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\hyphenation{sub-sti-tut-ing quad-rat-ic an-ni-hi-la-tion pro-ceed 
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\title{MONOPOLE NON-ANNIHILATION AT THE ELECTROWEAK SCALE---NOT!}
\author{Evalyn Gates, Lawrence M. Krauss\thanks{Also Department
of Astronomy. Research supported
in part by the NSF, DOE, and the Texas National Research Laboratory
Commission. Bitnet: Krauss@Yalehep.}\,, and John
Terning\thanks{Research supported by NSERC.}\\ Center for Theoretical 
Physics\\
Sloane Physics Laboratory\\
Yale University\\
New Haven, CT 06511}

\date{February 1992}

\begin{document}
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\begin{abstract}
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\normalsize
We examine the issue of monopole annihilation at the electroweak scale
induced by flux tube confinement, concentrating first on the simplest
possibility---one which requires no new physics beyond the standard 
model.
Monopoles existing at the time of the electroweak phase transition  may 
trigger $W$
condensation which can confine magnetic flux into flux tubes.  
However we show on very general grounds, using several independent
estimates, that such a mechanism is impotent. We then present several
general dynamical arguments limiting the possibility of monopole
annihilation through any confining phase near the
electroweak scale.

\end{abstract}

\vfill\eject

The ``monopole problem" has been with us since the advent of Grand 
Unified 
Theories (GUTs), which allow the formation of these non-singular stable
topological defects when a
semi-simple gauge group is broken to a lower symmetry group that 
includes an explicit
$U(1)$ factor.  These objects typically have a mass $m_M\simeq m_X /
\alpha$, where $m_X$ is the mass of the gauge bosons in the 
spontaneously broken
GUT theory, and $\alpha$ is the fine structure constant associated with 
the
gauge coupling of the theory.

Shortly after it was recognized that monopoles could result as stable
particles in spontaneously broken GUT models \cite{'tHooft}, and also 
that they 
would
be produced in profusion during the phase transition associated with the
GUT symmetry breaking in the early universe \cite{Kibble}, it was also 
recognized
that they posed a potential problem for cosmology.  Comparing 
annihilation rates
with the expansion rate of the universe after a GUT transition, it was 
shown
\cite{Preskill,Zeldovich} that the monopole to photon ratio would 
``freeze out"
at a level of roughly  $10^{-10}$.  Not only would such an initial
 level result in a cosmic
mass density today which is orders of magnitude larger than the present 
upper
limit, but direct observational limits on the monopole abundance in our
neighborhood are even more stringent \cite{parker}.

This cosmological problem was one of the main motivations for the 
original inflationary scenario\cite{guth}.  However one of the chief 
challenges 
to the original
inflationary solution of the monopole problem was the necessity of
having a reheating temperature which is high enough to allow 
baryogenesis, but
low enough to suppress monopole production.  In addition, recent 
work on large scale 
structure,
including observed galaxy clustering at large
 scales, large scale velocity flows, and the absence of any observable
anisotropy in the microwave background, has put strong
constraints on such models.

With the recent recognition that even something as exotic 
as
baryogenesis may be possible within the context of the standard 
electroweak
theory (supplemented by minor additions), it is worth examining the 
issue of
whether the monopole problem may be resolved purely through low energy 
physics.  A canonical method by which one might hope to achieve
complete annihilation is by confining monopole-antimonopole pairs in
flux tubes, such as might occur if $U(1)_{em}$ were broken during some
period. Proposals along this line, based on introducing 
 new physics have been made in the past,  eg.
\cite{LangPi,Lazarides}. The possibility that plasma effects might also play a
role in producing an effective confining potential at early times has also
been proposed\cite{Linde1,Linde2}. Most recently, the possibility that such a 
phase might briefly occur near the electroweak breaking scale, for multi-Higgs
models, has also been raised \cite{Sher}. By far the simplest  possibility,
however, is that flux tube confinement of monopoles might occur in the standard 
model at relatively low energies, unsupplemented by any new physics.  We explore
this issue in detail  here, and
then go on to examine the general dynamical obstacles facing any model
involving monopole confinement at the electroweak scale.
 

