields are Standard Model (SM) singlets
and vice versa.
%SM matter fields are also assumed to be singlets under this new
%$SU(2)_Z$ group. 
As such, the two sectors would be {\em incommunicado}
(except through gravity) if not for the fact they 
can communicate 
with each other, albeit {\em very weakly}, through
a ``messenger'' field which carries both SM and
$SU(2)_Z$ quantum numbers. 
%Through this ``messenger'', the two
%sectors can {\em communicate} with each other, albeit {\em very weakly}.
%Two versions are presented here: a non-supersymmetric(SUSY)
%and a supersymmetric $SU(2)_Z$.
%The first one 
%is supersymmetric containing one left-handed
%and one right-handed chiral superfields, both
%of which are SM singlets and triplets under
%$SU(2)_Z$, and another pair of left-handed and right-handed
%chiral superfields (messenger fields)
%transforming as fundamental representations
%under $SU(2)_Z$, singlets under $SU(2)_L$ and
%carrying an electric charge. The second one is a non-supersymmetric
%version containing two $SU(2)_Z$ triplets, SM singlets of fermions 
%(both left-handed and right-handed) and one $SU(2)_L$ doublet
%of scalar fields (messenger fields) which transform as a 
%doublet under $SU(2)_Z$.

%If we assume the $SU(2)_Z$ coupling to be of the same order as the
%$SU(2)_L$ coupling at some scale (either near the Planck scale or
%even in the TeV region), the choices of particle representations
%presented below ensure that the
%$SU(2)_Z$ coupling will get strong at {\em very low energies} of the
%order of $10^{-3}\,eV$. 
%We shall also point out that, beside
%the fact that the strong coupling regime of $SU(2)_Z$ 
%might provide an explanation for
%the dark energy, some of the particles involved in the two versions
%of the model could also become candidates for dark matter
%in the form of weakly interacting massive particles (WIMP).

Our model was based on a very simple observation: For an unbroken gauge group
(with a coupling comparable in magnitude to the $SU(2)_L$ gauge
coupling at high energy) to become confining at energy scales many orders
of magnitude below the QCD scale, its 
one-loop beta function has to be small compared with that of
$SU(3)_c$. 
%This means that its coupling would
%be ``walking'' instead of ``running''. 
%(Its dynamics would
%also be fairly different from that of QCD.)
%Let us recall that the one-loop 
%Renormalization Group (RG) equation looks like $\mu\,d\alpha/d\mu =
%- (\beta_0 /2\pi) \alpha^2$, with $\beta_0 = (11C_{2}(G)/3 -2/3\,\sum T_{i}(R)
%-1/3\,\sum T_{S}(R))$ for the non-SUSY case and
%$b_0 = (3C_{2}(G) -\sum T_{\chi}(R))$ for the SUSY case,
%where $T_{i}(R), T_{S}(R),T_{\chi}(R))$ are for chiral fermion,
%scalar, and chiral superfield representations.
%This means that, in order to {\em minimize} $\beta_0$, one looks for groups
%and representations with the smallest $C_{2}(G)$ and the largest
%$T_{i,S}(R),T_{\chi}(R)$, but in such a way that $\beta_0$ remains positive.

We propose:

\bi

\item An unbroken vector-like gauge group $SU(2)_Z$.

\item The initial value of the $SU(2)_Z$ gauge coupling is
similar to that of the SM $SU(2)_L$ coupling at some high energy scale
such as the early unification scale (PUT)\cite{BH} in the TeV range
(or even the GUT scale of O($10^{16}$ GeV). This assumption
makes the model fairly {\em predictive} as we shall see below
(although a more general case can be easily entertained and
will be dealt with elsewhere).
%scale of O($10^{16}$ GeV) or even in the TeV range (as
%in the early unification model (PUT)\cite{BH}).
%grows large at a scale $\Lambda_Z \sim 10^{-3}\,eV$.

\item Fermions (or chiral superfields in the SUSY case)
transform as adjoints of $SU(2)_Z$ and
singlets under the SM. We denote them by
$\psi^{(Z)}_{L,R} = (1,1,0,3)$ under $ SU(3) \otimes SU(2)_L
\otimes U(1)_Y \otimes SU(2)_Z$ for the
non-SUSY case, and similarly $\Phi^{(Z)}_{L,R} = (1,1,0,3)$ for the
chiral superfield. These are all {\em electrically neutral}
and the fermions would be qualified for the designation ``sterile
neutrinos''.

\item Messenger fields transform either as fundamentals
{\em or} as adjoints under $SU(2)_Z$ 
and possess SM quantum numbers. They are denoted by
$\varphi_{1/2}^{(Z)} =(\varphi_{1/2}^{(Z),0},\varphi_{1/2}^{(Z),-})=
(1,2,Y_{\varphi}=-1,\,2)$ or 
$\varphi_{1}^{(Z)} =(\varphi_{1/2}^{(Z),0},\varphi_{1/2}^{(Z),-})=
(1,2,Y_{\varphi}=-1,\,3)$ 
under $SU(3) \otimes SU(2)_L
\otimes U(1)_Y \otimes SU(2)_Z$ for the non-SUSY case and
$\chi_{L,R} =(1,1,Y_{\chi},2)$ for the SUSY case (the triplet
case is not allowed here if we want the model to be
asymptotically free).

%\item The vacuum energy density  is $\rho_V \sim \Lambda_{Z}^4$.

\ei

We split the discussion into two parts: Particle Physics and Cosmology.

