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\begin{document}
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\title{MODELS OF DYNAMICAL SUPERSYMMETRY BREAKING AND QUINTESSENCE.}
\author{P.~Bin\'etruy$^a$}
\address{$^a$LPTHE, Universit\'e Paris-XI, B\^atiment 211, F-91405
Orsay Cedex, France}
\maketitle
\begin{abstract}
We study several models of relevance for the dynamical breaking of
supersymmetry which could provide a scalar component with equation of
state $p=w\rho$, $-1<w<0$. Such models would provide a natural
explanation for recent data on the cosmological parameters.
\end{abstract}
\pacs{PACS numbers: 98.80.Cq, 98.70.Vc }
\vskip2pc]

\section*{Introduction}

There are increasing indications that the energy density of matter in the 
Universe is smaller than the critical density\cite{exp}. 
If one sticks to the inflation 
prediction of $\Omega_T=1$, then the natural question is the origin of the 
extra  component providing the missing energy density. An obvious candidate is 
a cosmological constant, whose equation of state is $p=-\rho$. This faces 
particle physics with the unpleasant task of explaining why the energy of the 
vacuum should be of order $(0.003\; {\rm eV})^4$, a task possibly even harder 
than the one of explaining why the cosmological constant is zero. In 
particular, it seems to require new interactions with a typical scale 
much lower than the electroweak scale, long range interactions that would 
have remained undetected.

It has recently been proposed to consider instead a dynamical time-dependent 
and spatially inhomogeneous component, with an equation of state $p=w\rho$,
$-1<w<0$. Such a component has been named ``quintessence'' by Caldwell, Dave 
and Steinhardt \cite{CDS}.   Indeed, present cosmological data seem to prefer
\cite{TW}, in the context of cold dark matter models, a value for $w$ of 
order $-0.6$. Several candidates have been proposed for this component: 
tangled cosmic strings \cite{SP}, pseudo-Goldstone bosons \cite{FHSW}.
Of particular relevance to some issues at stake in the search for a unified 
theory of fundamental interactions is a scalar field with a scalar potential 
decreasing to zero for infinite field values \cite{CDS,ZWS}. 

It has been noted that such a behaviour appears naturally in models of
dynamical supersymmetry breaking (DSB)\cite{ZWS}. Typically, the scalar
potential of supersymmetric models has many flat directions, {\em i.e.}
directions in field space where the scalar potential vanishes. Once
supersymmetry is broken dynamically, the degeneracy corresponding to the
flat direction is lifted but generally the flat direction is restored at
infinite values of the scalar field\footnote{In some cases, the field
value may be interpreted as the inverse coupling constant associated
with the dynamics responsible for supersymmetry breaking. An infinite field
value means a vanishing gauge coupling and thus restoration of
supersymmetry.}.
We are thus precisely in the situation of a potential smoothly
decreasing to zero at infinity. This is usually considered as a drawback
of spontaneous supersymmetry breaking models from the point of view of
cosmology: in the standard approach, the potential has a stable ground
state, where the potential is fine tuned to zero (in order to account
for a vanishing cosmological constant); but the initial conditions and
the subsequent cosmological evolution
may lead to a situation where the field misses the ground state and 
evolves to infinite values.

Dynamical supersymmetry breaking  is often favoured because it
can more easily account for large mass scale hierarchies such as
$M_W/M_P$ through some powers of $\Lambda/M_P$ where $\Lambda$ is the
dynamical scale of breaking. It is thus a natural question to ask whether the
corresponding models may account for quintessence. Indeed, in this case,
there is a fundamental reason why the scalar potential vanishes at
infinity: this is related to the old result that global supersymmetry
yields a vanishing ground state energy. And there may be reasons as to
why once it dominates, the contribution of the scalar field to the
energy density is very small (again through powers of $\Lambda/M_P$). 


In the following, we will discuss two models of dynamical supersymmetry
breaking which may be considered as representative of semi-realistic
models for high energy physics. One is based on gaugino condensation coupled to
the dynamics of a dilaton field, the other uses the condensation of
$N_f$ flavors in a $SU(N_c)$ gauge theory.

\section*{Models with a dilaton}

We start with a class of models, reminiscent of many superstring
models, where supersymmetry is broken through gaugino condensation 
\cite{gaugecond} along
the flat direction corresponding to the dilaton field. Indeed, in many
superstring models, the dilaton field $s$ does not appear in the
superpotential and thus corresponds to a flat direction in the scalar
potential. It couples to the gauge fields in a model-independent way: 
\begin{equation}
{\cal L} = -{1\over 4} s F^{\mu \nu} F_{\mu \nu}
\end{equation}
where $F_{\mu\nu}$ is the field strength corresponding to a generic
gauge  symmetry group $G$ and, throughout this article, $s$ is expressed
in Planck mass units. Thus the vacuum expectation value $<s>$ can 
be interpreted as the inverse of the  gauge coupling $1/g^2$ at the
string scale. Indeed, it
is directly related to the inverse of the string coupling constant
(see below). 
The interaction  corresponding to the gauge group $G$  becomes strong 
at a scale:
\begin{equation}
\Lambda = M_P e^{-1/2bg^2}=M_P e^{-s/2b_0} \label{eq:lambda}
\end{equation} 
where $b_0$ is the one-loop beta function coefficient of the gauge group $G$.
The corresponding gaugino fields are expected to condense:
\begin{equation}
<\bar \lambda \lambda> = \Lambda^3 = M_P^3 e^{-3s/2b_0}
\end{equation}
and they lead to a potential energy, quadratic in the gaugino
condensates, that scales like $e^{-3s/b_0}$. In the limit of infinite
$s$, that is of vanishing gauge coupling, the dynamics is inoperative
and one recovers the flat direction associated with the dilaton.

