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\title{Magnetic monopoles from gauge theory phase transitions}

\author{A.~Rajantie}
\affiliation{DAMTP, CMS, University of Cambridge, Wilberforce Road,
Cambridge CB3 0WA, United Kingdom}

\date{9 December, 2002}

\begin{abstract}
Thermal fluctuations of the gauge field lead to monopole formation 
at the grand unified phase transition in the early universe, even if
the transition is merely a smooth crossover. 
The dependence of the produced monopole density on various parameters 
is qualitatively different from theories
with global symmetries, and the monopoles have a positive correlation
at short distances. 
The number density of monopoles may be suppressed if the
grand unified symmetry is only restored for a short time by,
for instance, non-thermal symmetry restoration after preheating.

\end{abstract}
\pacs{PACS: 11.15.Ex, 11.27.+d, 74.60.Ge}

\preprint{DAMTP-2002-157}
\preprint
\maketitle

It is a generic property of grand unified theories (GUTs) that
magnetic monopoles of mass of the order $m_M\approx 10^{16}$~GeV 
exist~\cite{'tHooft:1974qc,Polyakov:ek}, and these
monopoles would have been produced in large 
numbers in the GUT phase transition at 
$T_{\rm GUT}\approx m_M$~\cite{Kibble:1976sj}.
Afterwards, pair annihilations can decrease the monopole density,
but
estimates show that the number density would still be
comparable to baryons~\cite{Preskill:1979zi}. 
Because the monopoles are $10^{16}$ times heavier than protons,
this would have caused the universe to collapse under its own weight
long ago.

This 
monopole 
problem, alongside with several other cosmological puzzles, was
wiped away by the theory inflation~\cite{Guth:1980zm},
as the monopole density would have been diluted to a negligible
level by
a period of accelerating expansion.
For this to solve the problem, the reheat temperature at which the
universe thermalizes
must be lower than $T_{\rm GUT}$. 
These constraints are even stronger
in models with non-perturbative effects such as
preheating~\cite{Kofman:1995fi}, since the GUT symmetry can be
temporarily
restored \cite{Khlebnikov:1998sz,Rajantie:2000fd} 
and topological defects formed
even if the reheat temperature is well below 
$T_{\rm GUT}$~\cite{Kasuya:1998td,Tkachev:1998dc}.
It is therefore important to understand how monopoles are formed
to estimate how strong the
bounds imposed by the monopole problem really are.


The symmetry broken at the GUT phase transition is a local gauge
invariance, whereas
most of the existing
literature on monopole
formation 
implicitly assumes a breakdown of a global 
symmetry. The Kibble (or Kibble-Zurek)
mechanism~\cite{Kibble:1976sj,Zurek:1985qw}, 
which forms the monopoles in the global case,
is ultimately based on the observations that the direction of the 
order parameter cannot be
correlated at infinitely long distances.
Because the direction of the order
parameter is not gauge invariant,
this argument cannot be used in GUTs.

Moreover, gauge symmetries cannot be spontaneously broken~\cite{Elitzur:im}.
For rather 
generic parameter values, there is actually no phase transition at
all, 
but simply a
smooth crossover between the phases~\cite{Polyakov:vu,Hart:1996ac}. 
Does this mean that the whole evolution could be
adiabatic and thereby there would
be no monopole formation at all?

In this letter, I will present an argument
that shows that monopoles are still formed. 
This result is based on causality and 
the conservation of magnetic charge. In
fact, Weinberg and Lee~\cite{Weinberg:uq,Lee:gt} have used somewhat
similar reasoning to constrain later annihilations of monopoles after
the phase transition in the context of the Kibble mechanism.
In our case, it is also important 
that the monopoles have long-range interactions,
which are screened by the presence of a non-zero monopole density, and
that there is no true phase transition, because it lets us discuss 
the monopole density and correlations and the screening of the
magnetic field even at and above the transition.
As there is, in this sense, 
a high density of monopoles above the transition, one
could say that we are describing annihilation rather than formation of
monopoles.
This is obviously a matter of taste, but in any case,
the monopoles do not correspond to localized energy
concentrations in the symmetric phase and cannot therefore be thought
of as particles.