%\bi

%\item 
{\bf (A) PARTICLE PHYSICS}:

%First, we start with a discussion on the particle physics aspect
%of the model. 
%As we shall see below, this proposal gives rise to a ``walking''
%$SU(2)_Z$ which will grow strong and break the chiral symmetry
%of the $SU(2)_Z$ fermions at some scale of O($10^{-3}$ eV). 
The basic one-loop RG equation is $\frac{\mu\,d\alpha_Z}{d\mu} = 
-\frac{\beta_0}{2\,\pi} \alpha_{Z}^2$
where $\beta_{0,NS} = (11C_{2}(G)/3 -2/3\,\sum T_{i}(R)
-1/3\,\sum T_{S}(R))$ for the non-SUSY case and
$\beta_{0,S} = (3C_{2}(G) -\sum T_{\chi}(R))$ for the SUSY case,
with $T_{i}(R), T_{S}(R),T_{\chi}(R))$ for chiral fermion,
scalar, and chiral superfield respectively. 
From the particle content listed above,
%For our proposal where ``matter fields''  
%and ``messenger fields'' (both left and right-handed)
%transform as adjoints and fundamentals of $SU(2)_Z$ respectively,
one has $C_{2}(G)=T_{\psi^Z}=T_{\Phi}=2$, $T_{\varphi}=
T_{\chi}=1/2$ or $T_{\varphi}=2$ (for the
Georgi-Glashow type of situation), giving
$\beta_{0,NS} =  (22-8 n_{\psi} -n_{\varphi})/3$
or $\beta_{0,NS} =  (22-8 n_{\psi} -4n_{\varphi})/3$, 
for the non-SUSY case, and
$\beta_{0,S} =  (6-4\,n_{\Phi}-n_{\chi})$, 
for the SUSY case. Here, $n_{\psi}$, $n_{\Phi}$,
and $n_{\chi}$ already include {\em both left and right-handed fields}.
We wish to make $\beta_0$ as small as
possible while keeping it positive. This leads to the following
minimal set:
(I) $n_{\psi}=2\,;\, n_{\varphi}=1$,
for the non-SUSY case and
(II) $n_{\Phi}=1\,;\, n_{\chi}=1$,
for the SUSY case. Above
all mass thresholds, one has $\beta_{0,NS}=5/3$ or $\beta_{0,NS}=2/3$, 
$\beta_{0,S} =1$,
to be compared with $\beta_{0,SU(2)_L}=10/3$ and $\beta_{0,SU(3)_c} =7$
in the SM with three families. 
%This is what we have referred to above
%In this sense, $SU(2)_Z$ is really ``crawling''. 
%To see what the scale
%where $\alpha_Z \approx 1$ might be, we will separately examine
%the non-SUSY and SUSY cases. 
%In what follows, 

Below, we will present a few numerical results, leaving a general
analysis for a longer version of the paper.

{\bf (I) Non-SUSY $SU(2)_Z$}:

Since $\varphi$ also carries the SM quantum number, it is safe and not
unreasonable to assume it has a mass of the order of the electroweak
scale $\Lambda_{EW} \approx 250 GeV$. 
%This is what we will do here
%For the sake of the arguments
%presented in this paper, we will assume $m_{\varphi} \sim
%\Lambda_{EW} \approx 250 GeV$ 
%and leave the cases where the mass can vary
%to another paper. 
We will assume
that the potential for $\varphi$ is such that $<\varphi>=0$
in order for $SU(2)_Z$ to be {\em unbroken}.

For the fermions $\psi^{(Z)}_{L,R}$ which are singlets with respect
to the SM model, one can write down a gauge-invariant mass term of the
type $\sum_{i=1}^{i=2} m_{i} \bar{\psi}^{(Z)}_{i}\,\psi^{(Z)}_{i}$.
For the purpose of this paper, we will deal with the case 
where $m_1 \ll m_2$.
%(A more general analysis will be presented elsewhere.)

%We now show that for $\alpha_Z \sim 1/20\,-\, 1/34$ at a scale
%$M \sim 10^{16}\,GeV$ typical of Grand Unified Theories (GUT),
%$\alpha_Z \sim 1$ at $\Lambda_Z \sim 10^{-3}\,eV$ provided
%$m_2 \sim O(MeV)-O(200GeV)$. 
%For $M \sim O(TeV)$ typical of early unification models, 
%$\alpha_Z \sim 1$ at $\Lambda_Z \sim 10^{-3}\,eV$ provided
%$m_2 \sim O(GeV)$.

Let $M$ be the starting scale (early unification scale or GUT).
The four basic equations which we will use are the following:
1) $\alpha_Z(\Lambda_{EW}) = \alpha_Z(M)/\{1+(\alpha_Z(M)/2\pi)
(K_2\,or K_3)\ln(\Lambda_{EW}/M)\}$ where $K_2 =5/3$
for $\varphi_{1/2}^{(Z)}$
%for an $SU(2)_Z$ doublet $\varphi$ 
and $K_2 =2/3$ for $\varphi_{1}^{(Z)}$;
%for an $SU(2)_Z$ triplet $\varphi$; 
2)
$\alpha_Z(m_2) = \alpha_Z(\Lambda_{EW})/\{1+(\alpha_Z(\Lambda_{EW})/\pi)
\ln(m_2/\Lambda_{EW})\}$; 3)
$\alpha_Z(m_1) = \alpha_Z(m_2)/\{1+(7 \alpha_Z(m_2)/3\,\pi)
\ln(m_1/m_2)\}$; 4)
$\Lambda_{Z}=m_{1} \exp\{-(3\pi/11\alpha_Z(m_1))(1-\alpha_Z(m_1))\}$.
If we assume $m_1=0$, then one can skip (3) and replace (4) by
$\Lambda_{Z}=m_{2} \exp\{-(3\pi/7\alpha_Z(m_2))(1-\alpha_Z(m_2))\}$.
%As one can readily see there, the two
%free parameters are $m_2$ and $\alpha_Z(M)$. 
%When we
%We will require $\alpha_{Z}(\Lambda_Z) \sim 1$ at
%$\Lambda_{Z} \sim 10^{-3}\,eV$.
 
