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\begin{document}
\title{Dark Matter from QCD-balls}
\author{Ariel R. Zhitnitsky}
\affiliation{Department of Physics
 and Astronomy, University of British Columbia,
  Vancouver, BC V6T 1Z1, Canada}
%\date{11.11.2002}

\preprint{}
\begin{abstract}
 We discuss a novel cold dark matter candidate which
 is formed from the ordinary quarks
during the QCD phase transition when  the  axion domain wall
undergoes an unchecked collapse
due to the tension in the wall. If a large number of quarks
is trapped inside the bulk of a  closed  axion domain wall, the collapse 
 stops due to the internal  Fermi pressure. In this case
the system in the bulk, may reach the critical 
density when it undergoes a phase transition to a color  superconducting phase
with the ground state  being the  quark condensate,
 similar to BCS theory. If this happens, the new state of matter
representing the diquark condensate with a 
large baryon number $B \sim  10^{32}$
  becomes  a stable soliton-like  configuration. Consequently, it may 
serve as a  novel cold dark matter candidate. We also discuss
a possibility that baryogenesis happens exactly at the same
 instant during the QCD  phase transition.


\end{abstract}
\pacs{11.27.+d , 14.80.Mz , 95.30.Cq , 95.35.+d}

\maketitle
\section{Introduction}
The presence of large amounts of non-luminous components in the Universe has been
known for a long time. In spite of the recent advances in the field ( see e.g. recent summary
 of the Snowmass 2001 P4 Working group, \cite{snowmass}), the mystery of the 
dark matter/energy remains: we still do not know what it is. The main goal of this letter
is to argue that the dark matter could be  nothing but well-known quarks
which however are not in the ``normal" hadronic phase, but rather
in some ``exotic", the so-called color superconducting (CS) phase. 

This is a novel phase in QCD
when light quarks form the condensate in diquark channels, and it  is  analogous to Cooper pairs 
of electrons in ordinary superconductors described by BCS theory. There existence 
of CS phase in QCD represents our first crucial
element for our scenario to work. The study of CS phase received a lot of attention   
last few years, see original papers\cite{cs_o},\cite{cs_n}
 and recent reviews\cite{cs_r} on the subject.
 It turns out that CS phase is realized when quarks are squeezed to the density which 
is not extremely large, but rather, only few times
nuclear density. It has been known that this regime may be realized in nature in neutron stars
interiors and in the violent events associated with collapse of massive stars or collisions 
of neutron stars, so it is important for astrophysics. The goal of this letter is to argue
that such conditions may occur in early universe during the QCD phase transition,
 so it might be important for cosmology as well.

 The force which squeezes quarks in neutron stars is gravity; the force which does
a similar job in early universe  during the QCD phase transition is 
a violent collapse  of a bubble formed from  the axion domain wall.
 If  number of quarks trapped inside of the bubble (in the bulk)
 is sufficiently large, the collapse 
 stops due to the internal Fermi pressure. In this case the system
in the bulk  may reach the critical 
density when it undergoes a phase transition to CS phase 
with the ground state  being the  diquark condensate. 
We shall call the configuration with a large number of quarks in color superconducting phase 
formed during the QCD phase transition as the QCD-ball.
Therefore, an existence of the axion domain wall
represents our second crucial element for our scenario to work.
We should note at this point that the axion field was introduced into the theory to explain 
the lack
of CP violation in the strong interactions. Later on the axion field
became one of the favorite candidates for the 
cold dark matter, see original papers \cite{PQ}-\cite{DFSZ}
 and recent reviews \cite{axion_r} on the subject. 
In the present scenario the axion field plays the role of squeezer rather than 
dark matter itself. In principle, it can be replaced by some other, yet unknown fields with 
similar properties, but  to be more concrete in estimates which   follow we shall use 
specifically  the axion field with known constraints on its coupling  constant.

We do not address the problem of formation of QCD-ball  in this letter. 
Instead we concentrate  on
the problem of stability of these objects. As we will show, once
such a configuration is formed, it will be extremely stable soliton like particle.
The source of the stability of the QCD-balls is related to the fact that
its mass $M_B$ 
%growth as $M_B\sim B^{8/9}$ for the large baryon (quark) charge $B$
becomes smaller than the mass of a collection of free separated nucleons with the same
baryon charge. The region of the absolute stability of the QCD-balls is determined by inequality
$m_N > M_{B}-M_{B-1}$ which is satisfied in some region of $B$, i.e. $B_{min}< B < B_{max}$.
The lower limit $B_{min}$ in this region determined  
 by inequality $m_N > M_{B}-M_{B-1}$ when the system becomes unstable with respect
 to decay to the nucleons. 
The upper limit $B_{max}$   is determined by the region of  applicability of our approach
when the baryon density   in the bulk becomes close to 
   the nuclear density, and therefore,  
 our calculation scheme (based
 on description in  terms of quarks )  becomes unjustified at this point.
Other approaches   based on consideration of hadronic rather than quark degrees of 
freedom have to be used in this regime. 
 It could  happen that some  metastable (or even stable) states may exist
 in this low-density regime. An analysis of such states would be an   interesting subject 
 for  future investigation due
 to the  phenomenological relevance of metastable states capable for
  production of  heavy elements observed in nature.
 However, the  corresponding analysis   is beyond
of the present letter and it shall not be considered here.

Therefore, if sufficiently large number of quarks (determined mainly
by the axion domain wall tension)  is trapped inside the 
axion bubble during its shrinking, it may result in formation of an absolutely stable 
 QCD-ball with the ground state being a diquark condensate.  
 Such QCD-balls, therefore,  may serve as the cold dark matter candidate which amounts
about 30\% of the total matter/energy of the Universe, $\Omega_{DM}\simeq 0.3 $\cite{snowmass}.

 Strictly speaking, the QCD-balls being the baryonic configurations, would behave like 
nonbaryonic dark matter. In particular, QCD-balls, in spite of their QCD origin, would not 
contribute
to  $\Omega_{B}h^2\simeq 0.02$ in nucleosynthesis calculations because 
the QCD-balls would complete the formation by the time 
when temperature reaches the relevant for nucleosynthesis
region   $T\sim 1 MeV$. Once QCD-balls are formed, their
 baryon charge  is accumulated in form of the diquark condensate, rather than in form
 of free baryons, and in such a form the baryon charge is not available for  nucleosynthesis.
Therefore, the observed relation $\Omega_B\sim\Omega_{DM}$ within an order of magnitude 
finds its natural explanation in this scenario: both contributions to $\Omega$ originated from  the 
same physics at the same instant during the QCD phase transition. As is known, this fact is extremely
difficult to explain in models that invoke a dark matter candidate not related to baryons.
In Section IV we shall also present some arguments to support the idea
that   the observed in nature asymmetry between baryons and antibaryons  may be  also originated 
from the same physics   during the QCD phase transition.
 More specifically, we shall argue that all
three Sakharov's criteria \cite{Sakharov} are satisfied 
during the instant when domain walls collapse, and the   observed baryon to
entropy ratio $n_B / s \sim 10^{-10}$ ($n_B$ being the net 
baryon number density, and $s$ the entropy density) finds its natural explanation
if it is originated at the QCD scale and if the QCD-balls indeed can accommodate
considerable amount of the dark matter as we propose. However an explicit mechanism for baryogenesis
is still lacking.

Before we continue the description of our proposal we would like to make few 
  comments on what have happened on the theoretical side during the last few years, 
which  are  crucial elements in our present discussions, and which were  not available
 to researchers earlier. 

First of all, there existence of the axion domain
walls, related to the symmetry under
discrete rotations of the so-called  $\theta$ angle $\theta\rightarrow\theta+2\pi n$
(which becomes a dynamical axion field $\theta(x)$)
has been known for a long time since \cite{Sikivie}.
  However, the structure of the domain wall considered
in  \cite{Sikivie} was a such that it has only one scale which is 
a typical width of order $m_a^{-1} \gg 1 fermi$. 
Therefore, the quarks, even if they were trapped inside the bubble at the very first moment,
 could easily penetrate through such domain wall configuration later on.
%without   noticing a weak axion field extended over a large distance $m_a^{-1} \gg 1 fermi$.
In this case the axion domain wall ( without support of
the fermi pressure from the bulk) would completely collapse.
What was realized only quite recently, is the fact that the axion domain walls have 
actually sandwich substructure
on the QCD scale $\Lambda_{QCD}^{-1}\simeq 1 fermi$. Therefore, the fermions
which are trapped inside the bubble at the very first instant,
can not  easily penetrate through  the  domain wall due to this  QCD scale substructure,
and will likely stay in the bulk, inside the bubble. In this case, 
the collapse of the axion domain wall stops due to the fermi pressure in the bulk.
The arguments ( regarding there existence of the
QCD scale substructure inside the axion domain walls) are based on analysis\cite{FZ}
of QCD in the large $N_c$ limit with inclusion of the  $\eta'$ field\footnote{
Uniqueness of the $\eta'$ field in this problem is related to the special
structure of interaction of  the axion field $\theta(x)$ and the singlet  $\eta' (x)$  field 
in low energy description of QCD when only a special combination $[\theta(x)- \eta'(x)]$  
is allowed to enter the low energy QCD Lagrangian. } and  independent
analysis \cite{SG} of supersymmetric models
where a similar $\theta$ vacuum structure occurs.
 