\vspace{.1 in}

\noindent{\bf  1. Monopole Confinement in the Standard Model:}

It has been known for some time that the electroweak vacuum
in the broken phase is unstable
in the presence of large ($\geq m_W^2/e$) magnetic 
fields\cite{Skalozub}.
The instability is due to the coupling between the magnetic field $H$
and the magnetic moment of the massive $W$ gauge bosons.  Due to this 
coupling 
the effective mass of the $W$ at tree level is 
\begin{equation}
        m_{W_{\rm eff}}^2 = m_W^2 - eH  
\end{equation}
where $e=g \sin\theta_W$  (all expressions are given in
Heaviside-Lorentz units for electromagnetism).  This effective mass 
squared becomes 
negative
for $H_c^{(1)}\geq m_W^2/e$.
The general resolution \cite{Skalozub} 
of the instability is the formation of a condensate of
$W$ and $Z$ bosons, which sets up currents that antiscreen the magnetic
field. The vacuum then acts as an anti-type II superconductor, and
the energy is minimized by the formation of a periodic network of 
magnetic flux tubes.  As we shall describe
in some length later, Ambj{\o}rn and Olesen have also shown,
at least for the special case $m_H=m_Z$, that if
the magnetic field increases above 
$H_c^{(2)} = {{m_W^2}\over {e \cos^2 {\theta}_W}}$, the full
$SU(2)_L \otimes U(1)$  symmetry is restored \cite{AO1990}. (Top quark
loops, for a sufficiently heavy top quark, might affect the field at
which symmetry is restored\cite{Linde3}, but this issue is not yet
resolved, and in any case will not affect the arguments presented here.)
Thus for an external magnetic field
$H_c^{(1)}< H < H_c^{(2)}$, the electroweak vacuum passes through a
transition region where a $W$ condensate exists and the magnetic
field is confined in a periodic network of flux tubes.

It is possible to imagine how such a phase might arise naturally in a 
way which
might lead to monopole-antimonopole annihilation at the electroweak 
scale
in the early universe (This idea has also been suggested elsewhere in 
the
literature \cite{Owen}).   First of all,
 the magnetic field
necessary to produce such a phase could come from the monopoles 
themselves, provided
the electroweak transition is second order, or sufficiently weakly
first order.   
In the approximation of a second order transition, the mass of the
$W$ boson generically has a temperature dependence of the form
\begin{equation}
m_W^2 (T) \approx m_W^2(0)[1 - T^2/T^2_c]
\label{m_W}
\end{equation}
where $T_c$ ($\approx 300$ GeV) is the critical temperature associated 
with electroweak breaking.
Thus, just below the transition temperature $T_c$, relatively small 
magnetic
fields could trigger $W$ condensation. (This argument carries through if the
transition is sufficiently weakly first order so the VEV of the scalar field is
small in the broken phase which nucleates.) A remnant density of GUT-scale 
monopoles could provide such 
a magnetic field.  Once the condensate forms,
monopoles would become confined to the network of flux tubes, whose 
width is
related to the $W$ mass, as we shall describe.
Once the width of the flux tubes is of the same order as the distance 
between
monopoles, the monopoles would experience a linear potential and
begin to move towards each other.  If the flux tubes exist for a 
sufficiently
long time, the monopoles could annihilate, and their density would
correspondingly decrease.

This picture is very attractive in principle.  However, we now 
demonstrate, using a series of arguments which probe this
scenario in successively greater detail, that the
parameters associated with such a transition at the electroweak scale 
generally preclude
it from being operational. Moreover, we present dynamical arguments 
relevant
for any scenario involving monopole annihilation via flux tube 
confinement at the
electroweak scale.

\vspace{.1 in}

\noindent{\bf 2. Kinemetic Arguments: Non-annihilation via Magnetic
Instabilities}:

(a) A Global Argument:  In figure 1, we display a phase diagram 
describing the
$W$ condensation picture discussed above, as a function of both 
temperature $T$
and
background magnetic field $H$. At $T=0$, for the case examined by 
Ambj{\o}rn and Oleson\cite{AO1990,AO1989a,AO1991}, in the region 
$1 < He/{m_W^2} <
1/\cos^2\theta_W$ a magnetic flux tube network extremizes the energy and 
both the $\phi$ (Higgs) and $|W|^2$ fields develop non-zero 
expectation 
values.  For finite
temperature the phase boundaries evolve as shown, in response to the 
reduction in the
$W$ mass with temperature, up to $T=T_c$, where they meet.  
Thus, the phase in
which flux tubes and a $W$ condensate are energetically 
preferred falls in between 
these two curves.   