For comparison, let us recall that the $SU(2)_L$
coupling at $M_Z$ is $\alpha_{2}^{-1}(M_Z) \sim 29.6$. At
$2\,TeV$, it is roughly (depending on unknown particle threshold)
$\alpha_{2}^{-1}(2\,TeV) \sim 28.2$. At $M= 2 \times 10^{16}\,GeV$,
$\alpha_{2}^{-1}(M) \sim 45$ for the non-SUSY case and
$\alpha_{2}^{-1}(M) \sim 24$ for the SUSY case with three
generations.
%where
%$\alpha_{Z}(\Lambda_Z) \sim 1$, this fixes a
%relationship between $m_2$ and $\alpha_Z(M)$. We show below
%a couple of numerical examples (a more comprehensive
%analysis will be presented elsewhere).

In Table I, we list the results coming from the requirement:
$\alpha_{Z}(\Lambda_Z) \sim 1$ at $\Lambda_{Z} \sim 10^{-3}\,eV$,
for doublet (bi-fundamental) $\varphi_{1/2}^{(Z)}$ and 
triplet $\varphi_{1}^{(Z)}$.

\begin{table}
\caption{\label{tab:table1} Correlations between $m_1$, $m_2$
and $\alpha_{Z}^{-1}(M)$} 
\begin{ruledtabular}
\begin{tabular}{ccccc} 
&$m_1$&$m_2$&$\alpha_{Z}^{-1}(M)$ \\ \hline
GUT ($\varphi_{1/2}^{(Z)}$): $M= 2 \times 10^{16}\,GeV$&
$0$&$30\,MeV$&$30$ \\
&$1.5\,eV$&$30\,MeV$&$33$ \\
&$0$&$200\,GeV$&$34$ \\
&$1\,eV$&$200\,GeV$&$36.5$ \\ \hline
GUT ($\varphi_{1}^{(Z)}$): $M= 2 \times 10^{16}\,GeV$&
$1\,eV$&$200\,GeV$&$31.5$ \\ \hline
PUT ($\varphi_{1/2}^{(Z)}$): $M= 2\,TeV$&
$0$&$30\,GeV$&$25$ \\
&$1\,eV$&$30\,GeV$&$27.8$ \\
&$0$&$120\,GeV$&$25.6$ \\
&$1\,eV$&$120\,GeV$&$28.4$ \\ \hline
PUT ($\varphi_{1}^{(Z)}$): $M= 2\,TeV$&
$0$&$200\,GeV$&$25.6$ \\
&$1\,eV$&$200\,GeV$&$28.2$ \\
\end{tabular}
\end{ruledtabular}
\end{table}


%Notice the correlation between $m_2$ and $\alpha_Z(M)$.
%Some remarks are in order. 
%For the ``GUT'' case and with $m_2 \sim 200\,GeV$, the initial
%coupling is closer in value to the SM one at that scale for
%the doublet $\varphi$ case than for the triplet case. For the
%Early Unification (PUT) scenario at a scale of $\sim 2\,TeV$, both
%doublet and triplet (Georgi-Glashow type) cases 
%give the ``starting'' coupling which is 
%closer in value to the $SU(2)_L$ coupling
%near the electroweak scale. 
%(It is amusing to note that 
%the Georgi-Glasow model in 2+1 dimensions has been studied as
%a toy model for confinement due to topological charges.) 
%in order to understand the structure of its phase transition as
%a prototype for QCD. These studies might be useful for our model.) 
%For the GUT case and for $m_2 \sim 30\,MeV-
%200\,GeV$, the ``starting'' coupling is closer in value to the conventional
%(non-SUSY) GUT value. For the PUT case, 
%the ``starting'' coupling is closer in value to the $SU(2)_L$ coupling
%near the electroweak scale. ``Roughly speaking'', the values
%of the $SU(2)_Z$ coupling somewhat {\em parallel} those of the
%SM $SU(2)_L$ coupling.

%Note that if one assumes $m_1=m_2=0$ and in the absence of $\varphi$, 
%a simple calculation then shows that, for $M= 2 \times 10^{16}\,GeV$,
%$\Lambda_{Z} \sim  10^{-3}\,eV$ provided $\alpha_Z(M)=1/20$ which is
%quite {\em a bit larger} than a typical non-SUSY GUT value. 
From Table I, one notices
how important it is to have a ``large'' value for
$m_2$, a non-zero value for $m_1$ and to have the presence of 
$\varphi^{(Z)}$ 
if we wish $\alpha_Z(M)$ to have a value close to the SM one.

{\bf (II) SUSY $SU(2)_Z$}:

We will assume the mass of the $\chi$ superfield
is of the order of the electroweak scale. 
%namely
%$m_{\chi} = \Lambda_{EW}$ for both partners. 
We will further
assume that SUSY breaking is such that the scalar field
inside $\Phi^{(Z)}_{L,R}$ has a mass $m_S$ while the
fermionic partner remains massless. 
%The basic equations
%for this case are then given by:
%1) $\alpha_Z(\Lambda_{EW}) = \alpha_Z(M)/\{1+(\alpha_Z(M)/2\pi)
%\ln(\Lambda_{EW}/M)\}$; 2)
%$\alpha_Z(m_S) = \alpha_Z(\Lambda_{EW})/\{1+(\alpha_Z(\Lambda_{EW})/\pi)
%\ln(m_S/\Lambda_{EW})\}$; 3)
%$\Lambda_{Z}=m_{S} \exp\{-(3\pi/5\alpha_Z(m_S))(1-\alpha_Z(m_S))\}$.
%In this case, the two free parameters are $m_S$ and $\alpha_Z(M)$.
We obtain:
\{$\alpha^{-1}_Z(M)=22.5$; $m_S=1.5\,GeV$\},
\{$\alpha^{-1}_Z(M)=1/23.5$; $m_S=150\,GeV$\},
for $M= 2 \times 10^{16}\,GeV$, $\Lambda_{Z} \sim 1.4 \times
10^{-3}\,eV$,
%From the above results, one notices that the values of $\alpha_Z(M)$
%are close to 
to be compared with the typical SUSY GUT 
scale $\alpha^{-1}_{GUT} \sim 24$.
%in particular when $m_S$ gets ``large''.