The second important element  of our proposal 
is related to the recent advances\cite{cs_n},\cite{cs_r} in understanding of CS phase,
not available earlier.
The fact that the color superconducting phase may exist at high baryon density
was discussed a while ago\cite{cs_o}, however it was not a  widely accepted phenomenon
 until recent papers\cite{cs_n} where a relatively large superconducting gap 
$\Delta\sim 100 MeV$ with a large critical temperature $T_c\simeq 0.6 \Delta$ were advocated.

To conclude the Introduction we should remark here that the idea that some 
quark matter, such as strange quark ``nuggets" may play a role of the dark matter,
was suggested long ago\cite{Witten}, see also original papers\cite{Jaffe}  
and relatively recent review\cite{Madsen}
 on the subject. 
The idea that soliton-like
configurations may serve as a dark matter, is also not a new idea. 
Most noticeable example is being
Q-balls\cite{qball}. The idea that the dark matter may be just  solitons containing large 
 baryon  (or even antibaryon) charge is, again, an old idea\cite{Widrow}, see also
\cite{baryogenesis}. The new element of this proposal
 is an explicit demonstration that one can  accommodate all the nice
properties discussed previously \cite{qball}- \cite{baryogenesis} but  without invoking 
any new fields and particles (apart from the axion), such as 
any super-partners, new scalar fields, squarks,
etc,  which supposed to be the constituents filling the bulk of
a new soliton-like configurations. Rather, our QCD-balls formed from the ordinary quarks
which however are not in the ``normal" hadronic phase, but rather
in  color superconducting  phase when  squeezed quarks organize a single coherent state 
described by the diquark Bose--condensate, similar to BCS theory in ordinary superconductor.

In many respects ( in terms of phenomenology)
the QCD balls are similar to strangelets\cite{Witten}-\cite{Madsen}
 with few important   differences:\\
 1. In our proposal
the first order QCD phase transition is not required 
for the formation of  the QCD-balls. Axion domain walls of a large size (in comparison with
a typical QCD scale) are able to  form  the large bubbles. These bubbles,
 filled by $u, d, s$ quarks, play the same role as the bubbles formed during
the first order phase transition as discussed in\cite{Witten}.\\
2.   The Stability of strange quark matter at  zero external pressure, as described 
in\cite{Witten}-\cite{Madsen}, is highly  model dependent result. In particular,
the stability of strangelets is very sensitive  to the
magnitude of the bag constant within MIT bag model calculations.
The idea  which is advocated in the present letter has a new element, the external pressure
due to the axion domain walls. With  this new element  the  stability 
of the system is very likely to occur in very wide region of the parametric space
even in the models which would not support strangelets in the absence of the external pressure.\\
3. There is a maximum size
 of the QCD-ball above which such an object can not be formed and can not be stable.
This is due to the fact that for very large system the axion domain wall pressure
becomes a negligible factor  which can not stabilize the system.\\
 4.The property on a maximum size mentioned
 above has a profound
phenomenological consequence. Indeed,  a general objection which is considered
 as a strong argument against of there existence of strangelets
in the form of a  dark matter, is as follows. Even if one assumes that such state of dark matter exists,
 the strangelets can collide with a   ordinary neutron star which results in formation of a quark
star.  In such a case all neutron starts would be transformed into quark stars long ago.
 This is definitely in  contradiction with the  observations that there are well studied
 ordinary neutron stars which have   typical masses, sizes etc. they supposed to have.
In our proposal, when the maximal 
size of the QCD ball is determined by the external axion domain wall
pressure, such a problem does not occur at all. 
 

\section{QCD-balls} 
Crucial for our scenario is the existence of a squeezer, axion domain wall which will be formed
 during the QCD phase transition. As is known, there are many types of the axion domain walls, 
depending on a model. We assume that the standard problem of the 
domain wall dominance is resolved in  some way as discussed previously
in the literature, see e.g.\cite{axion_r},\cite{Sikivie1},
and we do not address this problem in the present paper\footnote{It is widely accepted that
the domain walls in the so-called, N=1 axion model will be eaten up by the axion strings at
a very high rate. That is true for the axion walls bounded by strings.
 However, if a  domain wall is formed as a closed surface, the probability for such a wall to decay 
 is extremely small. Therefore, such domain walls   
 in $ N=1 $ model can play the same role in our scenario as stable domain walls in $N\neq 1$ models.
 Besides that,
 $ N=1 $ model  has a nice property that the domain wall dominance problem is automatically resolved.}.
We also assume that the probability
of formation of a closed bubble made from the axion domain wall
is non-zero\footnote{We do not attempt to develop 
a quantitative theory of the formation 
of the QCD-balls in this letter; It is sufficient for our following discussions  
 that this probability is finite.}. 
 We also assume that quarks which are trapped in the bulk, can not
 easily escape  
 the interior when the bubble  is shrinking. 
 In different words, the axion domain wall is not 
transparent due to the QCD sandwich structure of the wall  as 
 discussed in \cite{FZ},\cite{SG}. 
 The collapse is halted due to the Fermi pressure.
 Therefore, we assume that  a large number of quarks remains in the bulk,  inside of the bubble
 when the system reaches the equilibrium. 
\subsection{Equilibrium}
The equilibrium is reached  when the Fermi pressure cancels 
 the surface tension and pressure due to the  bag constant $E_B$.
To put this condition on the quantitative level, we 
represent the total energy $E$ of a QCD-ball with the fixed baryon charge $B$, 
in the following way,
 \be
\label{1}
E = 4\pi\sigma R^2+ \frac{g\mu^4}{6\pi}  R^3  +\frac{4\pi}{3}E_B  R^3 ~~~~~~~~
\\ \nonumber
B= g V\int^{\mu}_0\frac{d^3p}{(2\pi)^3}= \frac{2g}{9\pi}\mu^3R^3,~~
\mu=\left(\frac{9\pi B}{2 g R^3}\right)^{\frac{1}{3}}, \nonumber
\ee
where we assume the quarks to be massless, and 
relativistic fermi gas to be  non-interacting for a moment, see corrections
due to the interactions below.
In this formula
 $\mu$ is the Fermi momentum of the system  to be expressed in terms of the fixed
baryon charge $B$ trapped in the bulk, $R$ is the
size of the sysytem, $g$ is the degeneracy factor, $g\simeq 2N_cN_f =18$
 for massless degrees of freedom, $E_B$ is  bag constant 
 which describes the  difference in vacuum energy
between the interior and exterior and 
which is a phenomenological way to  simulate the confinement, 
finally, $\sigma\simeq f_a m_{\pi}f_{\pi}$ is  the axion domain wall tension
with $f_a\sim (10^{10}-10^{12} ) GeV$ being constrained by the axion search experiment. 

In  what follows, it is convenient to introduce dimensionless scaling variable $x$,
as follows, $x\sqrt[3]{B}= R \sqrt[4]{E_B}$ such that energy per quark $\epsilon_{tot}\equiv E/B$
can be expressed in the following simple way in terms
of dimensionless parameters $x$ and $\sigma_0$,
 \be
\label{2}
 \epsilon_{tot}(x)\equiv\frac{E}{B} = E_B^{1/4}\left(
 4\pi \sigma_0 x^2+ \frac{3}{4x}  \sqrt[3]{\frac{9\pi}{2g}}
  +\frac{4\pi}{3}x^3
\right) \\   \nonumber
x\equiv R\frac{E_B^{1/4}}{B^{1/3}}, ~~~~~~~~~~~
\sigma_0\equiv \frac{\sigma}{B^{1/3}E_B^{3/4}}. ~~~~~~~~~~~
\ee
The minimization of this expression $ {\d\epsilon_{tot}(x)}/{\d x}|_{x=x_0}=0$ determines 
the stability radius $x_0$ which  fixes the energy of the system
at the equilibrium, $\epsilon_{tot}(x_0)$. In particular, 
if one neglects $\sigma_0$ in eq. (\ref{2}) originated from  the axion domain wall tension, one reproduces
the well known results, $x_0\simeq 0.48, ~\epsilon (x_0)\simeq 1.9 E_B^{1/4}$. Such a relation means
that if  $E_B$ is relatively small  such that  the energy per quark is less than
$m_N/3$, the configuration becomes an absolutely stable state of matter\cite{Witten}-\cite{Madsen}.