We have followed a very crude approach and there are, of course, many
possible refinements: one may include supergravity corrections, the
effect of other scalar fields such as moduli, as well as corrections
which may be needed to stabilize the potential for small values of $s$ 
(that is in the regime of strongly coupled string \cite{BD}). For
example, in a given model \cite{BGW}, the potential reads, in term of 
the field $\ell$ which precisely describes the string gauge coupling:
\begin{eqnarray}
V(\ell)&=&{M_P^4 \over 16 e^2 \ell}\{(1+f-l{df\over d\ell})
(1+{2\over 3}b_0 \ell)^2-{4\over
3}b_0^2\ell^2)\} \nonumber \\
& & \hskip 1cm \times \; e^{g-3(f+1)/2b_0\ell},
\end{eqnarray} 
where $f(\ell)$ and $g(\ell)$ appear as non-perturbative contributions
to the K\"ahler potential. The dilaton $s$ is related to the field
$\ell$ as $s=(1+f)/2\ell$. One recovers, in the limit of large $s$
(small string coupling $\ell$) a leading  behaviour in  $e^{-3s/b_0}$. 

Since there are obvious power law corrections to this behaviour, we will
consider a toy model of a dilaton field $s$ with a Lagrangian:
\begin{equation}
{\cal L} = -{1 \over 4 s^2} \partial^\mu s \partial_\mu s - V(s),
\end{equation}
where
\begin{equation}
V(s) = V_0(s) e^{-3s/b_0}.
\end{equation}
The non-canonical kinetic term for $s$ is caracteristic of the string
dilaton and accounts for the non-flat K\"ahler metric.

The cosmological evolution of the $s$ field is described by the
following set of equations ($\kappa = 1$):
\begin{eqnarray}
{\ddot{s} \over 2 s^2} - {\dot{s}^2 \over 2 s^3} &+& 3 H {\dot s \over 2
s^2} + {dV \over ds} = 0 \nonumber \\
H^2 &=& {1 \over 3} (\rho_B + \rho_s)
\end{eqnarray}
where $\rho_B$ is the background energy density associated with matter
($w_B=0$) or radiation ($w_B=1/3$) and $\rho_s =
\dot{s}^2/(4s^2) + V(s)$.

If we first consider that $V_0(s)$ is a constant and solve these equations 
assuming that $\rho_B$ dominates for some time, there exists a scaling
solution with the following behaviour:\footnote{ For a similar analysis,
although in a different context, see Ref. \cite{BCC}.} 
the field $s$ evolves down the
exponentially decreasing potential as $(t/t_1)^{{1- w_B \over 1+
w_B}}$ as long as $s$ remains smaller than $s_1 \equiv {2b_0 \over 3} 
{{1+ w_B \over 1- w_B}}$, reached at $t=t_1$; for larger values, 
there exists a scaling solution \cite{PR,CLW,FJ} 
where the field  evolves logarithmically as
$s=s_1 + (2b_0/3) \ln(t/t_1)$. The ratio $\rho_s/\rho_{tot}$ starts at
$3(1-w_B)^2/16$ for $t \le t_1$ and from then on slopes down to zero 
as $(b_0^2/6s^2)(1+w_B)$ for large values of
$s$. Finally, $w_s = p_s/\rho_s$ starts at a value of $1$ and
decreases monotonically towards $w_B$ as $s$ increases. There is
therefore no hope of using the dilaton for the  dynamical component
of quintessence since $w_s$ never reaches a negative value.
Power law corrections ($V_0(s) \propto s^\alpha$) do not change this
conclusion.  

This might be in some sense a welcome conclusion since the vacuum
expectation value $<s>$ provides, after renormalisation down to low
energy, the fine structure constant $1/\alpha$. A sliding dilaton would
make the fine structure constant vary with time at an unacceptable rate
\cite{alpha}. 

Similar conclusions can be reached with other types of weakly coupled
scalar particles, such as the moduli of string theories. For example, in
a model with several gaugino condensates and a modulus field $t$
describing the radius of the six dimensional compact manifold, the 
scalar potential scales for large values of $t$  as \cite{BGW}:
\begin{equation}
V= \sum_a t^{{b+b_a \over b_a}} e^{-\pi {b-b_a \over 3 b_a} t}
e^{-2<s>/b_a}
\end{equation}
where the sum runs over the different condensates (one for each group
$G_a$, with corresponding beta function coefficient $b_a$). We have
fixed the dilaton field $s$ at its ground state value. Let us note
that, although the modulus $t$ definitely 
cannot be used for quintessence (since,
as above, the corresponding $w_t$ reaches asymptotically $w_B$), a large
value of $<s>$ may contribute to giving a small contribution from $t$ to
the vacuum energy.  

\section*{A model of fermion condensates}

We now turn to a model  which yields inverse powers of fields in the
potential, a welcome situation for quintessence models \cite{ZWS}.
It is based on the gauge group $SU(N_c)$ and has $N_f\le
N_c$ flavors: quarks $Q^i,\; i=1 \cdots N_f$ in
fundamentals of $SU(N_c)$ and antiquarks  ${\tilde Q}_i,\; i=1 \cdots
N_f$  in antifundamentals of $SU(N_c)$.