While the actual magnetic field due to the presence of a density of 
monopoles and
anti-monopoles will be complicated and inhomogenous, we first 
approximate it by a
homogenous mean field $H_m$, whose precise value is not important for 
this discussion.
(As we will later show, given the remnant density of monopoles predicted 
to result from a
$GUT$ transition, the value of this field will be well below the zero 
temperature
critical field $m_W^2 (0)/e$ at the time of the electroweak phase 
transition.)  As the
universe cools from above $T_c$, this background magnetic field will 
eventually cross the
upper critical curve for the existence of a flux tube phase.  

\begin{sloppypar}
We now imagine that immediately after this happens, flux tubes form, and 
monopole
annihilation instantaneously begins. We shall later show that this is 
far from the actual
case.  Nevertheless, this assumption allows us to examine constraints on 
monopole annihilation
even in the most optimistic case.   As monopole-antimonopole 
an\-ni\-hi\-la\-tion pro\-ceeds, the 
mean back\-ground mag\-ne\-tic field falls quickly.  At a certain point 
this 
mean field will fall
{\it below} the lower critical curve, and if it is this background field 
which governs the
energetics of $W$ condensation, the $W$ 
condensate will then become unstable, the
magnetic field lines will once again spread out, and 
monopole-antimonopole annihilation will cease. 
As can be seen in the figure, the net reduction in the magnetic field 
expected from this period of
annihilation will be minimal. Quantitatively the final field (neglecting 
dilution due to expansion
during this period) will be a factor of $\cos^2{\theta}_W$ smaller than 
the 
initial field.  This is hardly
sufficient to reduce the initial abundance of monopoles by the many 
orders of magnitude required to
be consistent with current observations. 
\end{sloppypar}

(b) A Local Argument:  The above argument points out the central problem 
for a
monopole annihilation scenario based on magnetic field instabilities at 
the electroweak
scale.  In order to arrange for flux tubes to form, and confinement of 
monopoles to occur,
the field must be tuned to lie in a relatively narrow region of 
parameter space. 
Nevertheless, a potential problem with the above argument, even if it 
were less sketchy, 
is
that flux tube formation, and monopole annihilation, may more likely be 
related to local and not
global field strengths. For example, even if the globally averaged 
magnetic 
field is
reduced by annihilation, the local field between a monopole-antimonopole 
pair connected
by a flux tube may remain above the critical field, so that the
flux tube will presumably persist, and annihilation can proceed.  We now 
demonstrate that
even under the most optimistic assumptions about the magnitude of local 
fields, for
almost all of electroweak parameter space, local flux tube
formation at a level capable of producing a confining potential between 
monopole-
antimonopole pairs will not occur. We first consider the case for which 
solutions
(involving a periodic flux tube network) were explicitly obtained by 
Ambj{\o}rn and Oleson\cite{AO1989a}.