{\bf (III) Comments}:

Although the above results are but a sample of a more comprehensive
analysis, a pattern clearly emerges: The $SU(2)_Z$ gauge coupling
can start out at high energy with a magnitude {\em close} to 
the value of the $SU(2)_L$ coupling and subsequently
%either
%the value of the GUT coupling or, in the case of early unification,
%to the value of the $SU(2)_L$ coupling, and subsequently 
becomes large at a scale
which is {\em eleven orders of magnitude smaller than the QCD scale}.
When the lighter of the two $\psi^{(Z)}$ has a mass at around
$1\,eV$ and the heavier one with a mass of O($100\,GeV$), the
initial $SU(2)_Z$ gauge coupling is {\em remarkably} close
in value to that of $SU(2)_L$. If one simply starts
out with the assumption $\alpha_Z(M) \sim \alpha_2(M)$,  
%With a rather reasonable assumption on the initial value of the
%$SU(2)_Z$ gauge coupling, 
the sought-after scale
$\Lambda_{Z} \sim  10^{-3}\,eV$ arises {\em naturally}. 
%from the
%dynamics of this new ``strong'' interaction. 
Notice that
$\alpha_Z(M)$ is much closer to $\alpha_2(M)$ for the
early unification scenario \cite{BH} than for GUT as can be seen
from Table I. Furthermore, this
scenario is closer in spirit and in construction to the early unification
picture \cite{BH} where the gauge couplings of the various $SU(2)$'s
are assumed to be equal at the PUT unification scale of the order of
a few TeVs. 
%than to a GUT scenario.
%The cosmological
%implication of this feature will be discussed next. Let us
%briefly discuss the physics of the messenger fields and the
%interaction of the $SU(2)_Z$ particles with normal matter.

{\bf(IV)  Messenger fields and their implications}:

The manner in which the $SU(2)_Z$ matter communicates with normal
matter will depend on whether $\varphi^{(Z)}$ is a doublet or
a triplet of $SU(2)_Z$ for the non-SUSY scenario. 
%Let us
%denote these two separate cases by $\varphi_{1/2}^{(Z)}=
%(1,2,Y_{\varphi}=-1,2)$
%and $\varphi_{1}^{(Z)}=(1,2,Y_{\varphi}=-1,3)$
%(under $SU(3) \otimes SU(2)_L \otimes U(1)_Y \otimes SU(2)_Z$).

For $\varphi_{1}^{(Z)}$, one can write down a gauge-invariant
Yukawa coupling between $\varphi_{1}^{(Z)}$, the
normal leptons, and $\psi_{2}^{(Z)}$ (and {\em not}
$\psi_{1}^{(Z)}$, the reason of which will become clear below) as:
\be
\label{yukvarphi}
{\cal L}_{\varphi_{1}} = \sum_{i} g_{Yi}\,\bar{l}_{L}^{i}\,
\varphi_{1}^{(Z)}\,\psi^{(Z)}_{2,R}+ h.c. \,,
\ee
where $l_L$ is a normal lepton doublet and the sum is over
three generations. (This could be accomplished by assuming
a discrete symmetry in such a way that a similar coupling to
$\psi_{1}^{(Z)}$ is forbidden.)
A scattering process of the type
$l_i+\psi^{(Z)} \rightarrow \psi^{(Z)} +l_j$ via a heavy
$\varphi_{1}$ has an amplitude proportional to
$g^{*}_{Yi}g_{Yj}/m_{\varphi^{(Z)}}^2$. What the constraints
on the arbitrary Yukawa couplings $g_{Yi}$ might be is of
great interest for the Dark Matter search.
%and this will be dealt with in a separate paper.

For $\varphi_{1/2}^{(Z)}$ (as well as for
the SUSY $\chi_{L,R}=(1,1,Y_{\chi},2)$), no such Yukawa coupling
can be written down and the interaction of $SU(2)_Z$ matter
with normal matter proceeds through
its anomalous magnetic moment (obtained at two loops)
$\mu^{(Z)}_{1,2} \sim [2\,m_{e}\,m_{1,2}/m_{\varphi^{(Z)}}^2]
[\alpha_{Z}^2 \,/16 \pi^2]\times log terms \times \mu_{B}$,
where $\mu_{B}$ is the Bohr magneton. (Recall that, in this case,
$SU(2)_Z$ matter interacts with the messenger field by exchanging
$SU(2)_Z$ ``gluons''.)
Since $m_1 \sim 1\,eV$, 
%is at most
%$10^{-3}\,eV$ (the scale of $SU(2)_Z$ confinement), 
one can see
that $\mu^{(Z)}_{1}$ is completely negligible. However,
$\mu^{(Z)}_{2}$ can be as large as $10^{-8}\,\mu_B$ (in the
regime where $\alpha_{Z} =1$).
The Lagrangian for the interaction between
$\psi_{1,2}^{(Z)}$ and the photon can be written as \cite{krasnikov}
${\cal L}_{\psi_{1,2}^{(Z)}} = \mu^{(Z)}_{1,2}\,\bar{\psi_{1,2}^{(Z)}}
(\sigma_{\mu\nu}/2)\,\psi_{1,2}^{(Z)}\,F^{\mu\nu} + h.c.$
Notice that $\varphi_{2}^{(Z)}$ cannot decay into lighter normal
SM particles, neither through Eq. (\ref{yukvarphi}) nor through
its anomalous magnetic moment. As a result it is absolutely
{\em stable}