In  eqs.(\ref{1}, \ref{2})  we have neglected many important contributions   
which  can drastically change  the  results.   We shall review the role of these contributions
 below.
 The main goal of our discussions is an analysis of   how these contributions
can be incorporated    into eqs.(\ref{1}, \ref{2}). 
First of all, in eq. (\ref{1}) we neglected 
the quark-quark interaction on 
the Fermi surface, which brings the system into  superconducting phase for relatively
 large  baryon density \cite{cs_n}. 
 The corresponding contribution $\Delta E_{int}$
to  the total energy (\ref{1}) is negative and at asymptoticaly large $\mu$
is equal to\cite{Krishna}, 
\be
\label{3}
\Delta E_{int}=-\frac{3\Delta^2\mu^2}{\pi^2}\cdot(\frac{4\pi}{3}R^3)
\ee
 The negative sign of $\Delta E_{int}$
 is quite obvious: the formation of the diquark condensate
due to the quark-quark interaction lowers the energy of the system.
 For appropriate treatment of this term
 one should express $\mu $ as a function of $B, R$ according to the 
relation (\ref{1}) and substitute this 
 into  eq. (\ref{2}). In principle, one should also take into account that
 the superconducting gap $\Delta(\mu)$ also strongly varies with $\mu$ (and therefore, with $R$)
 in the relevant region of  $\mu$.
  However, in what follows  we shall ignore this dependence for numerical estimates  
  and shall treat $\Delta \simeq 100 MeV$ as constant. 
Our last remark regarding eq. (\ref{3}). This formula was derived for 
very large $\mu$. Nevertheless for illustrative purposes we shall    use the
 expression for $ \Delta E_{int}$ literally  for  relatively small $\mu$.
We shall see that 
in the relevant region of densities this correction $\Delta E_{int}$ does not exceed
$15\%$ which somewhat justifies the use of expression 
 (\ref{3}) for our numerical estimates which follow.

With all these reservations in mind, we account the additional contribution
to energy per quark,  describing  the quark-quark interaction on 
the Fermi surface by adding $\Delta\epsilon_{tot}^{int}$ into eq. (\ref{2}) in the following way
\be
\label{3a}
\Delta \epsilon_{tot}^{int}
 = -E_B^{1/4}\left(\sqrt[3]{\frac{4}{\pi}}\cdot\frac{\Delta^2}{\sqrt{E_B}}\cdot x\right) ,
\ee
where we expressed everything in terms of dimensionless parameter $\frac{\Delta^2}{\sqrt{E_B}}$ 
and dimensionless variable $x$.

  The next modification of eq. (\ref{1}) we want to consider
   is related to the  actual variation of the  bag ``constant" $E_B$ with $\mu$.
 To explain  the physical meaning of this effect, 
we remind the reader that the  bag ``constant" $E_B$ actually describes the  difference in vacuum energies
of the interior and exterior regions and 
which is a phenomenological
way to  simulate the confinement.
    The bag ``constant" $E_B$   contribution goes with the positive sign to $E$, see eq.(\ref{1}).
    The physical reason for this sign is obvious: the vacuum energy outside the bubble
    is lower than inside, thus the positive  contribution to $E$, 
in contrast with the interaction term, $-\frac{3\Delta^2\mu^2}{\pi^2}$
discussed above.

   Our main point is as follows:  the contribution related to 
    $E_B$ describes the  difference in vacuum energies 
   between the interior and exterior, and therefore, formally can be expressed
  in terms of difference of 
   vacuum condensates calculated at zero (exterior) and non-zero (interior)  baryon densities.
   The most important contribution to $E_B$ is due to the gluon condensate, 
   such that $E_B (\mu)\sim \la \frac{b\alpha_s}{32\pi}G_{\mu\nu}^2\ra_{\mu=0}-
   \la \frac{b\alpha_s}{32\pi}G_{\mu\nu}^2\ra_{\mu\neq 0}$ with $b=\frac{11}{3}N_c-\frac{2}{3}N_f$
where we used the well-known expression for the conformal anomaly in QCD in the chiral limit.
   We do not know $ E_B (\mu)$ as a function of $\mu$  
   for the  relevant region of the baryon density. However we do know
   the behavior of this quantity for relatively small densities corresponding to
   the nuclear matter densities\cite{Cohen}, 
\be
\label{condensate}
\frac{\la \frac{\alpha_s}{\pi}G_{\mu\nu}^2\ra_{\mu\neq 0}}{ \la \frac{\alpha_s}{\pi}G_{\mu\nu}^2\ra_{\mu=0}}
\simeq 
  1-\frac{( 0.65 GeV) \rho_N}{  \la \frac{\alpha_s}{\pi}G_{\mu\nu}^2\ra_{\mu=0}}\simeq
1-\frac{\rho_N}{(264 MeV)^3}
\ee
 where $\rho_N $ is
   baryon density, and the magnitude for the gluon condensate is known to be,
 $ \la \frac{\alpha_s}{\pi}G_{\mu\nu}^2\ra_{\mu=0}\simeq 1.2\cdot 10^{-2} GeV^4$.
 As expected the gluon condensate (and therefore,
   the absolute value of the vacuum energy) decreases when the baryon density increase.
  % We also know the bag constant for $\mu\rightarrow\infty$ 
  % when gluon condensate at large densities vanishes  and $E_B$ corresponds to the 
  % vacuum energy in the hadronic
  % phase, $E_B\simeq (150- 170 ~ MeV )^4$.
 Similar formulae are known for the 
   chiral quark condensate where for the small densities one can derive
   the following relation  $\frac{\la\bar{q}q\ra_{\mu\neq 0}}{\la\bar{q}q\ra_{\mu= 0}}
   =1-\frac{\sigma_N \rho_N}{m_{\pi}^2f_{\pi}^2}$ with sigma term measured
   to be $\sigma_N\simeq 45 MeV$ see \cite{Cohen} for the details.
One should emphasize here that the 
formula (\ref{condensate}) describing the 
variation of the gluon vacuum condensate at small 
baryon densities  $\rho_N $,
is a direct consequence of the QCD low energy theorems.
It is a firm result of QCD,  not based on any model dependent considerations,
and should be accepted as it is.

   More specific information on the
bag ``constant" $E_B$ contribution as function of $\mu$ (or, what is effectively the same, baryon density $\rho_N$)
in the entire region of   of $\mu$ can be calculated
   in some non-physical models such as  QCD with two colors, $N_c=2$\cite{ssz_u}.
Such a knowledge can not be literally used for our numerical estimates which follow,
however it can  be quite usefull for the modelling of the effect we are 
discussing: analysis of the 
variation of  the  vacuum energy (described by $\sim E_B$ in eq.(\ref{1}))  as function of baryon density.
 
Therefore, we want to model  two properties  discussed above
 in order to incorporate them into the corresponding  eq. (\ref{2}).
First,  the bag constant contribution must vanish
when the baryon density in the bulk vanishes. This corresponds
to the case when vacuum energy inside and outside of the bubble is the same, 
and therefore, it should be 
no an additional vacuum energy contribution to the equation for the equilibrium. Secondly,
the bag constant contribution should vary with density as we discussed above.
%approach the constant $E_B$
%for relatively large densities when the nonperturbative gluon
%vacuum energy inside the bubble is almost zero while it is large and 
%negative, $-E_B$, outside  the bubble. 

Our first parametrization
is motivated by analysis\cite{ssz_u} of the vacuum condensates
in QCD-like theories at finite baryon density as a function of $\mu$.
If we assume  a similar behavior  in real QCD than we 
   should replace   the bag constant $E_B$
by the expression $E_B\rightarrow E_B(1-\frac{\mu_c^2}{\mu^2})$ for $\mu \geq \mu_c$
and $E_B\rightarrow 0$ for $\mu \leq \mu_c$, where $\mu_c$ would correspond to a  magnitude
of the critical
chemical potential at  which the baryon density vanishes.
In  QCD, one expects that this is to happen at
$\mu_c\simeq 330 MeV$.