The area A of flux tubes forming due to the $W$ condensate can be 
obtained by
minimizing the classical field energy averaged over each cell in the 
periodic 
network in the presence of a background $H$ field \cite{AO1989a}:
\begin{equation}
%]|Expr|[(2<c$%A^$^:4;,=b""E(#;)min_>;, A,] <2^$^m^;)W^2^;,e><c" #|
%|(%$^f(";)12_$^;,d_^;)2;,x  ($;)cell_>;,,M  <2^$^m^;)W^4(";,2$|
%|^e_^;)2>;, A ,K<c!$1(%:7l:4 ,M <2^$^g_^;)2(%;,8=co$^s_^;)2:7;,=b""q>>|
%|:4 <c" #(#<c!$1($$^:7f_^:4;)2;, ,M$^:7f^:4;)0^2>$^;,d_^;)2;,x|
%|__>]|[
\overline{{\cal E}}_{min}\rm \mit A\rm ={{\mit m}_{W}^{\rm 2} \over \mit 
e}
\int_{ce\ell \ell }^{}{f}_{\rm 12}{\mit d}^{\rm 2}\mit x\rm -{{\mit 
m}_{W}^{\rm 4} \over
2{\mit e}^{\rm 2}}\mit A\rm +\left({\mit \lambda \rm -{{\mit g}^{\rm 2} 
\over 8co{s}^{2}\mit
\theta_W }}\right)\rm \int_{}^{}\left({{\mit \phi }^{\rm 2}-{\mit \phi 
}_{\rm
0}^{2}}\right)^2{\mit d}^{\rm 2}\mit x,
\end{equation}
where $f_{12}$ is the magnetic field, and $\lambda$ is the 
$\phi^4$-coupling
in the Lagrangian, and $\phi_0$ is the Higgs VEV.
Utilizing the topological restrictions on the flux contained in the flux 
tubes (containing
minimal flux $2 \pi/e$),
\begin{equation} 
%]|Expr|[(,<c" #(%$^:4;,=b""f(";)12_$^;,d_^;)2;,x  ($;)cell_>;, |
%|,]<c"$#(#<c$%!^A>:7= /7<c$%!(":4=b""dl>__> ,]2.Yn ,O3]|[
\int_{ce\ell \ell }^{}{f}_{\rm 12}{\mit d}^{\rm 2}\mit x\rm =\oint_{}^
{}\vec{\mit A}\rm \cdot \vec{\mit d\ell }\rm =2\pi \rm /e,
\label{E}
\end{equation}
this yields an expression for A, determined by the energy density
$\overline{{\cal E}}_{min}$, which is in turn a function of the external 
magnetic field:
\begin{equation}  
%]|Expr|[(%:4;,=b""A ,] <2(%2.Yn $^m^;)W^2("$^;,e_^;)2<c!=Q(-<c$%A|
%|^;,E> ,K$^m^;)W^4;,,O2$^e_^;)2;, ,M,K<c!$1(%:7l:4 ,M <2^$^g_^|
%|;)2(%;,8= co$^s_^;)2:7;,=b""q>>:4 <c" #(#<c!$1($$^:7f_^:4;)2;, |
%|,M$^:7f^:4;)0^2>$^;,d_^;)2;,x__>>>]|[
 A\rm ={2\pi \rm {\mit m}_{W}^{\rm 2} \over 
{\mit e}^{\rm 2}\left[{\overline{\mit \cal E}_{min}\rm +{\mit 
m}_{W}^{\rm 
4}/2{\mit 
e}^{\rm
2}-\left({\mit \lambda \rm -{{\mit g}^{\rm 2} \over 8co{s}^{2}\mit 
\theta_W }}\right)\rm
\int_{}^{}\left({{\mit \phi }^{\rm 2}-{\mit \phi }_{\rm 
0}^{2}}\right)^2{\mit d}^{\rm 2}\mit
x}\right]}.
\label{A}
\end{equation} 
Taking the Bogomol'nyi limit\cite{Bogo}
${\mit  \lambda \rm ={{\mit g}^{\rm 2} \over 8 {\cos}^{2}\mit
\theta_W }}$, corresponding to $m_H=m_Z$, the classical field equations 
simplify, and
the properties of the flux tubes can be derived.  In 
particular, one can show
\cite{AO1990} that the area of the flux tubes is restricted to lie in 
the range
\begin{equation} 2 \pi \cos^2 \theta_W < A m_W^2 <2 \pi . 
\label{Aineq}
\end{equation}  
From
our point of view, it is important to realize that this result is 
equivalent to the statement that a $W$ condensate can only 
exist between the two
critical values of the magnetic field 
\begin{equation}
%]|Expr|[('<2^$^:4;,=b""m^;)W^2^;,e> ,\$^H__\<2^$^m^;)W^2(&;,e| 
%| = \cos:7=b""q>]|[
 {{\mit m}_{W}^{\rm 
2} \over \mit e\rm
\cos^2 \mit \theta_W } > {\mit H}^{}\rm > {{m}_{W}^{\rm 2} \over \mit 
e}\rm.
\label{H}
\end{equation}
Moreover, it gives a one to one correspondence 
between the area of the
flux tube and the background magnetic field value in this range.  We 
shall use this
correspondence, both in the Bogomol'nyi limit and beyond, to examine the 
confinement properties of
such a flux tube network connecting monopole-antimonopole pairs.
 