%From $\beta_{0}$ given in the previous section, the inclusion
%of a SM non-singlet ``messenger field'' slows down the
%evolution of the $SU(2)_Z$ gauge coupling more than it would
%be if it were not present. 
%In fact, it helps bring the
%initial value of the coupling closer to the $SU(2)_L$ coupling
%at some unification scale. The extra bonus is that this
%In addition, it can interact with both SM and $SU(2)_Z$
%matters. This makes
%it possible to envision a way to ``detect'' the $SU(2)_Z$ matter.
%Without the messenger fields discussed above, the $SU(2)_Z$ particles
%would be {\em incommunicado} with normal matter, except through 
%gravity and it would be practically impossible to detect them.
%Although it is a possibility, it is however more attractive
%if these particles could interact with normal matter with a strength
%that could be much larger than the gravitational one. 

%Let us recall that the messenger
%fields $\varphi^{(Z)}=(1,2,Y_{\varphi},2\,or\, 3)$ for the 
%non-SUSY case or $\chi_{L,R}=(1,1,Y_{\chi},2)$
%for the SUSY case 
%(under $SU(3) \otimes SU(2)_L \otimes U(1)_Y \otimes SU(2)_Z$).
%which carry {\em both} $SU(2)_Z$ and $SU(2)_L \otimes U(1)_Y$
%quantum numbers can do just that. 
%Hence, $\psi^{(Z)}_{L,R} = (1,1,0,3)$
%or $\Phi^{(Z)}_{L,R} = (1,1,0,3)$ can interact with the messenger
%fields via $SU(2)_Z$ ``gluons''. These messenger fields in turn
%can interact with normal matter via $SU(2)_L \otimes U(1)_Y$
%gauge bosons. As a result, $\psi_{1,2}^{(Z)}$, which behaves like
%``sterile neutrinos'', interacts with normal matter through
%its anomalous magnetic moment (obtained at two loops)
%$\mu^{(Z)}_{1,2} \sim [2\,m_{e}\,m_{1,2}/m_{\varphi^{(Z)}}^2]
%[\alpha_{Z}^2 \,/16 \pi^2]\times log terms \times \mu_{B}$,
%where $\mu_{B}$ is the Bohr magneton. Since $m_1$ is at most
%$10^{-3}\,eV$ (the scale of $SU(2)_Z$ confinement), one can see
%that $\mu^{(Z)}_{1}$ is completely negligible. However,
%$\mu^{(Z)}_{2}$ can be as large as $10^{-8}\,\mu_B$ (in the
%regime where $\alpha_{Z} =1$).
%The Lagrangian for the interaction between
%$\psi_{1,2}^{(Z)}$ and the photon can be written as
%${\cal L}_{\psi_{1,2}^{(Z)}} = \mu^{(Z)}_{1,2}\,\bar{\psi_{1,2}^{(Z)}}
%(\sigma_{\mu\nu}/2)\,\psi_{1,2}^{(Z)}\,F^{\mu\nu} + h.c.$. This will
%be useful for the discussion given below. 

The messenger field $\varphi^{(Z)}$ carrying {\em both} $SU(2)_Z$
and SM quantum numbers can be searched for at colliders such as
the LHC.
As shown below,
$\psi_{2}^{(Z)}$ has all the properties for being a Dark Matter
candidate in the form of a WIMP. The previous discussion will
be useful in the search for the Dark Matter, the
phenomelogy of which will be presented elsewhere. Notice also
that, since both $\psi_{1,2}^{(Z)}$ are ``sterile neutrinos'',
$\psi_{1}^{(Z)}$ with a mass around $1\,eV$ looks fairly
suggestive of the LSND \cite{LSND} one, although it remains
to be seen if this is the case. 
%Whether or not
%it has anything to do with it will be another story.
%Without going into details, one can readily
%see that normal matter will interact with $SU(2)_Z$ ``matter''
%at higher orders, e.g. the scattering $\psi_{SM} + \psi^{(Z)} \rightarrow
%\psi_{SM} + \psi^{(Z)}$ proceeds at a three-loop level, whose
%amplitude is proportional to $\alpha_{Z}^2 \, \alpha_{W}^2 \,/16 \pi^2$,
%with $\alpha_{W}$ referring to the electroweak part. 
%This is {\em very
%feeble}. Could the ``heavy'' $SU(2)_Z$ ``matter'', i.e. the one with
%mass $m_2$, be a candidate for Dark Matter, especially in the form
%of a stable WIMP?

%Can any of these particles
%be detected experimentally? 
%The messenger fields are
%good candidates to search for since they carry SM quantum numbers
%and are assumed to have a mass around the electroweak scale
%$\Lambda_{EW} \approx 250 GeV$. Notice that since $SU(2)_Z$ is
%confining, the messenger fields will bind to form $SU(2)_Z$
%hadrons with a typical size of the order $10^3/eV \sim O(mm)$!
%Can one also ``observe'' $\psi_{1,2}^{(Z)}$? One possibility is
%to exploit the possibility that $\psi_{2}^{(Z)}$ has a
%non-negligible anomalous magnetic moment.