   As before, one should  express the corresponding contribution
to $\epsilon_{tot}$
 in terms of fixed baryon charge  $B$ and radius $R$, such that the 
   bag`` constant" contribution actually becomes a complicated  function of
   $B, R$.  In terms of dimensional parameter $x$ 
the corresponding contribution to (\ref{2}) is accounted for by the following replacement,
\be
\label{3b}
E_B^{1/4}\frac{4\pi}{3}x^3\Rightarrow E_B^{1/4}\frac{4\pi}{3}x^3\cdot
\left(1-(\frac{4}{\pi})^{2/3}\cdot\frac{\mu_c^2}{\sqrt{E_B}}x^2\right)
\ee
Let us emphasize: we are not attempting to solve a difficult  problem of 
evaluation of nonperturbative vacuum energy as a function of $\mu$ in QCD.
Rather, we want to make some simple estimates to account for this effect in order to analyze
the stability of QCD balls later in the text.
%Let us emphasize: we are not attempting to solve the quantitative problem of equilibrium 
%of large  chunks of matter between CS and hadronic phases
%when all corrections mentioned above( as well as many others which are   not even mentioned)
%are taken into account. In such a case
 % the equilibrium, for example, can be achieved even without the wall tension $\sigma$
%\footnote{ Such a problem has been 
%recently discussed in ref.\cite{Krishna}
%where  the interface region between 
%nuclear matter and CFL (color flavor locking) 
%superconducting phase was analyzed.}.  Rather, we are attempting to estimate the initial 
%size $R_0$ of the QCD ball which is formed
%due to the collapse of the axion domain wall halted by the fermi pressure. 
%As we mentioned earlier,  this estimate
%is   more   qualitative rather than quantitative one. 
%Other contributions, such as quark-quark interaction $\sim \Delta^2\mu^2$
 %or the bag ``constant", $E_B$ will change the numerical estimates for $R_0$ (and, therefore,
%for $E$)  however
%they can not replace the main players of the game, the fermi pressure $P_{f}$ which is
%responsible for the stopping the collapse of the  domain wall with pressure $P_{\sigma}$.

We want to be confident that 
the results on stability of QCD balls (to be discussed later)  are not sensitive to the
 specific parameterization  (\ref{3b}) motivated by the study  of
QCD with two colors. Therefore, we would like to have a different,
 independent parameterization of the same effect to  be used in our stability analysis.
We   make use of eq.(\ref{condensate}) which is valid   for small densities $\rho_N$.
This formula gives us an idea about typical variation of vacuum condensates 
when the baryon density changes. We assume that 
the vacuum energy difference in QCD (the bag ``constant" contribution in eq. (\ref{2}))
can be expressed in terms of different vacuum condensates with the
typical scale for the variation  given by eq.(\ref{condensate}).

We want to implement the QCD property  (\ref{condensate}) into the MIT bag model. 
If the phenomenological numerical magnitude for the bag constant $E_B$
were closed to the numerical value for the vacuum energy 
$ \la \frac{b\alpha_s}{32\pi}G_{\mu\nu}^2\ra\simeq (340 MeV)^4$ we could use 
eq (\ref{condensate}) literally,  such that 
  the bag constant contribution can be parameterized
as follows, $E_B(\rho_N) \simeq E_B\frac{\rho_N}{(264 MeV)^3}$.
Unfortunately, these two are very different numerically, and we will introduce
the corresponding correction factor
 $r\equiv \sqrt[4]{\la \frac{b\alpha_s}{32\pi}G_{\mu\nu}^2\ra  / E_B}\simeq (340 MeV)/(150 MeV)\simeq 2.25$
in our implementation of QCD property 
(\ref{condensate}) into the MIT bag model, see below.


 Still,  formula  $E_B(\rho_N) \sim  \rho_N $
  can not be used literally  for our purposes
 because we need an expression for the bag ``constant" contribution
which goes to constant $E_B$ at large densities,
$E_B(\rho_N) \rightarrow E_B$. A simple model which satisfies this requirement
 is to have the following replacement,
\be
\label{3d}
E_B(\rho_N) \simeq E_B\frac{r^3\rho_N}{(264 MeV)^3}
\Rightarrow\frac{E_B}{\left(1+   \frac{(264 MeV)^3}{r^{3}\rho_N}\right)},
\ee
where we introduced the correction factor $r$ mentioned above which scales
all dimensional factors according to their dimensionality. 
   As before, one should  express  the bag ``constant" contribution proportional to
(\ref{3d})
 in terms of a  fixed baryon charge  $B$ and radius $R$ with the following transition
to dimensionless variable $x$. We shall analyse the corresponding equation 
(\ref{2}) with improvements (\ref{3d}) in the next subsection. To anticipate the events,
one should mention that our two models (\ref{3b}, \ref{3d}) describing the effect 
 of the bag ``constant" variation with baryon density    
 lead to the similar results, see below.

The next approximation
we have made in eqs.(\ref{1}, \ref{2})  is related to the  assumption
of  a  thin-wall approximation for the domain wall. This 
  may not be well justified assumption because the typical width   of the domain wall
 and the size of QCD ball could be  the same order of magnitude,
 such that  thin-wall approximation 
is failed.  
However, we neglect these
complications at this initial stage of study. Nevertheless,
 we  do not expect that this effect can drastically change our qualitative results which
follow.
  
   We also neglected  in eqs.(\ref{1}, \ref{2}) all complications   
related to the finite magnitude of the quark masses, first of all $m_s$, which 
result in additional $K$ condensation along with diquark condensation in CFL phase\cite{Thomas}. 
Last, but not least. At this stage we assume that baryogenesis occurs prior
the QCD phase transition, such that there is an excess of quarks in comparison with
  antiquarks  such that no   annihilation  occurs in the  system
(see, however, some speculations in Section IV 
on possibility for baryogenesis to take place at the same instant during
the QCD phase transition).
With all these reservations regarding eqs.(\ref{1}, \ref{2}) in mind 
we  express the   energy of a QCD-ball per baryon charge $B$ in units
of $\sqrt[4]{E_B}$ , 
as follows
 \be
\label{4}
 y(x)_{tot}\equiv 
E_B^{-1/4}\epsilon_{tot}(x)  = 
  \frac{4\pi}{3}x^3\left(1-(\frac{4}{\pi})^{2/3}\frac{\mu_c^2}{\sqrt{E_B}}x^2\right)\\ \nonumber
+\left(
 4\pi \sigma_0 x^2+ \frac{3}{4x}  \sqrt[3]{\frac{\pi}{4}}
-\sqrt[3]{\frac{4}{\pi}}\cdot\frac{\Delta^2}{\sqrt{E_B}}\cdot x
\right). 
\ee
In this formula, in comparison with eq.(\ref{2}),
  we took into account  the effect describing  the quark-quark interaction on 
the Fermi surface given by eq. (\ref{3a}) and the effect of the variation
of the vacuum energy with baryon density, given by eq. (\ref{3b}).
 
The equilibrium condition $ \partial \epsilon_{tot} (x=x_0)/ \partial x=0$ determines the 
 radius $x_0$ of the QCD ball with   baryon charge $B$. We shall analyze this
condition in the next subsection; now we want to constraint $x_0 \leq \bar{x}$ 
to be considered in such an
analysis from the condition that the baryon density should be 
relatively large. In this case our treatment of the problem 
by using the quark degrees of freedom, eq.(\ref{4}), rather than hadronic degrees of freedom,
is justified.
The baryon number density  $\rho_N $ for the QCD ball configuration  is given
 by\footnote{Our normalization for the baryon charge corresponds to $B=1$ for the quark, 
thus factor $B/3$ in eq. (\ref{density}).}, 
\be
\label{density}
 \rho_N \equiv \frac{B}{3V}  = \frac{ E_B^{3/4}}{4\pi x^3} \gg n_0,~~~
n_0\simeq (108 MeV)^3,
\ee 
which gives upper limit $\bar{x}$ above which our approach is not justified\footnote{Let us emphasize,
the eq.(\ref{density}) constraints our approach based on quark degrees of freedom;
it does not tell us much whether QCD balls with densities close to the nuclear densities may 
or may not exist.
To answer the last question one should study the problem using the nucleon degrees of freedom
which is not a subject of this work.}. 
Numerically, with our choice of parameters, see below,  $\bar{x}\simeq 0.6$,
and therefore, any solution  $x_0$ of the
 equilibrium condition $\partial \epsilon_{tot} (x=x_0)/ \partial x=0$  
must satisfy to the constraint $x_0\leq \bar{x}\simeq 0.6$. 