\begin{sloppypar}
Magnetic monopoles are formed at the $GUT$ transition with a density
of about one monopole per horizon volume.  This corresponds to a value
of ${n_M\over s}=10.4 {g_{*}}^{1/2} (T_{GUT}/M_{Pl})^3 \sim 
{10^{2}}(T_{GUT}/M_{Pl})^3$, 
where $n_M$ is the num\-ber
den\-si\-ty of mo\-no\-poles, $g_{*}$ is approximately the number of 
helicity 
states in the
radiation at the time $t_{GUT}$, $M_{Pl}$ is the Planck mass,
and $s$ is the entropy of the universe 
at this time.
Since $T_{GUT}$ could easily exceed $10^{15}$ GeV for SUSY GUTs, it is 
quite possible
that the initial monopole abundance left over from a GUT transition is 
${n_M\over s}
>10^{-10}$. Preskill has shown that in this case monopoles will 
annihilate shortly after
the GUT transition until ${n_M\over s} \sim 10^{-10}$\cite{Preskill}, 
and this value
remains constant down to the electroweak scale. Since 
$s=(2\pi^2/45)g_{*} T^3$, the
mo\-no\-pole
 num\-ber den\-si\-ty at the elec\-tro\-weak transition ($T_c\sim 300 
GeV$) of 
$\approx$ 0.13 $GeV^3$ (assuming
$g_{*}(T_{c}) \approx 100$) corresponds to a mean intermonopole 
spacing of $L \approx$ 2
$GeV^{-1}$. \footnote{Note that monopoles actually begin to dominate the mass
density of the universe somewhat before this temperature if they start with an
initial value of ${n_M\over s} >10^{-10}$. As we later describe, this can
have dramatic effects upon the expansion rate at this temperature.}  From this,
we can calculate the mean magnetic field  produced  by the monopoles with Dirac
charge $h = 2\pi/e$. In general, because the monopole background is best 
described as a ``plasma" involving both monopoles and antimonopoles, the mean
magnetic field will  be screened at distances large compared to the
intermonopole spacing. However,  because we will demonstrate that even under the
most optimistic assumptions,  monopole-antimonopole annihilation will not in
general occur, we ignore this mean field long-range  screening, and consider the
local field in the region between a monopole-antimonopole pair to be 
predominantly that of nearest neighbors, i.e. a magnetic dipole.  While the
field is not uniform in  the region between the monopole and antimonopole, we
will be interested in the minimum value of the  field here.  We shall make the
(optimistic) assumption that if this field everywhere exceeds the  critical
value $m_{W}^2/e$ on the line joining the two monopoles, that an instability of
the type  described above, involving a
condensate of $W$ fields and an associated magnetic flux tube, 
can occur along this line.  
\end{sloppypar}

    For a monopole-antimonopole pair separated by a
distance $L$, the minimum field will be halfway between them, and will
have a magnitude $H=2h/{\pi L^2} =4/{eL^2}$.  For this field to exceed 
the 
minimum
Ambj{\o}rn-Oleson field $m_{W}^2/e$ then implies the relation: 
$L<2/m_{W}$. 
For a
value $m_{W}=81 GeV$ this relation is manifestly not satisfied for the 
value of $L$
determined above. However, assuming a second order transition, as we 
have described, the $W$
mass increases continuously from zero as the temperature decreases below 
the critical
temperature, implying some finite temperature range over which the 
(fixed) background field
due to monopoles will lie in the critical range for flux tube 
formation.  In
this case, the magnetic field would enter this range from {\it 
above}.  In order that the magnetic field 
lie in the range given by 
inequality (\ref{H}), we find
 \begin{equation} 2/m_{W} > L> 2
\cos\theta_W/m_{W}. 
\label{L1}
\end{equation}    

Nevertheless, even if a
flux tube forms connecting the monopole-antimonopole pair, this will not 
result in a
confining linear potential until the width of the flux tube $2r < 
L$.  A bound on 
this width
can be obtained from the lower bound on the area of the flux tube 
(equation (\ref{Aineq})):  

\begin{equation} 2r > 2\sqrt2 \cos\theta_W/m_{W} ~, 
\label{width}
\end{equation} when the 
magnetic field is at
its upper critical value of $m_W^2/e\cos^2\theta_W$. This implies the 
constraint 
\begin{equation} L>  2\sqrt 2 \cos\theta_W/m_{W} ~.
\label{L2}
\end{equation}

As can be seen, inequalities (\ref{L1}) and (\ref{L2}) are mutually 
inconsistent.  Hence, there  appears to be
no region in which both a Ambj{\o}rn-Oleson type superconducting phase 
results, and
at the same time monopole-antimonopole pairs experience a confining 
potential.  We expect
the situation will be similar to the quark-hadron phase transition
when the
transition is second order.  In that case, it is impossible to 
distinguish between a
dense plasma of confined quarks and a gas of free quarks, because the 
mean interquark
spacing is small compared to the confinement scale.  Here there will be 
no physical impact
of a short superconducting phase, because the confinement scale is 
larger 
than the distance
between monopoles required to trigger the phase transition.  We expect 
no significant
monopole annihilation during the short time in which this phase is 
dynamically favored as
the $W$ mass increases.