As we shall see, 
the monopoles formed in a gauge theory
will have positive correlations at short distances, 
and there can even be
clusters of monopoles of equal sign. 
This is the opposite of what the Kibble mechanism would predicts.
The number density of monopoles will also be
qualitatively different from the Kibble mechanism.

Strictly speaking, these calculations correspond to
thermal phase transitions, but 
I will also discuss what we can expect to happen in 
the non-thermal
case~\cite{Khlebnikov:1998sz,Rajantie:2000fd,Kasuya:1998td,Tkachev:1998dc}
based on them.
Also, I will only study the formation of the initial monopole density
and
ignore the subsequent annihilations of
monopoles, which may be very different from the Kibble case
because of different monopole correlations.

Let us start by briefly reviewing the standard Kibble mechanism
\cite{Kibble:1976sj}.
For simplicity, we shall discuss the SU(2) symmetry group only, but
the same arguments should apply to SU(5), SO(10) or other possible
GUTs.
The Lagrangian of the theory is
\begin{equation}
{\cal L}={\rm Tr}\partial_\mu\Phi\partial^\mu\Phi
-m^2{\rm Tr}\Phi^2-\lambda\left({\rm Tr}\Phi^2\right)^2,
\end{equation}
where $\Phi$ is in the adjoint representation, and we are
assuming that at zero temperature, the SU(2) symmetry is broken. To
leading order, this means $m^2<0$.

We shall consider this theory at a non-zero temperature $T$. When the
temperature is high enough, the SU(2) symmetry is unbroken. We are asking
what happens if we start from thermal equilibrium in the
symmetric phase and gradually decrease the temperature so that the
symmetry gets broken. It only takes a small change in temperature
near the critical temperature $T_c$ to cause the phase transition, and
this effect is mainly due to the effective mass parameter's changing
from positive to negative. Therefore, we shall
simply consider keeping $T$ fixed and varying $m^2$.

In the high-temperature phase, the
field $\Phi$ vanishes on the average. In the broken phase, it would
ideally have a non-zero constant value $\Phi(\vec{x})=\Phi_0$, where
${\rm Tr}\Phi_0^2=\phi^2/2=-m^2/2\lambda>0$, but
this would require ordering of the
field at infinite distances, which cannot be achieved in finite
time. Instead, the field can only be roughly constant inside finite
domains, whose size $\hat{\xi}$ is determined by the maximum
correlation length $\xi$ and limited by the fact that $\xi$ cannot
grow faster than the speed of light, $\dot\xi<c$.
In fact, the maximum growth rate of $\xi$ is typically much less than
$c$, and depends on the critical dynamics of the
system~\cite{Zurek:1985qw}.

After the transition, 
we can imagine that the system consists of domains of radius
$\hat\xi$ inside each of which the field is constant, but between which
it is totally uncorrelated. At each point where four of these domains
meet, there is a fixed, non-zero probability that the field cannot
smoothly interpolate between the domains without vanishing at a
point. This point is a monopole, and therefore this scenario predicts
a monopole density 
\begin{equation}
n_M^{\rm Kibble}\approx \hat\xi^{-3}.
\label{equ:kibblepred}
\end{equation}
Furthermore, we can easily see that there must be a strong negative
correlation between monopoles at short distances: Imagine a sphere
centred at a monopole. If the radius is greater than $\hat\xi$, each
point at the sphere is uncorrelated with its centre and therefore
insensitive to whether there is a monopole inside or not. Therefore,
the average winding number must be zero, and that means that there
must be an antimonopole at distance $\approx\hat\xi$ from each monopole.