\subsection{Stability of QCD balls} 
  As expected,  
the equation describing the equilibrium $\partial \epsilon_{tot} (x=x_0)/ \partial x=0$  
has a nontrivial solution (minimum) in a large region of parametrical space
deterimed by parameters $E_B, \sigma, \Delta, \mu_c, B$. It is not our goal
to have a complete analysis of this allowed region of solutions.
 Rather, we shall make a specific
choice for all parameters except the baryon number $B$ and analize the stability condition
as a function of $B$. We shall also comment on results with $\sigma = 0$
 corresponding to pure QCD configuration without any involvement of the axion field
(case considered previously in MIT bag model, \cite{Witten}- \cite{Madsen}).
The first step is to calculate 
the point $x=x_0$
 which is determined by equation
$\partial \epsilon_{tot} (x=x_0)/ \partial x=0$.   
The next step is to analyze the stability of the obtained configuration
as a function of external parameters.
Condition when the QCD-ball becomes an absolutely stable object can be 
derived from the following arguments. 
 Total energy per quark $\epsilon_{tot}(x_0)$ in eqs. (\ref{2}, \ref{4}) is a combination
of two factors: the first one, $\epsilon_{QCD}(x_0)$, is due to the strong interactions;
the second factor, $\epsilon_{axion}(x_0)$ is mainly due to the axion domain wall 
tension\footnote{the QCD contribution to $\sigma$ due to the $\eta'$ and pions 
is suppressed by a factor $f_{\pi}^2/f_a^2\ll 1$.}, i.e.
$\epsilon_{tot}(x_0)=\epsilon_{QCD}(x_0)+\epsilon_{axion}(x_0)$, with
$\epsilon_{axion}(x_0)\equiv E_B^{1/4}\left(4\pi \sigma_0 x_0^2\right)$
and $\epsilon_{QCD}(x_0)$ is determined by rest  of terms in eq. (\ref{4}).
Absolute stability of the system implies that nucleon can not leave a system
because the energy of the configuration with baryon charge $B$
is smaller than the energy of configuration of charge $B-3$ plus energy
of a nucleon with baryon charge $B=3$ and energy of the  axion emission.
It is quite obvious that the axion domain wall 
with a typical correlation length $\sim m_a^{-1} \gg \Lambda_{QCD}^{-1}$
  can not produce nucleons by itself when it shrinks due to the nucleon emission, 
instead it emits axions.
Therefore, the term 
$\epsilon_{axion}(x_0)\equiv E_B^{1/4}\left(4\pi \sigma_0 x_0^2\right)$
  is responsible for the emission of axions
rather than production of nucleons. As a result of this, this term should be ignored
for the analysis of the nucleon production.
The relevant term which describes the emission of    nucleons
is the one related to the QCD  physics
i.e. $\epsilon_{QCD}(x_0)$. Therefore, the condition
when configuration becomes an
absolutely stable one is determined from the following inequality
\be
\label{stability}
 \epsilon_{QCD}(x_0)< \frac{m_N}{3},~~ \frac{\partial \epsilon_{tot} (x)}
{\partial x}|_{x=x_0}=0, ~~ x_0 < \bar{x},
\ee
 where the last condition follows from (\ref{density}).

To analyse eq.(\ref{stability}) we shall accept the following
magnitudes for the dimensional parameters:
\be
\label{parameter}
\Delta \simeq 100 MeV;~~\sigma\simeq 1.8\cdot 10^8 GeV^3; \\ \nonumber
\mu_c\simeq 330MeV;~~
E_B\simeq (150 MeV)^4.
\ee
Having  these external parameters fixed, we left with the only one unknown 
number, the baryon charge $B$, which eneters $\sigma_0$ in our dimensionless 
parametrization (\ref{2},\ref{4}). We shall treat $\sigma_0$ as a free parameter
and our goal is to find the region of $\sigma_0$ when conditions (\ref{stability})
are satisfied.
As we discussed above, we shall use two different models to account
the effect of  the  variation of the bag
constant contribution with   density, see eqs. (\ref{3b}, \ref{3d}).

Having defined our stability condition (\ref{stability}), external 
 parameters (\ref{parameter}) and  
two simple models  accounting the effect of  the  variation of the bag
constant, eqs. (\ref{3b}, \ref{3d}), we reduce our problem to 
analysis of  dimensionless functions, $y_{QCD}^{(1)}(x)$ and 
$y_{QCD}^{(2)}(x)$ defined as follows, see eqs. (\ref{3b}, \ref{3d}, \ref{4}),
  \be
 y_{tot}(x)\equiv y_{QCD}^{(1,2)}(x)+y_{axion}(x);~~~
y_{axion}(x)\equiv
 4\pi \sigma_0 x^2  \nonumber \\
\label{5a}
y_{QCD}^{(1)}\equiv \frac{0.69}{x} +4.2x^3\frac{1}{1+6 x^3}-{0.48}{x},~~~~~~~~~~~~~
\ee
\beq
\label{5b}
y_{QCD}^{(2)}\equiv \frac{0.69}{x} +4.2x^3(1-5.68 x^2)-{0.48}{x}, ~~~~~~~
\eeq
where three consequent terms describe: 
the fermi pressure,  the bag constant contribution accounting the
variation of the vacuum energy with the baryon density
(\ref{3d},\ref{3b}), and, finally, the quark-quark
 interaction on the fermi surface (\ref{3a}) correspondingly.
Stability condition (\ref{stability}) in dimensionless variables becomes
\be
\label{stability1}
 y_{QCD}^{(1,2)}(x_0)< \frac{m_N}{3\sqrt[4]{E_B}}\simeq 2.1,~~ \frac{\partial y_{tot} (x)}
{\partial x}|_{x=x_0}=0.
\ee
Before we  discuss some specific numerical results which follow from analysis of eqs.
(\ref{5a} -\ref{stability1}),
we would like  to list some general  model-independent properties of the solutions. 
We believe that the properties listed below are quite  common
features of the QCD balls, which
  likely to remain untouched even
in a more general treatment of the problem when many additional effects are included
(some of these effects were mentioned above).

a). 
As we already mentioned, in the absence of the axion field, $\sigma\equiv 0$, the problem
was extensively discussed earlier using MIT bag model,\cite{Witten}-\cite{Madsen}.
Our original remark here is: when a variation of the vacuum energy with density is taken
into account, a stable solution {\it disappears} provided that
 a typical QCD scale for
 the vacuum variation (\ref{3b},\ref{3d})  is used.  The physical reason for that behavior
 is quite obvious: a density -dependent vacuum energy is not a sufficiently strong
  squeezer to equilibrate the fermi pressure. Only when   a typical scale for
 the variation is reduced (in comparison with what we assumed
in eqs.(\ref{3b},\ref{3d}) ) by an order of magnitude, the solution starts to reappear.
 Specifically, we checked that the equilibrium is possible for $\sigma\equiv 0$
 if coefficient $5.68$ in (\ref{5b}) describing the vacuum energy variation 
 is replaced by $0.5$.  
 Therefore, we incline to accept that there is no solution for such a
 configuration (strange quark nuggets,\cite{Witten}-\cite{Madsen}) in QCD
 if no external pressure (such as gravity or axion domain wall) is applied.
 It is certainly  not a very new result: special study 
 on stranglets reveals\cite{Alberico} a strong  model dependence of 
the stability of strange quark matter. In particular, the Nambu Jona- Lasinio 
 model does  not support any kind of strangelets\cite{Ratti}.

b). The negative result on the absence of a stable  solution 
with $\sigma\equiv 0$   should not dissapoint
the reader; just opposite: observation of ``normal" (not strange) neutron stars
is a strong argument  suggesting that such kind of dark matter can not exist. If it existed, 
all neutron stars would transfered to  ``quark stars" long ago. However, as we shall see in a moment,
stable QCD balls can exist if non-zero pressure due to the axion domain wall is present.
In this case, the QCD balls can not esquire an arbitrary large  size because an additional
pressure $\sim \frac{2\sigma}{R}$ becomes too weak at large  $R$; therefore, at some point
the situation gets similar to $\sigma =0$ case where strange quark matter,
as we argued, is not supported.  
 
c). In general, one expects there existence of a 
 minimal and maximal sizes (baryon charges $B_{min}$ and $B_{max}$) 
for the QCD balls in the region of stability. The minimal charge $B_{min}$
corresponds to the maximum $\sigma_0^{max}\sim B^{-1/3}_{min}$
 when solution at the equlibrium 
satisfies the stability requirement (\ref{stability}). When $B < B_{min}$ ,
 $\sigma_0 > \sigma_0^{max}$
  becomes too large such that quark energy per baryon charge becomes large enough to escape,
and nucleons can leave the system. On the other hand, the maximum possible charge, $B_{max}$,
corresponds to the minimum value of  $\sigma_0^{min}\sim B^{-1/3}_{max}$
 when the baryon density at the equlibrium
(\ref{density}) becomes too low to justify our approach based on the  quark degrees of freedom.
At lower baryon densities some metastable states may form; they could decay to some heavy elements
which might be of interests for astrophysics. 
However the corresponding study
would  require  an analysis of the system in terms of nuclear degrees of freedom,
which  is beyond  the scope of the present work. When $\sigma_0$ becomes even smaller, 
the problem is essentially equivalent to $\sigma =0$ studied earlier  where
stable solutions are not expected to occur.