{\bf (B) COSMOLOGICAL IMPLICATIONS}:

%In the Particle Physics discussion, we have presented a model where
%a new interaction grows strong at around $10^{-3}\,eV$. 
%When this
%happens, one expects a phase transition from an unconfined state
%to a confined state or from a chiral symmetric phase of the
%$SU(2)_Z$ fermions to a broken phase to occur. 
%In a discussion of the early
%universe, the average energy dependence of the running coupling
%is usually replaced by the temperature $T$.
%Hence the temperature
%when the $SU(2)_Z$ coupling grows large is around 
%$T_c \sim 10^{-3}\,eV$, which is close to the present
%epoch. The associated phase transition is discussed below.
%The aim of this paper is to suggest that the physics
%related the stage when $SU(2)_Z$ interactions grow strong provides
%a candidate explanation for the origin of Dark Energy and
%perhaps also for Dark Matter. 
%First, we present a ``brief
%thermal history'' of our model, concentrating, for simplicity,
%on the non-SUSY case.

{\bf (I) $SU(2)_Z$ ``brief thermal history''}:

At $T \gg m_i$, where $m_i$ is a generic particle mass, all
normal matter and matter that carries $SU(2)_Z$ quantum numbers
are in thermal equilibrium and are characterized by
a common temperature. The fact that $SU(2)_Z$ matter is
in thermal equilibrium with normal matter is because the
messenger fields, e.g. $\varphi^{(Z)}$, carry both SM and
$\varphi^{(Z)}$ quantum numbers and, therefore, can interact
with normal matter as well as with the $SU(2)_Z$ ``gluons'' and
fermions $\psi^{(Z)}$. (Let us recall
that $\varphi^{(Z)}$ also serves
the purpose of slowing the evolution of the $SU(2)_Z$ coupling
so that it can grow strong at $\sim meV$ if 
$\alpha_Z(M) \sim \alpha_2(M)$.) 

When $T<m_{\varphi^{(Z)}}$, the 
now-non-relativistic messenger field $\varphi^{(Z)}$ begins
to annihilate each other with their number density
decreasing like $n \sim (m_{\varphi^{(Z)}}\,T)^{3/2}
\exp(-m/T)$. For the charged messenger
$\varphi^{(Z),+}$, the dominant annihilation cross section
is the electromagnetic one which goes like $\sigma \sim
\alpha^2/T^2$ which grows with decreasing $T$. Their relic
density would be negligible. For $\varphi^{(Z),0}$, it can
decay into either 2 $Z$ bosons or into lighter SM particles
via 2 $Z$ bosons. Again, its present relic density would be 
negligible.

For $\psi_{2}^{(Z)}$ with mass $m_2$, the situation becomes
interesting when $T<m_2$. Notice that, since 
$m_{\varphi^{(Z)}} > m_2$, $\psi_{2}^{(Z)}$ is a 
{\em stable particle}. 
The dominant annihilation cross section
(into 2 $\psi_{1}^{(Z)}$ or 2 $SU(2)_Z$ gluons) 
behaves as $\sigma_{Z} \sim \alpha_{Z}^{2}(T)/m_{2}^2$. 
As we have seen above, the
crucial feature of the $SU(2)_Z$ model is that its coupling
basically almost ``parallels'' that of the SM $SU(2)_L$.
%In the particle physics discussion above, we noticed that
%the larger $m_2$ becomes, the closer the initial value 
%$\alpha_Z(M)$ is to that of $SU(2)_L$. Take $m_2=200\,GeV$
%for example. 
From the above analysis, one can make the
approximation $\alpha_Z(T \sim M_Z) \sim \alpha_{2} (T \sim M_Z)$.
This implies that $\sigma_{Z}$ is naturally of the order of the
weak cross section for $m_2 = O(100\,GeV)$. This will have
an interesting implication concerning the nature of the dark
matter as shown below. 

After $\varphi^{(Z)}$ and $\psi_{2}^{(Z)}$ decouple, the only
relativistic particles left in the $SU(2)_Z$ sector are
the $SU(2)_Z$ 'gluons'' and $\psi_{1}^{(Z)}$. As we have
mentioned earlier, these particles interact {\em very weakly} with
normal matter and {\em decouple soon} after $\varphi^{(Z)}$ and 
$\psi_{2}^{(Z)}$ went out of equilibrium. Their temperature
$T_Z$ would go like $R^{-1}$. It is important to compare
$T_Z$ with $T$ for the normal matter at the time of Big Bang
Nucleosynthesis (BBN) \cite{kolb}. The SM number of degrees of freedom
(including $\varphi^{(Z)}$) for three families above the top quark
mass is $g_{\ast} = 427/4 + 8= 459/4$ and between $m_e$ and 
$m_{\mu}$ is $g_{\ast}^{'} = 43/4$. Because of entropy
conservation, the relationship between $T_Z$ and $T$
at $m_e < T < m_{\mu}$ is simply $T_Z = (43/459)^{1/3}T
\sim 0.45\,T$. During that period, $g_{\ast}^{SU(2)_Z}=
33/2$ and $\rho_{SU(2)_Z}/\rho_{SM} = (66/43)(43/459)^{4/3}
\sim 0.07$ and one can see that the presence of $SU(2)_Z$
``matter'' will have a negligible effect on BBN. After $e^{\pm}$
decoupling, one obtains $T_Z = [(43/459)(4/11)]^{1/3}T
\sim T/3$. 
%This is more or less how the temperature of the
%$SU(2)_Z$ plasma tracks that of the normal matter one after
%$e^{\pm}$ decoupling.