This result has been derived in the Bogomol'nyi limit, when $m_H=m_Z$.  
What about going
beyond this limit? First, note that the energy density of the external 
magnetic
field, ${\cal E} = H^2/2$, provides an upper bound on $
\overline{\mit \cal E}_{min}$. Then from equation (\ref{A})
one can show that as long as  $\lambda >  g^2/8\cos^2\theta_W$ 
($m_H>m_Z$), the flux tube area, for a fixed value of the field, is {\it
larger} than it is in the Bogomol'nyi limit.   While we have no
analytic estimate of the upper critical field, and hence no lower bound 
on the flux tube
area, the scaling between area and magnetic field will still be such 
that for a given
monopole-antimonopole spacing, and hence a given magnetic field 
strength, 
the area of the
corresponding flux tube will be larger than in the Bogomol'nyi limit. 
Hence the
inconsistency derived above will be exacerbated.  Only in the narrow 
range $m_Z/2\stackrel{<}{\sim} m_H<m_Z$
(still allowed by experiment) is there a remote possibility that even in 
principle, flux tube
areas may be reduced sufficently so that confining potentials may be 
experienced by monopoles
triggering a $W$ condensate.  However, in this range, the energy 
(\ref{E}) can be reduced by
increasing $\phi$, so we expect that instabilities arise in 
this range which are likely
to make a $W$ condensate unstable in any case.

\vspace{.1 in}

\noindent{\bf 3. Dynamical Arguments Against Annihilation}:

 Even if a confining potential may be
achieved through flux tube formation, there are dynamical reasons to 
expect monopole
annihilation will not be complete.  These arguments apply to any
scenario involving a confining phase for monopoles, and suggest that 
estimates based on the efficacy of monopole annihilation 
may be
overly optimistic. In 
the first place,
we can estimate the energy of a 
monopole-antimonopole
pair separated by a string of length $L$.  For a long flux tube of 
radius 
$r$,
considerations of the electromagnetic field energy trapped in the tube 
imply a net energy
stored in the flux tube of
\begin{equation}
E= {L \over 2\alpha r^2 }.   
\label{kappa}
\end{equation}
  Considering the 
case when $L
\approx 2r$, when confinement would first begin, we find the energy 
associated with the
string tension is $E = {2 \over  \alpha L } \approx$ 130 $GeV$.  This is 
significantly smaller
than the mean thermal energy associated with a transition temperature 
$T_c \approx$ 300
$GeV$.  Hence, if the string tension does not vary significantly over 
the 
period during
which the magnetic field exceeds the critical field, the 
string tension exerts
a minor perturbation on the mean thermal motion of monopoles, and
will not
dramatically affect their dynamics.  The only way this would not be the 
case would be if
the monopole-antimonopole pair moved towards one another at a rate which 
could keep 
the magnetic
field between them sufficiently large so as to track the increase in the 
minimum critical
field as $m_W(T)$ increased to its asymptotic value.  However, this 
cannot in general
occur, because thermal velocities are sufficiently large so as to swamp 
the motion of the
monopole-antimonopole towards each other. Using the mean thermal 
relative
velocity of  monopoles at 
$T=T_c$,  we can calculate
how much time, $\delta t$, it would take to traverse a distance equal to 
the
initial
 mean distance between monopoles. This time becomes an ever smaller fraction of
the horizon time, $t$, as the universe expands. Since the  thermal velocity is
$\ll 1$, non-relativistic arguments are sufficient.  
We find $\delta t/t
\approx 6 \times 10^{-4}$, for $m_M \approx 10^{17} GeV$, and $T_c 
\approx$ 300 $GeV$.  During such a
small time interval, $m_W(T)$ remains roughly constant, and hence so 
does the string
tension. We find that during the time $\delta t$ the flux tube induced 
velocity of the monopole-antimonopole pair
remains a  small 
fraction of the mean thermal 
velocity, for $m_M >10^{15} GeV$.  Thus, monopoles and 
antimonopoles will
not in general move towards one another as $m_W$ increases.  Since 
$r(T)$ will not change
significantly between $H_c^{(1)}$ and $H_c^{(2)}$ as $m_W$ increases, if 
the mean 
inter-monopole spacing
remains roughly constant, monopole annihilation will, on average, not 
proceed before the
field drops below its critical value.   

What about the more general case  of
a brief superconducting phase which might result if $U(1)_{em}$ is 
broken for a
small temperature range around the electroweak scale
\cite{Sher}?  In this  case, the flux tube
area is not driven by the strength of the background magnetic field, and 
hence is not tied
to monopole-antimonopole spacing.  Nevertheless, dynamical arguments 
suggest that annihilation,
even in this case, may be problematic.  We describe three obstacles
here: (a) as above, the field energy 
contained in
the  string
may not be enough to significantly alter the dynamics of a thermal
distribution of monopoles;  (b) even in the event that this energy is
sufficiently large, the time required to dissipate this energy will in 
general
exceed the lifetime of the universe at the time of the  $U(1)_{em}$
breaking transition; (c) the time required for monopoles to annihilate 
even once
they have dissipated most of the string energy and are confined within a 
``bag"
may itself be comparable to the lifetime of the universe at the time of 
the
transition.  