Below the scale of dynamical breaking of the gauge symmetry $\Lambda$,
the effective degrees of freedom are the fermion condensate (``pion'')
fields $\Pi^i_j \equiv Q^i \tilde Q_j$. The dynamically generated
superpotential reads \cite{ADS}:
\begin{equation}
W= (N_c - N_f) {\Lambda^{{3N_c-N_f \over N_c-N_f}} \over (det \Pi)^{{1
\over N_c-N_f}}} .
\end{equation}
Usually, one allows a term linear in $\Pi$ in the superpotential in
order to stabilize this field. We will instead assume here that a
discrete symmetry ensures that no linear term is allowed by the
abelian symmetry. Let us note that this symmetry cannot be a continuous
gauge symmetry since this would yield in the scalar potential D-terms
with positive powers of $\Pi$ which would stabilize the field.

The effective Lagrangian  reads:
\begin{eqnarray}
{\cal L} &=& -{1\over 2} {\rm Tr} \; \left[ (\Pi^\dagger \Pi)^{-1/2}
\partial_\mu \Pi \partial^\mu \Pi^\dagger \right] \nonumber \\
& & + 2 {\rm Tr} \; \left[  (\Pi^\dagger \Pi)^{-1/2}{(\Lambda
\Lambda^\dagger)^{{3N_c-Nf \over N_c-N_f}} \over ({\rm Det} \;
\Pi^\dagger \Pi)^{{1 \over N_c-N_f}}} \right]
\end{eqnarray}
where the potential originates from the F-term for the field $\Pi$.
For simplicity, we will take   $\Pi^i_j$ to be diagonal and write
$\Pi^i_j \equiv \Phi^2 \delta^i_j$ with $\Phi$ real. One obtains:

\begin{equation}
{1\over 4N_f}{\cal L} = -{1 \over 2} \partial^\mu \Phi \partial_\mu \Phi + V(\Phi)
\end{equation}
where
\begin{equation}
V(\Phi) = \lambda {\mu^{4+\alpha} \over \Phi^\alpha},
\end{equation}
with $\mu = (\Lambda \Lambda^\dagger)^{1/2}$ and
\begin{equation}
 \alpha = 2 {N_c + N_f \over N_c - N_f}.
\end{equation}
The corresponding potential has been studied in Ref.\cite{PR} in the
case where $\rho_B$ dominates over the energy density $\rho_\Phi$ of the
$\Phi$ field. One obtains
\begin{equation}
{\rho_\Phi \over \rho_B} = \left( {a \over a_Q}
\right)^{6(1+w_B)/(2+\alpha)}.
\end{equation}
Hence $\rho_\Phi$ decreases less rapidly than $\rho_B$ until it
dominates it for values of the cosmic scale factor larger than
$a_Q$. Throughout this period (which must obviously include
nucleosynthesis), one has:
\begin{eqnarray}
\rho_\Phi &=& {2(2+\alpha) \over 4+ \alpha(1-w_B)} \left({3(1+w_B) \over
\alpha(2+\alpha) }\right)^{{\alpha\over 2}} \lambda {\mu^{4+\alpha} \over
M_P^\alpha} \left({a\over a_Q}\right)^{{6(1+w_B)\over 2+\alpha}} \nonumber
\\
\Phi &=& M_P \sqrt{{\alpha(2+\alpha) \over 3(1+w_B)}} \left({a \over
a_Q}\right)^{{3(1+w_B)\over 2+\alpha}}. 
\end{eqnarray}
The equation of state for the $\Phi$ field has \cite{ZWS}:
\begin{equation}
w_\Phi = -1 + {\alpha (1+ w_B)  \over 2+\alpha}. \label{eq:wphi}
\end{equation}
Thus, in a matter-dominated universe ($w_B=0$), $w_\Phi =
-1/2+2N_f/N_c$ which is  between $-1/2$ and $0$ for $N_f \le
N_c$. This provides a candidate for the dynamics of quintessence.

Once $\Phi/M_P$ has reached the value $\sqrt{{\alpha(2+\alpha)\over
3(1+w_B)}}$, we enter a different regime where $\rho_\Phi$
dominates the energy density. The field $\Phi$ slows down and one may
solve for ir neglecting the terms $\ddot{\Phi}$ in its equation of
motion and $\dot{\Phi}^2/2$ in $\rho_\Phi$. One obtains:
\begin{equation}
\Phi = \Phi_0 \left[ 1 + {1\over 2 \sqrt{3}} \alpha (4+\alpha) V(\Phi_0)
(t-t_0)\right]^{{2\over 4+\alpha}},
\end{equation}
where $\Phi_0$ is the present value for $\Phi$, and one obtains
\begin{equation}
w_\Phi \sim -1 + {\alpha^2 \over 3 \Phi^2}.
\end{equation}
If $\rho_\Phi$ at $a_Q$ is already close to the present value (this
occurs typically for $\mu \sim 10^{-12+30\alpha/(4+\alpha)}$ GeV), 
this second period is short ($a_Q \sim a_0$)
and $w_\phi$ will be given approximately by (\ref{eq:wphi}). For
simplicity, we will suppose from now on that this is so. In this case,
the value of $w_\Phi$ might prove to be too small to account for the
data \cite{SCP}.