{\bf (II) Dark Energy}:

We now discuss two possible scenarios: one having to do with
a first-order phase transition from an unbroken 
chiral symmetry to a broken
one, and another with an axion-like potential for $SU(2)_Z$.
If the present universe is dominated by a vacuum energy of
the order of $(\Lambda_Z)^4 \sim (10^{-3}\,eV)^4$ and is
entering an inflationary stage, how long will it last? The first-order
phase transition proceeds by the nucleation of bubbles of the
true vacuum with a rate $\Gamma = A \exp\{-S_{E}\}$, where $A$
is a prefactor and $S_E$ is the Euclidean action. If this rate
is small, the inflationary period can last for a long time.
(Notice also that $SU(2)_Z$ provides a rationale for the scale
in front of the flat potential of \cite{hung}.) 

{\bf (a) Chiral phase transition}:

%If the phase transition is first order, one expects
%the vacuum energy density to be of the order of $(10^{-3}\,eV)^4$
%which becomes the dominant one at the present time. Since this
%is a strong coupling theory, a study of the nature of the
%phase transition is highly complicated. 
Extensive studies of
chiral phase transition in QCD have been performed and they
appeared to suggest that it is of first-order. One might
expect a similar first-order phase transition for
$SU(2)_Z$. 
%(or even a strong first-order transition if
%the global symmetry of the $SU(2)_Z$ fermions is $U(1)_L \otimes
%U(1)_R$ \cite{appelquist}). 
There have also been studies of the
chiral phase transition by bubble nucleation, e.g. one in which
the linear $\sigma$ model is used \cite{chi}, and there it is
seen that, at $T=T_c$, the bubble nucleation
rate is negligible. A similar study for the $SU(2)_Z$ case is
under investigation. 
%The phase structure of the Georgi-
%Glashow model in 2+1 dimensions (also called the Polyakov model)
%has been explored at finite temperatures \cite{antonov} which
%also included one flavor of fermion in the fundamental
%representation.
%with the prime motivation being the study
%of the phase structure of QCD itself. 
Since one scenario for 
the messenger field involves $\varphi^{(Z)}$ in the adjoint 
representation, one can also investigate
the phase structure of the Georgi-
Glashow (Polyakov) model in 2+1 dimensions \cite{antonov}.
%that study would be even more relevant
%here, except that our fermions are in the adjoint representation
%(just like the original Georgi-Glashow model). 
%Until this
%is understood,
%Numerical studies on
%``walking technicolor'' models \cite{king} showed that
%the self-energy of the technifermions has a milder dependence
%of the momentum than the quark self-energy in QCD. Although
%the $SU(2)_Z$ interactions (and fermion representations) are
%not the same as in a generic ``walking technicolor'' model, it
%does share a similar ``walking'' feature, and
%we might expect a similar result concerning the fermion
%self-energy. This might have an impact on the nature of
%the $SU(2)_Z$ chiral phase transition.
%Briefly speaking, one writes an
%effective potential as $V_{eff}(\sigma)= U(\sigma) + V(0)$ with
%$V(0)$ being the vacuum
%energy of order $\Lambda_Z^4 \sim (10^{-3}\,eV)^4$. 
%The potential $U(\sigma)$ satisfies $U(\sigma=0)=0$ and
%$U(<\sigma> \sim \Lambda_Z) = -V(0)$. When $T<T_c$, the universe
%is ``stuck'' in the metastable vacuum where $<\sigma>=0$ and the
%energy density is dominated by $V(0)$. For $F/T \ll 1$, the inflationary
%stage will be effective until bubbles of the true vacuum fill
%up the entire universe.
%This transition can proceed by
%bubble nucleation with a rate $\Gamma = A \exp\{-F/T\}$, where
%$A$ is a prefactor and $F$ is the free energy. Inside the bubbles
%is the true vacuum where $<\bar{\psi}^{(Z)} \psi^{(Z)}> \neq 0$.
%the computation of the nucleation rate via this avenue 
%would be highly non-trivial and is beyond the scope of this paper. 
%There, one could make an assumption
%on the effective potential $V_{eff}(\sigma)$, write down
%the free energy $F$, minimize it to find the radius of the critical
%bubble and study $F$ as a function of $T$ for the critical bubble.

%We now discuss another possible scenario which is also
%related to the dynamics of $SU(2)_Z$ strong interactions.


{\bf (b) Axion-like potential}:

There has been interesting proposals to use the QCD (or QCD-like) axion 
\cite{axion} potential $V(a) =V_{0}[1-\cos\frac{Na}{v}] -
\eta\cos[\frac{a}{v} +\gamma]$ ($V_0 \sim \Lambda_{QCD}^4$)
as a model for the early inflationary scenario \cite{freese}.
%It has been observed that the axion $a$ acquires a potential with a
%$Z_N$ symmetry of the form $V(a) =V_{0}[1-\cos\frac{Na}{v}] -
%\eta\cos[\frac{a}{v} +\gamma]$, where the first term is the QCD
%instanton potential with $N$ degenerate vacuua and the second term
%is an explicit soft-breaking term of the $U(1)_{PQ}$ symmetry. The
%factor $V_{0} \sim (\Lambda_{QCD})^4$. 
%The required number of e-folds
%has been estimated in that model.
We would like to propose a similar scenario but, this time, as an
explanation for the origin of the dark energy. The $SU(2)_Z$
axion comes from a $U(1)_{``PQ''}$-like symmetry of the $SU(2)_Z$
fermions. For the purpose of this paper, we will only need
to consider the case of the $SU(2)_Z$ ({\em zophos}) instanton
potential with $N=1$. The $SU(2)_Z$ axion $a_Z$ (where 
$\phi_Z = \frac{1}{\sqrt{2}}(v + \rho) \exp(i\frac{a_Z}{v})$) has
a periodicity $a_Z \rightarrow a_Z + 2\pi\,v = a_Z + 2 \pi f_a$, 
where we assume
that the axion decay constant $f_a = O(v)$. (For this analysis, it
is irrelevant whether or not $f_a=v$ as long as they are of the same 
order.) We write
\be
\label{potential}
V(a_Z) = \Lambda_Z^4 [1-\cos\frac{a_Z}{v}] -\epsilon\,\frac{a_Z}{2\pi v}\,.
\ee
In Eq.(\ref{potential}), the first term on the right hand side
represents the potential with two degenerate vacuua and the
second term represents a soft-breaking term lifting that degeneracy.
The difference in energy density between the two vacuua is $\epsilon$.
We will assume that $\epsilon \leq \Lambda_Z^4$ but not too different
from it and at the same time have a small barrier between the two vacuua. 
In other words we assume $\epsilon = O(\Lambda_Z^4)$ and
try to see its consequence on the cosmology of the model. The false
vacuum of the ``axion'' potential is at $a_Z = 2 \pi v$ and the
true vacuum is at $a_Z=0$. The scenario goes as follows.