Numerically, we analyzed two models (\ref{5a}, \ref{5b}) which lead to the similar results.
In particular, for model (\ref{5a}) the maximum possible tension, $4\pi \sigma_0^{max}\simeq 10$
corresponds to the minimum baryon charge $B_{min} $. For  such $\sigma_0$
 the equilibrium is reached at $x_0\simeq 0.32$ 
wnen the  energy per quark $y_{QCD}^{(1)}(x_0)\simeq 2.1$
 hits the upper energy bound of the stability region (\ref{stability1}).
When $4\pi \sigma_0^{max}> 10$, the  energy per quark becomes too high such that nucleon can escape
and the system would decay. In physical units this solution corresponds 
to $B_{min}\simeq 10^{32}$  and  stabilization radius 
$ R_0= x_0\sqrt[3]{B}/\sqrt[4]{E_B}\simeq 10^{11} GeV^{-1}$. Energy per
quark for this configuration $\epsilon_{QCD}^{(1)}=y_{QCD}^{(1)}(x_0)\sqrt[4]{E_B}
\simeq 2.1\sqrt[4]{E_B}\simeq 320 MeV$ is smaller than constituent quark mass, 
as it should be\footnote{We remind that we discuss the QCD part  of energy only;
 the total energy of the configuration which includes the axion part is larger.}.

For the same model, the minimum possible tension, $4\pi \sigma_0^{min}\simeq 2$
corresponds to the maximum possible  baryon charge $B_{max}$.
According to the scaling $B\sim \sigma_0^3$, the maximum baryon charge 
$B_{max}=(\frac{\sigma_0^{max}}{\sigma_0^{min}})^3B_{min}\sim 10^{34}$ is 
two orders of magnitude larger than $B_{min}$.
 In this case
the equilibrium is reached at $x_0\simeq 0.52$ when the baryon density (\ref{density}) 
is  relatively low, and  close to the boundary when the quark based lore can not be 
trusted. 

Our second model (\ref{5b}) gives quantitatively similar results, 
and it is not worthwhile to discuss numerical details here. The most important 
features of the solution for this model remain the same: 
there is a region between $ B_{max} $ and $ B_{min} $
when solutions are stable; at $\sigma=0$ solution does not exist at all
provided that
 a typical QCD scale for
 the vacuum variation (\ref{3b},\ref{3d})  is used. 
 
However, one should take all these numerical estimates very  cautiously
because of a number approximations we have made in eqs. (\ref{1}, \ref{2})
discussed above. 
  Nevertheless, in what follows, mainly for
 the illustrative purposes, we shall stick with these numerical estimates. 
 
  Now we want to estimate the quark number density $n$ in the 
region between $ B_{max} $ and $ B_{min} $
when solutions are stable and our approach is justified,
\be
\label{7}
 n \equiv \frac{B}{V}  = \frac{3 E_B^{3/4}}{4\pi x^3}
\simeq (1.5 - 6.5 ) \cdot 3n_0,
\ee
where $3 n_0\simeq 3(108 MeV)^3,$ is the nuclear saturation density normalized with our convention
( $B=1$ for quarks), thus factor $3$ in front of the numerical
value $0.16 (fm)^{-3}\simeq (108 MeV)^3$. It is quite remarkable that the numerical value
for $n$ is in the   region where color superconductivity phase is   likely   to   realize,
and therefore, our treatment of the squeezed fermi system as quark 
dense matter (rather than ordinary nuclear matter)  is  justified a posteriori.

Few remarks are in order regarding eq.(\ref{7}). 
 First of all, the estimates presented above demonstrate that we are in the region
of the phase diagram where CFL phase is likeley to   realize. Therefore, our original
assumption is justified. Secondly,
    for   large $B\geq  B_{max}$
 our treatment of the system  is not valid anymore, and 
 a different type 
of QCD balls with an ordinay nuclear matter (instead of diquark condensate)  in the bulk
may be formed and could be even  stable in some regions of parametrical space.
Though this region of  large $ B\geq B_{max}$ could be an interesting region
from the phenomenological point of view, it shall not be discussed here\footnote{The 
corresponding proper treatment would require the knowledge of the dynamics
of the interacting
nuclear matter, which is not the subject of the present work. 
In principle such nuclear matter
could be also stable in some parametrical regions.}.
  However, even in this case when the QCD balls
made of nuclear matter, rather than quark dense matter,
we still expect that there   should exist
 a maximum size    above which the stability is not possible.
This follows from our analysis that stability can not be achieved without the external pressure
$P_{\sigma}$  due to  the axion domain $P_{\sigma}\sim 2\sigma/ R$ 
  which vanishes at very large $R$. 
 
Another factor which also constraints the size of the balls is related to the suppression
 of  large size closed axion domain walls during the formation stage. 
 It is clear that the formation of the large size closed domain walls is
 suppressed according to the
 Kibble-Zurek  mechanism\cite{Kibble},\cite{Zurek}, however 
 an explicit estimation for this effect is still missing.
 
As we mentioned in the Introduction, we do not address the problem of formation of QCD balls
in this letter, it will be a subject of a different work.  However we would like to
mention some relevant elements of a possible scenario of how QCD-balls, in principle,
can be formed after the QCD phase transition, at a temperature of
order 150 MeV which is much higher than the critical
temperature for quark pairing   estimated to be $\sim 0.6\Delta$.  
The main point is this: the axion domain wall with the QCD-scale substructure
as discussed in\cite{FZ} is very selective with respect to the momentum of the particles;
it is almost transparent for light $\pi$ mesons with 
large momentum  $k \geq m_{\pi}$ such that the  transmission coefficient is close to one.
Therefore, the  highly energetic pions  can easily penetrate through the domain wall and leave
 the system. 
At the same time, the  transmission coefficient is close to zero for slow-moving
particles such as baryons with $k \leq m_{\pi}$. Eventually, this  ``selective"  feature
 of the domain wall may cool down the system considerably. Due to the 
domain wall pressure it may reach the critical 
density when it undergoes a phase transition to a color  superconducting phase
with the ground state  being the  quark condensate. 
At this point we assume that the baryon number trapped in the bulk is sufficiently large.
If $B\gg B_{max}$, the  quarks will leave the system by forming
nucleons until the upper limit $B_{max}$ is achieved when   energy per unit baryon charge,
$\epsilon_{QCD}=\sqrt[4]{E_B}y_{QCD}(x_0) < \frac{m_N}{3}$, (\ref{stability1})
 is not sufficient for  quarks to form a nucleon and leave the system.
It is important to note that though reflection  coefficient for the domain wall
(taken as a separate coniguration) is never equals exactly  to one, 
the quarks can not leave the system because the energy per baryon charge in the bulk
is smaller than $m_N/3$. The property of the domain wall to have  
the reflection coefficient close
to one for the baryons was very important during the formation period
to keep the baryons in the bulk some time before the equlibrium is reached; it is less
relevant when the equilibrium is already reached. 

Some specific calculations are required before any statements regarding a possibility
to form the  QCD balls after the phase transition can be made. At this moment we simply assume
that this is possible and we do not see any fundamental obstacles which would
prevent the  formation of such objects.
 
\subsection{QCD- balls versus Q-balls}
In this subsection we would like to mention a striking  resemblance
of the QCD-ball (which is the subject of this letter) and Q-ball\cite{qball} which is 
a nontopological soliton 
associated with some conserved global $Q$ charge.  
In both cases, 
a soliton mass as function of $Q$  has behavior, similar  to our eq. (\ref{3}), and therefore, 
it may become a stable configuration for relatively large $Q$ charge. Therefore, an effective
 scalar field theory
with some specific constraint on potential (when
$Q$ ball solution exists) is realized for QCD in high density regime by 
formation of the diquark scalar condensate which plays the role
of the effective scalar field.
The big difference, of course, that underlying theory for QCD-balls is well known, it is QCD
 with no free parameters, in huge contrast with the theory of   Q-balls. 
Formal similarity becomes even more striking if  one takes into account that the ground state of the
CFL phase in QCD  is determined by the diquark condensate with the following  time 
dependence $\sim e^{i2\mu t}$,
\be
\label{diquark}
    \langle \Psi_{La}^{i\alpha}\Psi_{Lb}^{j\beta}\rangle^* \sim
    \langle \Psi_{Ra}^{i\alpha}\Psi_{Rb}^{j\beta}\rangle^* 
    &\sim (e^{i2\mu t})\cdot\epsilon^{ij}\epsilon^{\alpha\beta c}\epsilon_{abc}
    ~~,  
\ee
with $\Psi$ being the original QCD quark fields, and $\mu$ being the chemical potential of the system,
see formula (40) from ref. \cite{FZ2}.
As is known, such time-dependent phase is the starting point in construction of the 
Q balls\cite{qball}.
 In the expression (\ref{diquark}) we  explicitly show the structure
for the diquark condensate corresponding to CFL (color-flavor locking) phase\cite{cs_r}
with   ($\alpha$, $\beta$, etc.) to be flavor,
($a$, $b$, etc.) color  and  ($i$, $j$, etc.) spinor indices correspondingly.
Of course, there are many differences in phenomenology between Q balls\cite{qball}  and QCD-balls. 
For example, in CFL phase the baryon symmetry is spontaneously broken, and corresponding Goldstone
massless boson
carries the baryon charge. However, the evaporation of this massless particle into hadronic phase
from the surface of the QCD-ball is not possible, because hadronic phase does not support such excitation.
This is in contrast with phenomenology of Q-balls, where the theory is formulated in terms
of one and the same scalar $\phi$ field, such that evaporation of $\phi$ particles 
from the surface of the Q-ball is possible if some conditions are met.
In spite of many differences, the  analogy with Q-balls is quite useful and can be used for analysis
of different experimental bounds  on QCD-balls, which is the subject of the next section.