Let us now turn our attention to the gauge theory. 
The gradients in the Lagrangian are replaced by covariant derivatives
$D_\mu=\partial_\mu+igA_\mu$,
\begin{eqnarray}
{\cal L}&=&
-\frac{1}{2}{\rm Tr}F_{\mu\nu}F^{\mu\nu}+
{\rm Tr}[D_\mu,\Phi][D^\mu,\Phi]
\nonumber\\&&
-m^2{\rm Tr}\Phi^2-\lambda\left({\rm Tr}\Phi^2\right)^2,
\end{eqnarray}
where $F_{\mu\nu}=(ig)^{-1}[D_\mu,D_\nu]$.

To a good approximation, the phase structure of this theory is given
by a three-dimensional effective theory~\cite{Rajantie:1997pr}, 
which depends on two
parameters, $g^{-2}(T/T_c-1)$ and
the ratio of the coupling constants $\lambda/g^2$.
There is a line of first-order transitions at
small $\lambda/g^2$~\cite{Coleman:jx}, which ends at a
second-order point. For large $\lambda/g^2$, the two phases are smoothly
connected to each other~\cite{Hart:1996ac}, and we shall concentrate
on this case. We shall still assume that
$\lambda\approx g^2$ and use that to simplify estimates.
In this case, correlation lengths are finite at any parameter values,
apart from the low-temperature limit. Even though there is no true
phase transition, it is still possible to find an approximate
crossover point~\cite{Hart:1996ac}, 
which separates a ``symmetric'' and a ``broken''
phase, and we will use this terminology although it is not quite precise.

In the broken phase, 
$m_M> T$, and we can treat the monopoles as point-like
particles. Therefore, we have the standard expression for the equilibrium
monopole density
\begin{equation}
n_M^{\rm eq}\approx
(m_MT)^{3/2}\exp\left(-\frac{m_M}{T}\right).
\label{equ:eqdens}
\end{equation}
The monopole mass $m_M$ is roughly $m_M\approx \phi/g \approx
(-m^2/\lambda g^2)^{1/2}$.
When $m^2$ decreases further, $m_M$ grows
rapidly, which suppresses $n_M^{\rm eq}$.

If the monopoles did not have long-range interactions, they
would be essentially uncorrelated and behave very
much like the magnetic field in the Abelian Higgs 
model~\cite{Hindmarsh:2000kd,Rajantie:2001ps}, 
albeit in three rather than two
dimensions. 
There is, however, a magnetic Coulomb interaction between the
monopoles, and we shall see that it suppresses their production.
This interaction gives rise to correlations, which are
reflected in the screening of the magnetic field by the 
monopoles~\cite{Polyakov:vu,Davis:2001mg}, in
analogy with the Debye screening of the electric field.

The 
magnetic screening length
$\xi_B$ can be defined as the decay rate of the correlator of 't~Hooft
field strength operators~\cite{'tHooft:1974qc}. In equilibrium, 
it is approximately
\begin{equation}
\xi_B\equiv 1/m_B\approx\sqrt{\frac{T}{n_Mq_M^2}}
\approx\sqrt{\frac{g^2T}{n_M}},
\label{equ:debye}
\end{equation}
where $q_M=4\pi/g$ is the magnetic charge of a monopole.
Correspondingly, the magnetic charge-charge correlator is
\begin{equation}
\langle\rho_M(\vec{x})\rho_M(\vec{y})\rangle
\approx q_M^2n_M\left(\!
\delta(\vec{x}\!-\!\vec{y})-\frac{m_B^2}{4\pi|\vec{x}\!-\!\vec{y}|}
e^{-m_B|\vec{x}\!-\!\vec{y}|} \!\right)\!.
\label{equ:eqcorrcoord}
\end{equation}
Using Eq.~(\ref{equ:eqdens}), we find that the
equilibrium screening length behaves as
\begin{equation}
\xi_B\approx gT^{-1/4}m_M^{-3/4}e^{m_M/2T}.
\label{equ:eqxiB}
\end{equation}
This expression is only valid in the broken phase, but we know that
above the crossover, the only relevant scale is $g^2T$~\cite{Hart:1996ac} and
therefore we expect $\xi_B\approx (g^2T)^{-1}$. 
Since the screening length is always well defined, we can
actually use Eq.~(\ref{equ:debye}) to define the monopole density $n_M$
also in the symmetric phase.