(i) As $T_Z$ drops below $T_c \sim \Lambda_Z \sim 10^{-3}\,eV$, the universe
is trapped in the false vacuum with $a_Z = 2 \pi v$. The total energy
density is dominated by $\epsilon = O(\Lambda_Z^4)$. 
%and the present
%universe is entering an inflationary stage. 
%Actually, the
%microwave background temperature which corresponds to $T_c$
%is roughly $3\,T_c$ as discussed above.

(ii) The first order phase transition to the true vacuum at $a_Z=0$
proceeds by bubble nucleation. 
%The rate of the nucleation of the
%true vacuum bubble is written as $\Gamma = A \exp\{-S_E\}$ and
The Euclidean action $S_E$, in the thin wall limit, can be
computed by looking at $\tilde{S}=\int_{a_Z = 2 \pi v}^{a_Z=0}
\sqrt{2\,\Lambda_Z^4\,[1-\cos\frac{a_Z}{v}]} da_Z= 8\,
v\,\Lambda_{Z}^{2}$ giving
	\be
\label{action}
S_E =  \frac{27\,\pi^2\,\tilde{S}^4}{2\,\epsilon^3} \geq
5 \times 10^{5}\,(\frac{v}{\Lambda_Z})^4 \,.
\ee
The scale $v$ of $U(1)_{``PQ''}$ breaking is unknown but even
when it is of the order of the electroweak scale, one would
still get $S_E \geq 5 \times 10^{61}$ which is huge. This will
assure us that the transition will take a very long time from
the present epoch to complete. 
%The analysis will be presented
%elsewhere.

{\bf (III) Dark Matter}:

There are several possibilities concerning Dark Matter candidates
in our model. To be more specific, we will concentrate on the
non-SUSY case. 
%and leave the SUSY case for future considerations.

%Let us remind ourselves of some important facts. 
A combination
of various data (WMAP, etc..) gives a constraint for the non-baryonic
dark matter density as follows: $\Omega_M\,h^2 = 0.135_{-0.009}^{+0.008}$,
for $h \approx 0.72$ giving $\Omega_{M} \sim 0.26$.
%There are many candidates for Dark Matter. 
(See \cite{silk} for an excellent recent review.)
%\cite{silk} gives a panorama of the subject. 
Among the candidates
for Dark Matter, there is an interesting one that goes by the name
of WIMP (for weakly interacting massive particle). An approximate
solution to the Boltzman equation gives \cite{kamionkowski} 
$\Omega_{\chi}h^2
\sim 3 \times 10^{-27}\,cm^3\,s^{-1}/<\sigma_{A}v>$, where $\chi$
denotes a generic WIMP, $\sigma_A$ the annihilation cross section
and $v$ the relative velocity. It is generally noticed 
\cite{kamionkowski}that
$\Omega_{\chi}h^2$ is of order unity if $\sigma_{A}$ is of a typical
weak cross section, e.g. $\sigma_{A} \sim \alpha^2/m^2$ with
$\alpha \sim O(0.01)$ and $m \sim O(100\,GeV$. In brief, a stable WIMP
with a weak cross section might fulfill the requirement of being
a plausible candidate for Dark Matter.

The $\psi_{2}^{(Z)}$ particle with mass $m_2 \sim
O(100\,GeV)$ is just such a candidate. It is {\em stable} (see
the comment made in the particle physics discussion). Its
annihilation cross section is typically of the order of the weak cross
section as we have discussed above. Let us recall that the fact
that it is so is because $\alpha_{Z}(T \sim m_2) \sim \alpha_2 (m_2)$, a
notable feature of our model.




\begin{acknowledgments}
I would like to thank Paul Frampton and Marc Sher for useful discussions.
This work is supported in parts by the US Department
of Energy under grant No. DE-A505-89ER40518. 
\end{acknowledgments}
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\end{document}

hep-ph0504060.extracted_bib
\begin{thebibliography}{99}
\bibitem{acceleration} S. Permutter {\em et al.}, Astrophys. J. {\bf 517},
565 (1999); A. Riess {\em et al.}, Astron. J. {\bf 116}, 1009 (1998).
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\bibitem{CTIO} M. Jarvis {\em et al.}, astro-ph/0502243.
\bibitem{BH} A.~J.~Buras and P.~Q.~Hung,
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Phys.\ Rev.\ D {\bf 68}, 035015 (2003);
A.~J.~Buras, P.~Q.~Hung, Ngoc-Khanh ~Tran, A. ~Poschenrieder, E. ~Wyszomirski,
Nucl.\ Phys.\ B {\bf 699}, 253 (2004).
\bibitem{krasnikov} S. N. Gninenko and N. V. Krasnikov, Phys. Lett.
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\bibitem{kolb} E. W. Kolb and M. S. Turner, {\em The Early Universe},
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