\section{Experimental bounds on masses and fluxes of QCD-balls}
In this section we adopt the results of paper\cite{qball_exp} to constraint 
the free parameter (charge $B$) of the QCD-balls.
In the paper \cite{qball_exp} the authors re-analyzed  the results of various experiments, 
originally not designed for the Q-ball searches, but nevertheless these experimental results 
were successfully used 
in \cite{qball_exp} to bound  different properties of the Q-balls. We actually repeat this analysis 
for a specific type of the QCD-balls when original quarks are in the CFL (color-flavor locking)
phase\cite{cs_r}.

As we mentioned earlier, at sufficiently large baryon density, the color superconductivity 
phenomenon takes place.
However, there are many different 
phases (as a function of parameters like $m_s$, number of light flavors, etc.) associated with
color superconductivity. In particular, for 3 degenerate flavors of light quarks, the CFL phase
with nonzero value for the diquark condensate (\ref{diquark}) is realized. Due to the fact that 
equal numbers of $u, d, s$ quarks
condensed in the system, the electric charge of the ground state is zero, i.e.
 no electrons required to neutralize
the system. This is quite important feature for the phenomenology of the QCD-balls
 we about to discuss.
Nature is less symmetric, and other CS phases could be realized. In particular, 
for relatively large $m_s$, along with diquark condensate, the $K$ condensate
 may also be formed\cite{Thomas}.
In the limit of  very large $m_s$, QCD becomes effectively a theory with two light quarks.
In this case, the Cooper pairs are $ud-du$ flavor singlets. This phase,   the so-called 2SC
(2 flavor super-conductor ) phase is a phase with non-zero electric charge. Electrons
 neutralize the system,  
however, all properties, such as interaction cross sections, the rate of energy loss of QCD balls  
 in matter, 
 are very different for QCD-balls with quarks in CFL or 2SC phase.
In what follows, to avoid many complications,  we limit ourself with analysis of QCD balls 
where quarks are in the most symmetric CFL phase, in which case the QCD-ball has zero electric charge.

We assume, in analogy with\cite{qball_exp}, that a typical cross section of a neutral QCD-ball
  with matter is determined by their geometrical size, $\pi R_0^2$.
In this case,   the only information we need to constraint the QCD-ball parameters,
is its size   and mass. We also assume that the QCD-balls is  the main contributor 
toward the dark matter in the Galaxy. Their flux $F$ then should satisfy
\be
\label{dark}
F < F_{DM}\sim \frac{\rho_{DM} v}{4\pi M_B} \sim 
7.2\cdot 10^5 \frac{ GeV}{M_B} cm^{-2}sec^{-1}sr^{-1},
\ee
where $\rho_{DM}$ is the energy density of the dark matter in the Galaxy, 
$\rho_{DM}\simeq \frac{0.3 GeV}{cm^3}$, and $v\sim 3\cdot 10^{-3}c$ is
 the Virial velocity of the QCD-ball.
We identify $M_B$ in the expression (\ref{dark}) 
with the total energy $E$  of the QCD ball at rest with given baryon  charge $B$.
The {\it  Gyrlyanda} experiments at Lake Baikal 
reported that the flux of neutral soliton-like objects has the bound
\cite{Baikal}
\be
\label{8}
F~ <~ 3.9\cdot 10 ^{-16} cm^{-2}sec^{-1}sr^{-1},
\ee
which translates to the following 
lower limit of the neutral QCD-ball mass $M_B$ and baryon charge $B$,
\be
\label{9}
M_B^{exp} ~ > ~ 2\cdot 10^{21}~~ GeV ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\
B^{exp} \simeq  (\frac{M_B}{\sigma^{1/3}})^{9/8} [\frac{3}{2}(8\pi c^2)^{\frac{1}{3}}]^{-9/8}
~ >~ 1.6\cdot 10^{20}. \nonumber
\ee
Similar constraints follow from the analysis of the {\it Baksan} experiment\cite{Baksan}
and analysis \cite{qball_exp}
of the Kamiokande Cherenkov detector\cite{Kam}, and we do not explicitly quote these results.
These experimental bounds are well below the critical line of the {\it absolute} stability
of the QCD-balls.    
 \section{Speculations. QCD-balls: Formation and Baryogenesis}
Complete theory of formation of the QCD-balls is   still lacking. Only such a theory would predict whether
QCD-balls can be formed in sufficient number to become the dark matter.
Such a  theory of formation of the QCD balls would answer on questions like this: 1. What is the 
probability to form  a closed axion domain wall with size $\xi$ during the QCD phase transition ? 2. How many quarks are trapped inside the domain wall at the first instant? 3. How many quarks
will leave the system and how many of them will stay inside the system while the bubble is shrinking? 4. 
What is the  dependence of  relevant parameters such as: size $\xi(t)$, baryon number density $n(t)$
and internal temperature $T(t)$ as function of time ? 5. Do these parameters fall into appropriate region
of the QCD phase diagram where the color superconductivity takes place? 6. What is the final density
distribution of the QCD-balls as a function of their size $R$( baryon charge $B$)after the formation
period is complete ?  

Clearly, we do not have answers on these, and many other important questions at the moment.
Therefore, we go in  an opposite direction in our analysis of the QCD balls (from bottom to the top)
and formulate the problem in the following
way. Let us assume that the QCD-ball is indeed a valuable dark matter candidate at the present epoch.
What can we say about their properties  during the formation period?
Before we continue, we would like to make one more remark regarding the 
 observed relation $\Omega_B\sim\Omega_{DM}$. 
Up to this moment we assumed that the baryogenesis problem is resolved before the QCD phase transition,
such that QCD-balls are formed in an asymmetric baryon environment. Now we go even further
and give some arguments supporting the idea that baryogenesis may  also  be originated at the QCD scale.

 As we already mentioned,  the most important argument 
is the observed relation $\Omega_B\sim\Omega_{DM}$
which  is extremely
difficult to explain in models that invoke a dark matter candidate not related to
 the ordinary quarks/baryons degrees of freedom.
In our scenario the dark matter made of 
ordinary quarks, which however, are not in the hadronic phase, but rather
in color superconducting phase. Therefore, the observed relation $\Omega_B\sim\Omega_{DM}$ 
could be a quite natural consequence of the underlying QCD physics.
Our second argument is the observation that all three Sakharov's criteria \cite{Sakharov} are satisfied 
during the instant when axion domain walls are shrinking in size to form the QCD balls.
Indeed, \\
1.The process takes place out of thermal equilibrium;\\
2.This process involves  strong  CP violation due to the axion-related  physics. In particular, 
the  effective $\theta$ parameter across the axion domain wall takes 
a non-zero value.\\
 3. Baryon current is exactly conserved in QCD in a big contrast with
baryogenesis mechanisms considered at electroweak scale; however the baryon symmetry is spontaneously
broken in CFL phase. In principle, it is  sufficient for a charge separation 
(rather than baryogenesis) scenario.