If $m^2$ is decreased at a constant rate, $\xi_B$ would have to grow
exponentially fast to stay in equilibrium,
but, obviously,
it cannot grow faster than the speed of light. In practice, 
it would grow much slower than this.
This means that sooner or later the growth rate 
$d\xi_B/dt$ needed for
the system to stay in equilibrium exceeds the maximum value, and the
system falls out of equilibrium. We shall denote the time when
this happens by $\hat{t}$. 

The screening length $\xi_B$ can
still keep on growing, but so slowly that we can ignore it if we are
only interested in finding an order-of-magnitude estimate for the initial
monopole density. Therefore, we define the freeze-out screening
length $\hat\xi_B$ as $\xi_B$ at the time when it falls out of
equilibrium,
\begin{equation}
\hat\xi_B=\xi_B(\hat{t}).
\end{equation}
At the time of the freeze-out, the monopole density is
\begin{equation}
\hat{n}_M\approx T/q_M^2\hat{\xi}_B^2
\approx g^2T / \hat{\xi}_B^2,
\label{equ:freezeoutdensity}
\end{equation}
and as $T\gg \hat{\xi}_B^{-1}$, the typical distance 
$\hat{d}\approx \hat{n}_M^{-1/3}$ between monopoles
and antimonopoles is much shorter than the screening length.

Even after the freeze-out, the monopole density will keep on
decreasing, but this is now due to pair annihilations at length scales
shorter than $\hat\xi_B$.
These annihilations smoothen the distribution of monopoles at short 
distances, but they cannot remove them completely~\cite{Weinberg:uq,Lee:gt}. 
To see this, consider a sphere of radius $\hat\xi_B$.
The annihilations may reduce the number of monopoles inside the sphere 
to the
minimum, but they cannot change
its net magnetic charge significantly. 
While the net magnetic charge is zero on the average, it fluctuates
with a root-mean-squared value of
\begin{equation}
Q_M(\hat\xi_B)
= \sqrt{\left\langle \left(\int^{\hat\xi_B} d^3x \rho_M(\vec{x})
\right)^2 \right\rangle}\approx \sqrt{T\hat\xi_B}.
\label{equ:rmscharge}
\end{equation}
Since the annihilations cannot
reduce the charge below this, the monopole
density cannot fall below
\begin{equation}
n_M\approx \frac{Q_M(\hat\xi_B)}{q_M\hat\xi_B^3} 
\approx q_M^{-1}\sqrt{\frac{T}{\hat\xi_B^5}}\approx 
g\sqrt{\frac{T}{\hat\xi_B^5}}.
\label{equ:omapred}
\end{equation}
We have not shown how to estimate $\hat\xi_B$, but
nevertheless, this expression is clearly different from the
Kibble-Zurek result (\ref{equ:kibblepred}), 
because of the explicit appearance of
$g$ and $T$.

Moreover, we can note that as long as $Q_M(\hat\xi_B)\gg q_M$, there
will be clusters of monopoles of equal sign, and the number of
monopoles in each of them can be large if $T\gg \hat\xi_B^{-1}$.
This means that there is a positive correlation between monopoles
at short distances, very much in the same way as in the case of
vortices in the Abelian Higgs model~\cite{Hindmarsh:2000kd,Rajantie:2001ps}
and in stark contrast with
the Kibble mechanism. 