(a) Consider the energy (\ref{kappa}) stored in the flux tube.  The 
radius, $r$,
will depend upon the  magnitude of the
VEV of the field responsible for breaking $U(1)_{em}$.  Until this field
achieves a  certain minimum
value, flux tubes will not be sufficiently thin to produce a confining 
potential for
monopoles.  Even if this VEV quickly achieves its maximum 
value, one must
investigate whether or not this field energy is large compared to the 
thermal energy at
that time, in order to determine whether the monopoles will be 
dynamically driven towards
each other.  As long as $r^{-1} \approx e{\phi_0} \approx eT_c$, where 
in this 
case $\phi_0$  represents the VEV of the field associated with 
$U(1)_{em}$ 
breaking
and $T_c$ represents the transition temperature, then $E
\gg  T$, so that the
condition of a confining potential is in general satisfied.
Nevertheless, one must also verify that this 
inequality is such that the Boltzmann tail of the monopole 
distribution with velocities
large enough to be comparable to this binding energy is sufficiently 
small (i.e. that
sufficiently few monopoles have thermal motion which is not
significantly affected by the confining potential). 
If we assume that such monopoles do not annihilate, then to avoid the
stringent limits on the monopole density today probably requires $E > 
O(25-30)
T$.  (This includes the fact that if monopoles annihilate, since they dominate
the energy density of the universe at this time, they will increase the
entropy by a factor of up to $10^5$, for $m_{M} \approx 10^{17} GeV$.)
Determining $L$ by scaling from the initial density, we find that if  $\phi_0 =
\rho  T_c$, then the ratio of the binding energy to the transition  temperature,
$E/T_c \approx 3800 \rho^2$, independent of $T_c$.  This implies a rather mild
constraint on the VEV of the field associated with  $U(1)_{em}$
breaking: $\rho > 0.09$.

(b) Monopole's must dissipate the large energy associated with the 
string
field energy if they are to annihilate.  There are two 
possible
ways in which this energy can be dissipated: thermal scattering, and the
emission of radiation
\cite{Preskill,Vilenkin}. Utilizing the estimates of energy loss by
radiation given by Vilenkin \cite{Vilenkin} we find that this process
requires $\approx 10^{15}$ times longer to dissipate the string energy 
than
the lifetime of the universe at the time of the 
transition.\footnote{This
calculation itself is probably an underestimate (unless the monopole
couples to massless or light particles other than the photon), since it  
assumes
the  photon is
massless, which it is not in this phase.}  Hence, we concentrate on the
possibility of dissipating the energy by thermal scattering.   We shall 
assume
here that $\rho \approx 1$, so that the initial average monopole-
antimonopole
pair energy is $\approx 3800 T$.  The energy loss by collisions with 
thermal
particles in the bath is \cite{Vilenkin} (Note that Callen-Rubakov
scattering might also play a role, although its efficiency as an 
energy loss mechanism for monopoles is questionable.  In any case
its cross section is also of this
order, and thus the estimates given here are appropriate even if this
is included.)
$ {dE}/dt \approx - bT^2 v^2$, where $b = 3 \zeta(3)/(4 
\pi^{2})
\sum {(q_i/2)}^2$, and the sum is over all helicity states of charged 
particles
in the heat bath.  At $T \approx$ 100 $GeV$, $b \approx 0.7$.  Recall
that at this time, monopoles will likely dominate the energy density
of the universe. Utilizing the appropriate relationship between temperature 
and
time resulting from the faster expansion rate in this case and setting
$\eta={n_M \over S}$, we find  
\begin{equation}   
E_{\infty}
 = E_i \exp \left[{-{0.04 M_{Pl}} \over  m_M} {\left({{T_i} \over
{\eta m_M}} \right)}^{1/2} \right]~. 
\end{equation}
For $m_M \approx 10^{17} GeV, E_{\infty} \approx 0.96 E_i$, so that
monopole-antimonopole pairs will apparently never dissipate their string
energy in a matter dominated epoch, and thus {\it{annihilation should not
proceed}}.  Even if somehow the universe were to become radiation dominated at
this time, either by monopole annihilation or some other mechanism, then we find
instead  \begin{equation} 
\ln {\left( {E_f \over  E_i}\right)} = {{0.03 b M_{Pl}} \over 2 m_M}
\ln {\left({{t_i} \over {t_f}} \right)}~.
\end{equation}
If we take $E_f$ to be the string energy
(\ref{kappa}) when $L=2r$, i.e.  the energy when  the string has become a
``bag", then even in this case the the time required to dissipate the initial
string  energy is O(50) $t_i$ for $m_M \approx 10^{17} GeV$.\footnote{Note that
even if the monopole annihilation were to keep the temperature relatively
constant for some period while it remained matter dominated, the time at which
the universe passes through any temperature could not be longer than that in a
universe which had remained radiation dominated throughout.}  Unless the phase
of broken $U(1)_{em}$ lasts for longer than this time (which depends
sensitively upon the monopole mass) not all the string
energy will be dissipated.  We have ignored here possible transverse motion of
the string.  This energy  must also be dissipated by friction, which may be
dominated by Aharanov-Bohm  type scattering\cite{alford}.\footnote{This latter
issue  has been raised in concurrent work  by R. Holman, T. Kibble, and
S.-J. Rey\cite{sj}, which concentrates on this mechanism while pointing out the
potential inefficiency of monopole annihilation.  Our results suggest that
dissipation of energy in the longitudinal modes of the string can take even
longer, thus providing stronger constraints on models, and further
support that monopole annihilation is not automatic in a
confining phase. }    In any case, even under the optimistic assumption
that monopoles somehow annihilate sufficiently (reducing the initial abundance by
a factor of 2) to result in a radiation dominated universe, this latter
result is still a rather severe constraint on the temperature range over which
the $U(1)_ {em}$ breaking phase must last. 