However, larger values for $w_\Phi$ may be obtained by complicating
slightly the model and introducing other fields. As an example, we will
assume the presence of a dilaton field, much in the spirit of the
models of the previous section (although the dilaton is this time 
not sliding but
fixed at its ground state value). The dynamical scale $\Lambda$ is
expressed in terms of the dilaton through (\ref{eq:lambda})
with $b_0 = (3 N_c-N_f)/(16 \pi^2)$. This induces a new term in the
scalar potential:
\begin{equation}
\delta V= 4 s^2 |F_s|^2,
\end{equation}
with
\begin{equation}
F_s = {dW\over ds} = -8 \pi^2 {\Lambda^{{3N_c-N_f \over N_c-N_f}}\over
({\rm Det}\; \Pi)^{{1 \over N_c-N_f}}}.
\end{equation}
that is an extra term of the form $\mu^{4+\beta}/\Phi^\beta$ with
\begin{equation}
\beta = {4N_f \over N_c-N_f}.
\end{equation}
Since $\beta<\alpha$, this term dominates for large values of the
condensate $\Phi$ and, for $w_B=0$,
\begin{equation}
w_\Phi  = -1 +{2N_f \over N_c+N_f},
\end{equation}
which precisely lies between $-1$ and $0$: taking for example $N_c=5$
and $N_f=1$ yields $w_\Phi = -2/3$.

There could be other contributions to the $F$-term auxiliary field for
$S$, say $F_0$ (which will contribute to supersymmetry breaking). If so,
the leading term in $\delta V$ for large $\Phi$ is $F_s^\dagger F_0 +
F_s F_0^\dagger$ and $\beta=2N_c/(N_c-N_f)$, in which case $w_\Phi =
-1+ N_f/N_c$. This time, one may even obtain $w_\Phi= -2/3$ with $N_c=3$
($N_f=1$).

Strictly speaking, the leading term is $|F_0|^2$ and thus of the
cosmological constant type. But this is an artifact of global
supersymmetry and it is well-known that, by going to supergravity, we
may cancel this cosmological constant term, while keeping a
non-vanishing contribution $F_0$ to the $F$-term of the $S$ field.
Such a study goes beyond the framework of this paper. This stresses
however an important fact: even if we deal here with a dynamical
component ($\Phi$) which may account for a cosmological constant type
behaviour of the cosmological parameters, it is important that the
$\Phi$ energy density eventually dominates over all other forms and 
thus that these other components do not produce a significant 
cosmological constant of their own.
Thus, the cosmological constant remains a problem for all
other components.

Likewise, the amount of supersymmetry breaking due to the fact that
$\Phi$ has not reached an infinite value (and thus its $F$-term is not
vanishing) is not sufficient to account for the amount of
supersymmetry-breaking observed in nature. There must be other sources
({\em e.g.} $F_0$ in our example) which may produce unwanted amounts of
cosmological constant if care is not taken.

In other words, there is still a ``cosmological constant problem''  in
the models studied here (that is to say, from the point of view of the 
quantum theory) but the interest of such models lies in the fact that they can
successfully account for the recent cosmological data on supernovae of
type Ia, if confirmed. 

\vskip .8cm
{\bf Acknowledgments}
\vskip .5cm
I wish to thank Alex Vilenkin for raising my interest in the issue of
quintessence, the Berkeley Lab Theory group  for hospitality while
part of this work was done, and Reynald Pain for valuable discussions.

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\section{The model}

The antisymmetric tensor field $B_{\mu\nu}$ which appears among the
massless 
modes of the closed string plays a fundamental role in the cancellation
of 
the anomalies. Indeed Green and Schwarz \cite{GS} had to introduce a 
counterterm of the form $B \wedge F \wedge F \wedge F \wedge F$ in order 
to cancel the gauge anomalies of the 10-dimensional theory, $F$ being
the 
gauge field strength. Compactifying six dimensions to get to four
spacetime 
dimensions and replacing three of the field strength with compact
indices by 
their background values yields a term proportional to $B \wedge F$. 
It is well-known that in four spacetime dimensions an antisymmetric
tensor 
is dual to a pseudoscalar $a$: $\epsilon_{\mu\nu\lambda\tau}
\partial^{\nu} 
B^{\lambda \tau}    \sim \partial_\mu a$. The coupling 
is then simply $A^\mu \partial_\mu a$.

In this formulation, the pseudoscalar belongs to the same 
(chiral) supermultiplet 
as the dilaton field $s$ and they form a complex scalar field $S=s+ia$.
This 
superfield couples in a model independent way to the gauge fields
present in 
the theory; one has in particular:
\begin{equation}
{\cal L} = -{s \over 4 M_{_P}} \sum_a F^a_{\mu\nu} F^a_{\mu\nu}
           +{a \over 4 M_{_P}} \sum_a F^a_{\mu\nu} \widetilde
F^a_{\mu\nu},
\end{equation}
where $M_{_P}$ is the reduced Planck scale, the index $a$ runs over 
all gauge groups and
\begin{equation} \widetilde F^{a\mu\nu}\equiv {1\over 2}
\varepsilon^{\mu\nu\rho\sigma}
F^a_{\rho\sigma}.\label{deftilde}\end{equation} Thus $a$ has
axion-like couplings and is indeed called the string axion.  And the
vacuum expectation value of the dilaton $\langle s\rangle $ yields the
gauge coupling $1/g^2$.