We can reach the same conclusions by studying the time evolution of
the magnetic charge correlator in the Fourier space. We define $G(k)$
by
\begin{equation}
\langle \rho_M(\vec{k})\rho_M(\vec{q})\rangle
=q_M^2 G(k)(2\pi)^3\delta\left(\vec{k}+\vec{q}\right),
\end{equation}
and from Eq.~(\ref{equ:eqcorrcoord}), we find
\begin{equation}
G(k)=\frac{T}{q_M^2}\frac{m_B^2k^2}{k^2+m_B^2}.
\label{equ:eqG}
\end{equation}
As there is no transition, we expect that
\begin{equation}
G(k)\approx Tk^2 / q_M^2
\label{equ:eqGlowk}
\end{equation}
in the symmetric phase where $m_B$ is large, and that
$G(k)$ approaches zero at $T\ll T_c$.

Causality implies that very long-wavelength (low $k$) 
correlations can only change
slowly~\cite{Weinberg:uq,Lee:gt}, and 
we can give a rough upper bound for the rate of change,
\begin{equation}
\left|\frac{d \ln G(k)}{dt}\right| \lsim k.
\label{equ:adiabgen}
\end{equation}
Using Eq.~(\ref{equ:eqG}), this becomes
\begin{equation}
\frac{k^2}{k^2+m_B^2}\frac{d\ln m_B^2}{dt}\lsim k.
\label{equ:adiabcond}
\end{equation}
Below the transition, $\ln m_B\approx \sqrt{-m^2}/g^2T$, and if we keep on
decreasing $m^2$, then sooner or later Eq.~(\ref{equ:adiabcond}) 
ceases to be satisfied
for $k$ less than some critical value $\hat{k}$. The modes
with higher $k$ keep on decreasing and we approximate the final
correlator by
\begin{equation}
G(k)\approx \frac{Tk^2}{q_M^2}\exp\left(-\frac{k^2}{2\hat{k}^2}\right).
\label{equ:freezeoutG}
\end{equation}
A Gaussian fall-off like this would follow naturally from diffusion, but
our conclusions do not depend on the precise form of the correlator,
as long as it has a relatively sharp cutoff at $\hat{k}$.
The corresponding monopole density is given by
\begin{equation}
n_M\approx \left(\int \frac{d^3k}{(2\pi)^3}G(k)\right)^{1/2}
\approx q_M^{-1}\sqrt{T\hat{k}^5},
\end{equation}
which agrees with Eq.~(\ref{equ:omapred}) if we identify $\hat{k}=1/\hat\xi_B$.

We can also find the monopole-monopole correlator in coordinate space
by taking the Fourier transform of Eq.~(\ref{equ:freezeoutG}),
\begin{equation}
G(r)\approx\frac{T\hat{k}^5}{q_M^2}\frac{e^{-r^2\hat{k}^2/2}}{(2\pi)^{3/2}}
\left(
3-r^2\hat{k}^2\right),
\end{equation}
and it is indeed positive at distances
$r\lsim \sqrt{3}/\hat{k}$.