(3)  Even if the string energy can dissipated so that the mean distance 
between
monopole-antimonopole pairs is of order of the string width, they will 
be
confined in a ``bag", and one must estimate the actual time it takes for 
the pair
to annihilate in such a ``bag" state. (The monopole ``crust", of 
characteristic size
${m_W}^{-1}$, is assumed to play a negligible role here. 
In any case,
inside this ``bag" it is quite possible that the electroweak symmetry 
may 
be
restored, in  which case such a
crust would not be present.)  In a low lying s-wave state, the 
annihilation time
is very short.  However, in an excited state, involving, for example, 
high
orbital angular momentum (on the bag scale), this may not be the case, since the
wave  function
at the origin will be highly suppressed. We provide here one approximate
estimate for the annihilation time based on the observation that the 
Coulomb
capture distance $ a_c \approx 1/4\alpha E$ is 8 times smaller than the 
``bag"
size, for a monopole whose ``bag" energy  is inferred from equation 
(\ref{kappa})
with $L=2r$.  It is reasonable to suppose that annihilation might 
proceed
via collapse into a tightly bound Coulomb state.  
Thus, for the sake of argument one might roughly  
estimate
a lower limit on the annihilation time by utilizing the Coulomb capture 
cross 
section\cite{Preskill}
inside
the ``bag".  This capture time is  $\tau \approx (4 e
/3\pi T) (m_M/T)^{11/10}$, and is slightly longer 
than
the lifetime of the universe at temperature $T \approx 300 GeV$, for 
$m_M=10^{17}
GeV$. 
  Again, this suggests that the time during which the 
$U(1)_{em}$ 
breaking
phase endures must be long compared to the lifetime of the universe 
when this phase begins\footnote{If one imagines that because of the
monopole outer crust, emission of scalars is possible, the
capture cross section may be increased\cite{Preskill} to $ \approx 
(T_c)^{-2}$.
This  would
decrease the capture time by a significant amount ($\approx 10^{6}$).  
However,
once again, this requires that the scalars are light, otherwise phase 
space
suppression might be important.}.  If capture into a Coulomb state has 
not
occured  by the time the $U(1)_{em}$ breaking phase is over, 
previously confined monopole pairs separated by more than the 
Coulomb capture distance will no longer be bound.
The annihilation rate  for these previously confined pairs 
compared to the expansion rate will remain less than order
unity, so that monopoles will again freeze out. 

These considerations suggest that monopole-antimonopole 
annihilation by
flux tube formation at the electroweak scale is far from guaranteed. In
particular, monopole confinement triggered by monopole induced 
magnetic fields seems unworkable. More generally, in any confining 
scenario,
dissipation of the initially large flux tube energies requires times 
which are
generally long compared to the horizon time at the epoch of electroweak
symmetry breaking. (If the universe remains matter dominated
during this phase, it appears impossible to dissipate the string energy.)  At
the very least this places strong constraints on the minimum range  of
temperatures over which a confining phase for monopoles must exist.


\vfill\eject

\noindent \medskip\centerline{\bf Figure Captions}
\vskip 0.15 truein
Fig. 1. Phase diagram for $W$ condensation as a function of 
external
magnetic field and temperature assuming a second order electroweak phase
transtion.




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\end