An abelian symmetry with gauge field $A_\mu$ may seem to 
have (mixed) anomalies: under $A_\mu \rightarrow A_\mu + \partial_\mu
\alpha$
$$ \delta {\cal L} = -{1 \over 2} \delta_{_{GS}} \alpha \sum_a
F^a_{\mu\nu} 
\widetilde F^a_{\mu\nu}.$$
But this can be cancelled by an apropriate shift of the string axion
$a$.
Since there is a single model-independent axion, 
only one abelian symmetry, henceforth 
referred to as $U(1)_X$, may be pseudo-anomalous.

The kinetic term for the dilaton-axion fields is described at the 
supersymmetric level by the D-term of the K\"ahler function $K= -
\ln (S + \bar S)$. This may be modified to include the Green-Schwarz 
term $A^\mu \partial_\mu a$ \cite{U1}: 
$K= -\ln (S + \bar S - 4\delta_{_{GS}} V)$
where $V$ is a vector superfield describing the anomalous $U(1)_X$
vector supermultiplet. This D-term includes other terms, such as a mass
term 
for the $A_\mu$ gauge field. The constant $\delta_{_{GS}}$ may be
computed 
in the framework of the weakly coupled string and is found to be
\cite{U1}:
\begin{equation}
\delta_{_{GS}} = {1 \over 192 \pi^2} \sum_i X_i,
\end{equation}
where $X_i$ are the charges of the different fields under $U(1)_X$.

We are therefore led to consider the general class of models described
by the Lagrangian (restricted here to the bosonic fields)
\begin{eqnarray} {\cal L} &=& 
-(D_{\mu}\Phi_i )^\dagger (D^{\mu}\Phi_i)\nonumber\\
& & -{1\over 4g^2} \left( F_{\mu\nu} F^{\mu\nu} -{a\over
M_{_P}}F^{\mu\nu}
\widetilde F_{\mu\nu}\right)\nonumber \\
& & -\delta_{_{GS}}^2  M_{_P}^2 A_\mu A^\mu +
\delta_{_{GS}}  M_{_P} A^\mu \partial _\mu a \nonumber\\
& & -{1\over 4} \partial _\mu a \partial^\mu a -{1\over 4} m^2 
a^2-V(\Phi_i) 
,\label{lag}\end{eqnarray}
where we have set the dilaton field to its vacuum expectation value 
$\langle s\rangle = 1/g^2$, we have included a mass term for the axion,
without specifying its origin,
and we have introduced scalar fields  $\Phi_i$ carrying the 
integer charge $X_i$ under the $U(1)_X$ symmetry; the covariant
derivative is 
defined by 
\begin{equation}
D^{\mu}\Phi_i \equiv (\partial^\mu - iX_i A^{\mu})\Phi_i
,\label{defderiv}\end{equation}
and the potential $V(\Phi_i)$ by
\begin{equation}
V(\Phi_i) \equiv {g^2\over 2} (\Phi_i^\dagger X_i \Phi_i +\delta_{_{GS}}
M_{_P}^2)^2 ,\label{pot}
\end{equation}
The Green-Schwarz coefficient $\delta_{_{GS}}$ and the axion field $a$
have 
been rescaled by a factor $g^2$. 

The lagrangian ~(\ref{lag}) is invariant under the following local
gauge transformation with gauge parameter $\alpha(x^\mu)$
\begin {eqnarray} \Phi_i &\rightarrow& \Phi_i e^{iX_i\alpha} \nonumber
\\
A_{\mu} &\rightarrow& A_{\mu} + \partial_{\mu}\alpha \label{gauge}\\
a&\rightarrow&a+2M_{_P}\delta_{_{GS}}\alpha,\nonumber
 \end {eqnarray}
the transformation of the term $(a/ 4g^2 M_{_P})F^{\mu\nu}
\widetilde F_{\mu\nu}$ cancels the variation of the effective 
lagrangian due to the anomaly, namely $\delta {\cal L} = -(1/2g^2
)\delta_{_{GS}}\alpha F^{\mu\nu} \widetilde F_{\mu\nu}$ (assuming we
are also transforming the fermions of the theory not written
explicitely in ~(\ref{lag})).  Making a rigid gauge transformation
with parameter $\alpha = 2\pi$ without changing $a$ as a first step
but transforming the other fields (including the fermions), leads us
to interpret $a$ as a periodic field of period $4\pi
\delta_{_{GS}}M_{_P}$ through the redefinition $a
\rightarrow a-4 \pi \delta_{_{GS}}M_{_P}$ which leaves
the lagrangian invariant. It is also manifest in the following rewriting
of the kinetic term and of the axionic $\theta$-term in ${\cal L}$ where
it
is clear that $a$ behaves like a phase:

\begin {eqnarray}
{\cal L}_{kin,\theta}&=&-\frac{1}{4g^2}\left(F^{\mu\nu}F_{\mu\nu} -
\frac{a}{M_{_P}}F^{\mu\nu}\widetilde F_{\mu\nu}\right) \nonumber\\
& &  -\partial^{\mu}\phi_i\partial_{\mu}\phi_i
 -
\phi_i^2X_i^2\left(\frac{\partial^{\mu}\eta_i}{X_i}-A^{\mu}\right)^2\nonumber\\
& & - {M_{_P}}^2\delta_{_{GS}}^2\left(\frac{\partial^{\mu}a}
{2M_{_P}\delta_{_{GS}}}-A^{\mu}\right)^2
\end {eqnarray}
where we have set $\Phi_i\equiv\phi_{i}e^{i\eta_i}$ ($\phi_i$ being
 the modulus of $\Phi_i$).\\
 Let us now work out the Higgs mechanism in
 this context. We consider for the sake of simplicity a single scalar
field 
$\Phi$ of negative charge $X$ and we drop consequently the $i$ indices.
  ${\cal L}_{kin,\theta}$ can be rewritten

\begin{eqnarray}
\lefteqn{{\cal L}_{kin,\theta}=-\left[ M_{_P}^2\delta^2_{_{GS}} 
+ \phi^{2}X^{2}\right]\times}\nonumber
\\
& &\left[A^\mu-\frac
 {\frac{1}{2}M_{_P}\delta_{_{GS}}\partial^\mu a+\phi^2X\partial^\mu
\eta}{M_{_P}^2\delta_{_{GS}}^2+\phi^2X^2}\right]^2\nonumber\\
&& -\frac{\phi^2M_{_P}^2\delta^2_{_{GS}}X^2}{
M_{_P}^2\delta_{_{GS}}^2 + \phi^2X^2}\left[\frac
{\partial^{\mu}a}{2M_{_P}\delta_{_{GS}}}-\frac
{\partial^\mu\eta}{X}\right]^2 \nonumber\\
&& +\frac{\delta_{_{GS}}}{2g^2}\left(\frac{\phi^2X^2}{M_{_P}^2
\delta_{_{GS}}^2+\phi^2X^2}\left[\frac{a}{2M_{_P}\delta_{_{GS}}}
-\frac{\eta}{X}\right]\right.
\nonumber\\
&&\left.+ \frac {\frac{1}{2}M_{_P}\delta_{_{GS}}a+\phi^2X \eta}
{M_{_P}^2\delta_{_{GS}}^2+\phi^2X^2}\right) F_{\mu\nu}\widetilde
F^{\mu\nu}
-\frac{1}{4g^2}F_{\mu\nu}F^{\mu\nu} .
\end{eqnarray}

The linear combination appearing in this last equation
\begin{equation}
\frac{a}{2M_{_P}\delta_{_{GS}}}-\frac{\eta}{X}
\end{equation}
is the only  gauge invariant linear combination of $\eta$ and $a$ 
(up to a constant).
 The other one 
\begin{equation}
\ell \equiv \frac {\frac{1}{2}M_{_P}\delta_{_{GS}}a+\phi^2X \eta}
{M_{_P}^2\delta_{_{GS}}^2+\phi^2X^2}
\end{equation}
has the property of being linearly independent of the previous one
and of transforming under a gauge transformation ~(\ref{gauge}) as
$\ell \rightarrow \ell + \alpha$.  We now assume explicitely that
$\Phi$ takes its vacuum expectation value $\langle \Phi^\dagger \Phi
\rangle \equiv \rho^2$ in order to minimize the potential~(\ref{pot}):
\begin{equation}
\rho^2=-\delta_{_{GS}}M_{_P}^2/X.
\end{equation}
 We are left, among other fields, with a massive scalar Higgs field
 corresponding to the modulus of $\Phi$ of mass $m_{_X}$ given by
\begin{equation}
m_{_X}^2 = 2g^2\rho^2X^2=-2\delta_{_{GS}}Xg^2M_{_P}^2
\end{equation}
and we define
\begin{equation}
\hat a \equiv \left[\frac{a}{2M_{_P}\delta_{_{GS}}}-
\frac{\eta}{X}\right]\frac{\sqrt{2}\rho M_{_P}\delta_{_{GS}}X}
{\sqrt{M_{_P}^2\delta_{_{GS}}^2+\rho^2 X^2}}\label{hata}
\end{equation}
and 
\begin{eqnarray}
F_{a}^2 &= &\frac{1}{128\pi^4}\frac{M_{_P}^2 g^{4}}{\rho^2 X^2}
\left( M_{_P}^2 \delta^2_{_{GS}}+\rho^2X^2\right) \nonumber\\
 &=&\frac{1}{128\pi^4}M_{_P}^2 g^4
\left(1+\left(\frac{m_{_X}}{M_{_P}}\right)^2\frac{1}{2g^2X^2}\right)
\end{eqnarray}
so that with $\rho$ being set:
\begin{eqnarray}
\lefteqn{{\cal L}_{kin,\theta} = \nonumber} \\
&& -\left[ M_{_P}^2\delta_{_{GS}}^2
+\rho^2X^2\right]\left[A^{\mu} -\partial ^\mu \ell\right]^2
-\frac{1}{2}\partial^{\mu}\hat a \partial_{\mu} \hat a \nonumber \\
&&+\left[\frac{\hat
a}{32\pi^2F_{a}}+\frac{\delta_{_{GS}}}{2g^2}\ell\right]F_{\mu\nu}
\widetilde F^{\mu\nu}-\frac{1}{4g^2}F_{\mu\nu}F^{\mu\nu}
 \end{eqnarray}
we can now make a  gauge transformation to cancel $\partial ^\mu \ell $
by setting $\alpha = -\ell + \beta$ where $\beta$ is a constant
parameter.
This leaves us with 
\begin{eqnarray}
\lefteqn{{\cal L}_{kin,\theta}= -\frac{m^2_{_A}}{2g^2}A^{\mu}
A_{\mu}-\frac{1}{2}\partial^{\mu}\hat a \partial_{\mu} \hat a
}\nonumber\\ &&+\frac{\hat a}{32\pi^2F_{a}}F_{\mu\nu}
\widetilde F^{\mu\nu}-\frac{1}{4g^2}F_{\mu\nu}F^{\mu\nu} ,
\end{eqnarray} 
where $m_{_A}$ given by
\begin{eqnarray}
m_{_A}^2 &=& 2g^2\left[\rho^2X^2+M^2_{_P}\delta_{_{GS}}^2\right]\nonumber \\
&= &m_{_X}^2 \left[
1+\left(\frac{m_{_X}}{M_{_P}}\right)^2\frac{1}{2g^2X^2}
\right] \label{mA}
\end{eqnarray}
is the mass of the gauge field after the symmetry breaking.
The remaining symmetry
\begin{equation}
\hat a \rightarrow 
\hat a  + \frac {32\pi^2F_a}{2g^2}\delta_{_{GS}}\beta
\end{equation}
is the rigid Peccei-Quinn symmetry which compensates 
for the anomalous term arising from a
 rigid phase transformation of parameter $\beta$. 