Let us then estimate the
monopole density produced in the GUT phase transition
using only causality to limit the growth of $\xi_B$.
At high temperatures, the effective mass parameter of the theory is 
$m^2(T)\approx g^2(T^2-T_{\rm GUT}^2)$.
Because of the expansion of the universe, the temperature is
decreasing at the rate $dT/dt\approx -T^3/M_P$, where $M_P\approx
10^{19}~{\rm GeV}$ is the Planck mass. Near $T_{\rm GUT}$,
we can therefore approximate
\begin{equation}
m^2\approx -g^2\frac{T_{\rm GUT}^4}{M_P}t.
\end{equation}
Below $T_{\rm GUT}$, the monopole mass grows as
\begin{equation}
m_M\approx \sqrt{\frac{t}{g^2M_P}}T_{\rm GUT}^2.
\end{equation}
From Eq.~(\ref{equ:eqxiB}) we see that the growth rate of $\xi_B$ 
is
\begin{equation}
\frac{d\xi_B}{dt}\approx \frac{T_{\rm GUT}^{11/4}}{gm_M^{7/4}M_P}
e^{m_M/2T_{\rm GUT}}=
\frac{T_{\rm GUT}}{gM_P}x^{-7/4}e^x,
\end{equation}
where we have introduced the dimensionless variable $x=m_M/2T_{\rm GUT}$.
We require that this is equal to $1$ for the freeze-out scale, 
and find
$x\approx \ln(gM_P/{T_{\rm GUT}}),
$
and consequently
$
\hat{\xi}_B\approx g^2 M_P/T_{\rm GUT}^2.
$
Then, Eqs.~(\ref{equ:omapred})
tells us that the monopole density is
\begin{equation}
n_M\approx g^{-4}T_{\rm GUT}^{11/2}M_P^{-5/2},
\label{equ:GUTpred}
\end{equation}
which we can compare with the prediction of the Kibble mechanism under
the same circumstances~\cite{Einhorn:ym},
\begin{equation}
n^{\rm Kibble}_M\approx g^2T_{\rm GUT}^4/M_P.
\end{equation}
The two results differ
by a factor of $g^6(M_P/T_{\rm GUT})^{3/2}$, which is not particularly
large for realistic GUTs.
On the other hand, Eq.~(\ref{equ:rmscharge}) 
shows that the typical number of monopoles in a cluster is
\begin{equation}
N_M^{\rm net}=\frac{Q_M}{q_M}\approx g^2\sqrt{\frac{M_P}{T_{\rm GUT}}}.
\end{equation}
This combination is, again, of order one, which means that there is a
possibility of forming small clusters.

The estimate in
Eq.~(\ref{equ:GUTpred}) 
is obviously not
very precise. If the transition is fast enough, and
this may actually be the case in the GUT transition, the
approximation in Eq.~(\ref{equ:eqdens}) 
that the monopoles are point particles is
not justified. One will then have to use a field theory description,
and probably calculate the monopole correlator numerically using
lattice Monte Carlo simulations. It
is also likely that the monopoles move diffusively rather
than at a constant velocity.
In any case, the estimate illustrates how monopoles form in a
transition that starts from thermal equilibrium, and confirms that 
GUTs suffer from 
a monopole problem, as
earlier studies based on global theories have suggested. 



It is also interesting to apply this same picture to 
cases where the GUT symmetry is restored only briefly
after inflation, either because of ``non-thermal''
fluctuations~\cite{Khlebnikov:1998sz,Rajantie:2000fd,%
Tkachev:1998dc,Kasuya:1998td} or because the reheat temperature is
slightly above $T_{\rm GUT}$.
Then, causality and charge conservation would suppress
long-wavelength fluctuations of the magnetic charge.

The estimated monopole density depends on the low-momentum behaviour
of $G(k)$ given in Eq.~(\ref{equ:eqGlowk}). Because of charge
conservation, the monopoles and antimonopoles must be produced in
pairs, and even if they move at the speed of light, the leading term
in $G(k)$ grows as $G(k)\sim n_Mk^2t^2$. It will therefore take at
least the time $t_{\rm eq}\approx(g^2T/n_M)^{1/2}\approx 
\xi_B$ to achieve the form
(\ref{equ:eqGlowk}). This conclusion can also be reached by
considering the time it takes for the pairs to reach the typical
equilibrium size $\sim\xi_B$.

This means that if the GUT symmetry is restored only very briefly, for
a period shorter than $t_{\rm eq}\approx (g^2T_{\rm GUT})^{-1}$, the number
density of monopoles will be suppressed. In reality, the equilibration
process is probably significantly slower, and therefore $t_{\rm eq}$
can be much larger, perhaps even so large that the bounds on the
reheat temperature disappear completely. 
In any case, a more careful analysis of the
dynamics is needed to estimate how strong the suppression is in
practice and whether it solves the monopole problem in the case of
non-thermal symmetry restoration.




The author would like to thank
Mark Hindmarsh, Tom Kibble and Andrei Linde
for useful discussions, and PPARC, 
Churchill College and the ESF Programme ``Cosmology in the
Laboratory'' for financial support.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{thebibliography}
\end{document}