Few comments regarding that three criteria are in order.
First of all,  the   collapse of the axion domain walls with formation of the QCD-balls is clearly
an out of equilibrium process.
Secondly, regarding CP violation in this process.  
 We are making the assumption that the strong
 CP problem is cured by the axion field. 
 At temperature $T \simeq  T_c$, the axion is not yet in its ground state, and thus 
the axion field, $\theta(T_c)$
might be of order unity. 
Therefore, CP violation is expected to be order of one at this instant.
Note that as long as the initial value $\theta(T_c)$ is the same in the entire 
observed Universe, the sign of the baryon asymmetry will also be the same. 
This will occur if the Universe undergoes inflation either after or during the Peccei-Quinn symmetry breaking,
which is the standard assumption in the axion- related physics. 
Therefore we  explore the possibility that the baryon asymmetry may have been generated at the QCD scale
 via nonperturbative processes, without the need to introduce any new physics beyond the standard model, 
except for a solution of the strong CP problem. 
Finally, since baryon number is globally conserved in QCD, the 
only way to produce a baryon asymmetry is via charge separation, not charge
production mechanism.  Therefore, the third criteria is somewhat modified in comparison with the 
original formulation. However, the idea that the spontaneous (rather than explicit) breaking
of the baryon symmetry can be  responsible for the baryogenesis, has been known for a
 while, see \cite{Widrow}
where a simple toy model  was suggested to explain the phenomenon
(see also work by Brandenberger et al in\cite{baryogenesis} on the subject). 
It is not a goal of this letter to discuss a specific mechanism for the baryogenesis; 
rather we wanted to argue that both phenomena: dark matter and baryogenesis could be
 originated at the same instant. 


Having made these assumption we ask the following question: what consequence we can 
 derive from these assumptions? Are they self consistent assumptions?
First of all, consider the following ratio
\be
\label{r1}
\left(\frac{dark ~matter~ number~ density}{baryon~ number~ density}\right)
\simeq \frac{m_N\Omega_{DM}}{M_B\Omega_B },
\ee
 where $M_B$ is the mass of the QCD- ball.
We combine this formula with known expression for the baryon to photon ratio,
$n_B/n_{\gamma}\simeq 5\cdot 10^{-10}$ 
to estimate the following ratio
\be
\label{r2}
 \left(\frac{dark ~matter~ number~ density}{entropy~ density}\right) \sim 5\cdot 10^{-9}
 \frac{m_N}{M_B},
\ee
where we assume $\Omega_{DM} / \Omega_B \sim 10$. On the other hand, 
the energy density $\rho_B$   of the QCD-balls will redshift as matter, i.e. $\rho_B\sim T^3$
not as radiation $\sim T^4$.
Hence, the QCD-balls contribute to the dark matter of the Universe as follows
\be
\label{r3}
\Omega_{DM}\simeq \frac{\rho_{B}(t_{eq})}{\rho_{rad}(t_{eq})} 
\sim~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\
 \left(\frac{dark ~matter~ number~density}{entropy~ density}  \right)\cdot \frac{M_B}{T_{eq}}
\sim 5\cdot 10^{-9} \frac{m_N}{T_{eq}},\nonumber
\ee
where  $t_{eq}(T_{eq})$ is the time (temperature) of equal matter and radiation, and
$\rho_{rad}$ is the energy density of radiation which, until $t_{eq}$, dominates
the total energy density. In obtaining (\ref{r3}) we used the  ratio (\ref{r2}) which remains 
approximately a fixed number while  the Universe is cooling, and approximated the 
 energy density for radiation as $\rho_{rad}\simeq sT$,
instead of using exact formula  $\rho_{rad}=\frac{\pi^2}{30}g(T)T^4$ and 
 $s=\frac{2\pi^2}{45}g(T)T^3$ for entropy density with $g(T) $ being the 
number of massless degrees of freedom.
Using the phenomenological values for $m_N\sim 10^{9}eV$ and $T_{eq}\sim 10 eV$
we immediately see that this large scale factor $ \frac{m_N}{T_{eq}}$ in eq. (\ref{r3}) almost exactly 
overwrites small factor $n_B/n_{\gamma}$ such that $\Omega_{DM}\sim 1$.
Of course this estimation does not prove anything
because we started from the assumption that QCD-balls is the main contributor to the dark matter.
Nevertheless, the relation (\ref{r3}) is quite nontrivial because we used three different  
independent observational  numbers to derive it.

Therefore, the point we want to make
is:  our assumption that the dark matter is originated at the QCD scale from ordinary quarks
 fits very nicely
with phenomenological ratio $n_B/n_{\gamma}$ provided that baryogenesis is also
originated at the same QCD scale $T_c$, and also $\Omega_{DM} / \Omega_B \sim 1$ within the order of magnitude.
In this case the smallness of the ratio
$n_B/n_{\gamma}$ is understood as a small scale factor  $\frac{T_{eq}}{m_N}$
describing the difference in  evolution of matter and radiation from 
the QCD phase transition, $T_c$ until $T_{eq}$.

If baryogenesis (separation of charges) indeed happens at this instant, than the dark matter
may be antimater (as well as matter) made of ordinary quarks in the ``exotic" color superconductor
 phase. 
Finally, we want to use eq.(\ref{r2}) to estimate the absolute value 
for the dark matter number density $n_{DM}$ after the QCD-balls are formed soon after
the QCD phase transition at $T\sim T_c$,
\be
\label{r4}
n_{DM}\sim 5\cdot 10^{-9}\frac{2\pi^2}{45}g_*T_c^3 \frac{m_N}{M_B},
\ee
which for the baryon  charge $B\sim 10^{32}$  and effective
massless degrees of freedom, $g_*\simeq 10$
can be estimated as 
\be
\label{r5}
r T_c\equiv n_{DM}^{-1/3}T_c \simeq 3.5\cdot 10^{13},~~ r\sim  10 ~cm,
\ee
where $r$ has the  physical meaning of an average distance between QCD-balls after they formed.
As expected, average distance $r$ is much smaller than the horizon radius $R_H^{QCD}$
at the QCD phase transition, $r\sim 10^{-5}R_H^{QCD}$.
It is quite remarkable that $r$ is much larger than the size of the
 QCD-ball, see   eq.(\ref{2}), 
such that QCD-balls become well separated soon after they formed.
Besides that we expect that the QCD ball size should be related, through dynamics,
 to the correlation length $\xi$ of the original axion field, or what is the same, to 
the  typical wall
separation at the instant of formation, which we expect to be order (or somewhat larger)
of  inverse axion mass, $\xi \geq m_a^{-1}$. We also expect that the spatial extend of a typical
closed wall at the instant of formation has the same order of magnitude 
$\xi ~$ \cite{Kibble, Zurek}.  Initial size of a closed wall $\sim\xi$
eventually (after some shrinking as a result of tension, and after
some expansion as a result of evolution of the Universe) 
determines  the size of the QCD-balls. However, the dynamics of this transition
is quite complicated, and we are not able to derive a relation between initial domain wall size 
distribution and QCD-ball size distribution at the later stage.
  Close numerical values for the QCD ball size
  and $\xi\sim m_a^{-1}$ also suggest that these parameters are related somehow.
 Therefore, it is at least possible,
that the decay of the axion domain wall network may result in formation of the QCD-balls
with their nice properties discussed in this letter.
\section{Conclusion}
In this letter we argue that the QCD-balls could be 
a viable cold dark matter candidate which
 is formed from the ordinary quarks
during the QCD phase transition when  the  axion domain walls
form.    As we argued 
the system in the bulk     may reach the critical 
density when it undergoes a phase transition to a color  superconducting phase
in which case the new state of matter
representing the diquark condensate with a 
large baryon number $B$
  becomes  a stable soliton-like  configuration. 
We also speculate that the baryogenesis may happen exactly at the same
 instant during the QCD  phase transition. In which case the dark matter may be 
actually antimatter.
The scenario is no doubt lead to important consequences for cosmology and astrophysics,
which are not  explored yet.
In particular,  some unexplained events, such as Centauro events, or  even the Tunguska-like
events (when no fragments or chemical traces have ever been recovered),
can be related to the very dense QCD balls.


 If this is the case, the arrival directions
should correlate with the dark matter distribution and show the halo asymmetry.
Also: recent observation\cite{NASA}  suggests
that the matter in some stars could be  even denser than nuclear
matter. It could be also   related to the very dense QCD balls.
Last but not least, the recent detection\cite{Teplitz} of two seismic events with epilinear
(in contrast with a typical epicentral ) sources may also be related to the 
very dense QCD balls.

 Therefore, the ``exotic", dense color superconducting phase in QCD, might be much  
more common state of matter in the Universe  than the ``normal" hadronic phase we know.
In conclusion, qualitative as our arguments are, they suggest that
 the dark matter   could be  originated at the QCD scale.
We have seen that it is, at least conceivable, without fine tuning of parameters, 
to obtain a reasonable relation between $\Omega_{DM}, ~\Omega_B $ and 
 the baryon to entropy ratio, which, otherwise,  is very difficult to understand
if these quantities do not have the same origin.  
\section*{Acknowledgments}
I am thankful to Robert Brandenberger,    Michael Forbes,   
Sasha Kusenko, Shmuel Nussinov, Krishna Rajagopal, Pierre Sikivie, Paul Steinhardt 
and Fran Wilczek  for useful discussions, comments and  remarks.
This work was supported in part by the National Science and Engineering
Research Council of Canada. 
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