To summarize we have seen that in the presence of the axion the gauge
boson of the pseudo-anomalous symmetry absorbs a linear combination
$\ell$ of the axion and of the phase of the Higgs field. We are left
with a rigid Peccei-Quinn symmetry, the remnant axion being the other
linear combination $\hat a$ of the original string axion and of the
phase of the Higgs field.

\section{Pseudo-anomalous U(1) strings}

Cosmic strings can be found as solutions of the field equations
derivable from Eq.~(\ref{lag}) provided the underlying U(1) symmetry
is indeed broken, which implies that at least one of the eigenvalues
$X_i$ is negative. This is the first case we shall consider here, so
we shall in this section assume again only one field $\Phi$ with charge
$X$, with $X<0$. Assuming a Nielsen-Olesen-like solution along
the $z-$axis~\cite{NO}, we set, in cylindrical coordinates,
\begin{equation} \Phi = \phi (r) \hbox{e}^{i\eta}, \ \ \ \ \eta =
n\theta, \end{equation}
for a string with winding number $n$. This yields the following
Euler-Lagrange equations
\begin{equation} \Box a = 2\delta_{_{GS}} M_{_P} \partial_\mu
A^\mu -{1\over 2g^2 M_{_P}}F_{\mu\nu}\widetilde F^{\mu\nu} + m^2 a,
\label{Boxa}\end{equation}
\begin{equation} \Box \phi = \phi (\partial _\mu\eta-XA_\mu)^2 
+g^2X\phi (X\phi^2 +\delta_{_{GS}} 
M_{_P}^2),\label{Boxphi}\end{equation}
\begin{equation} \partial_\mu [\phi^2 (\partial^\mu \eta - X A^\mu)]
= 0,\label{Boxeta}\end{equation}
\begin{eqnarray}{1\over g^2} \partial_\mu ({a\over M_{_P}}\widetilde 
F^{\mu\nu} -
F^{\mu\nu})&=&\delta_{_{GS}} M_{_P} \partial
^\nu a -2\delta_{_{GS}}^2 M_{_P}^2 A^\nu \nonumber\\ & &
+2X\phi^2 (\partial^\nu\eta-XA^\nu),\label{BoxA}\end{eqnarray}
from which the string properties can be derived.

Eq.~(\ref{BoxA}) can be greatly simplified: first we make use of
Eq.~(\ref{deftilde}), which implies $\partial_\mu \widetilde
F^{\mu\nu} =0$, and then we derive Eq.~(\ref{BoxA}) with respect to
$x^\nu$. This gives, upon using Eqs.~(\ref{Boxa}) and (\ref{Boxeta}),
\begin{equation} F_{\mu\nu}\widetilde F^{\mu\nu} = 2m^2 M_{_P} g^2
a,\label{FFa}\end{equation}
and subsequently
\begin{equation} {1\over g^2}\partial_\mu F^{\mu\nu} = {1\over M_{_P}}
\widetilde F^{\mu\nu}\partial_\mu a + {\cal J}^\nu +
J^\nu,\label{BoxA2} \end{equation}
where the currents are defined as
\begin{equation} J^\mu = -2 X\phi^2 (\partial^\mu\eta -
XA^\mu),\label{J}\end{equation}
and
\begin{equation}{\cal J}^\mu = -\delta_{_{GS}}  M_{_P}
(\partial ^\mu a-2\delta_{_{GS}} M_{_P}A^\mu).
\label{calJ}\end{equation}
Eqs.~(\ref{Boxa}) and (\ref{Boxeta}) then simply express those two
currents conservation $\partial \cdot J = \partial \cdot {\cal J} =0$,
when account is taken of Eq.~(\ref{FFa}).

The standard paradigm concerning the strings obtained in this simple
model states that the presence of the axion makes
the string global in the following sense: even for a
vanishing $a$, $A_\mu$ behaves asymptotically in such a way as to
compensate for the Higgs field energy density  (i.e.,
$A_\mu \to -\partial_\mu\eta/X$) and therefore yields an energy
per unit length
which diverges asymptotically. It should be clear however that
the behaviour of $a$ could  be different; indeed, it could
as well compensate for this divergence as we will now show. In
this case then, a divergence is still to be found, but this time
at a small distance near the string core, so that the total energy 
is localised in a finite region of space. 
This is in striking contrast with the case of a global
string where the divergent behavior arises because the energy is not
localized and a large distance cut-off must be introduced. 

In order to examine the behaviour of the fields and the required
asymptotics, we need the stress energy tensor
\begin{equation} T^\mu_\nu = -2g^{\mu\gamma} {\delta {\cal L}\over
\delta g^{\gamma\nu}}+\delta^\mu_\nu {\cal L},\end{equation}
which reads explicitely
\begin{eqnarray} T^{\mu\nu}&=&2[\partial^\mu\phi\partial^\nu\phi
-{1\over 2}g^{\mu\nu}(\partial \phi)^2]\nonumber \\ & & +{1\over g^2}
(F^{\rho\mu} F_\rho ^{\ \nu}-{1\over 4}g^{\mu\nu}F\cdot F)
%-{a\over 2M_{_P}}F^{\rho(\mu} \widetilde F_\rho^{\ \nu)}\nonumber
\\
& & -{1\over 2} g^2 g^{\mu\nu} (X\phi^2+
\delta_{_{GS}}M_{_P}^2 )^2\nonumber\\
& & +{1\over 2\delta_{_{GS}}^2 M_{_P}^2 } [{\cal J}^\mu {\cal
J}^\nu -{1\over 2} g^{\mu\nu} {\cal J}^2]\nonumber\\ & & +{1\over 2
X^2
\phi^2} [J^\mu J^\nu -{1\over 2} g^{\mu\nu} J^2]-{1\over 4} m^2 a^2
g_{\mu\nu}
\nonumber\\
\label{Tmunu}\end{eqnarray}
where account has been taken of the field equations. The energy per
unit length $U$ and tension $T$ will then be defined respectively as
\begin{equation} 
U=\int d\theta\, rdr T^{tt} \ \ \hbox{ and }\ \ 
T=-\int d\theta\,rdr T^{zz},\label{UT}
\end{equation}
The question as to whether the corresponding string solution is local
or global is then equivalent to asking whether these quantities are
asymptotically convergent ({\em i.e.} at large distances).

It can be seen on Eq.~(\ref{Tmunu}) that only the last two terms can
be a potential source of divergences. The Nielsen-Olesen~\cite{NO}
solution for the very last term consists in saying that $A_\mu$ is
pure gauge, namely $\lim_{r\to\infty} D_\mu \Phi = 0$, so that, as
already argued, $\lim_{r\to\infty} A_\mu = -\partial_\mu\eta/X$. With
this solution, setting $a=0$ implies that the second to last term in
Eq.~(\ref{Tmunu}) should diverge logarithmically for
$r\to\infty$. However, at this point, it should be remembered that $a$
can be interpreted as a periodic field of period $4\pi
\delta_{_{GS}}M_{_P}$ (as long as a cosine-like mass term is not
included as is usually the case at very low temperatures if this axion
is to solve the strong CP problem of QCD) and therefore can be
assigned a variation along $\eta$. In fact, setting
\begin{equation} a = {2 \delta_{_{GS}}M_{_P} \over X} \eta,\label{a}
\end{equation}
a perfectly legitimate choice, regularizes the integrals in
Eqs.~(\ref{UT}), at least in the $r\to\infty$ region.

The solution (\ref{a}) turns out, as can be explicitely checked using
Eqs.~(\ref{Boxa}) and (\ref{FFa}), to be the only possible non trivial
and
asymptotically converging solution. In particular, no dependence in the
string internal coordinates ($z$ and $t$ in our special case) can be
obtained. This means that the simple model used here cannot lead to
current-carrying cosmic strings~\cite{witten}~\cite{peter}. Moreover,
the stationnary solution (\ref{a}) shows the axion gradient to be
orthogonal to $\widetilde F^{\mu\nu}$, i.e., $\partial_\mu a
\widetilde F^{\mu\nu}=0$. Therefore, Eqs.~(\ref{Boxa}-\ref{BoxA2})
reduce to the usual Nielsen-Olesen set of equations~\cite{NO}, with
the axion coupling using the string solution as a source term. It is
therefore not surprising that the resulting string turns out to be
local.

The total energy per unit length (and tension) is however not finite
in this simple string model for it contains the term
\begin{equation} U = \hbox{f.p.} + 2 \pi \int {dr\over r}
({\delta_{_{GS}} M_{_P} n\over X} - \delta_{_{GS}} M_{_P}
A_\theta )^2,\end{equation} (f.p. denoting the finite part of the
integral) so that, since $A_\theta$ must vanish by symmetry in the
string core, one ends up with
\begin{equation} U = \hbox{f.p.} + 2 \pi  ({\delta_{_{GS}}
M_{_P} n\over  X} )^2 \ln ({R_A\over r_a}),\label{div}\end{equation}
where $R_A$ is the radius at which $A_\mu$ reaches its asymptotic
behaviour, i.e., roughly its Compton wavelength $m_{_{A}}$ given in
(\ref{mA}), while $r_a$ is defined as
the radius at which the effective field theory (\ref{lag}) ceases to
be valid, presumably of order $M_{_P}^{-1}$; the correction factor is
thence expected to be of order $|\ln (\delta_{_{GS}} g^2)|$, which is of
order unity for most theories. Hence, as claimed, the strings in this
model can be made local with a logarithmically divergent energy. The
regularization scale $r_a$ is however a short distance cut-off, 
solely dependent on the microscopic
structure and does not involve neither the interstring distance nor
its curvature radius. In particular, the gravitational properties of
the corresponding strings are those of a usual Kibble-Vilenkin
string~\cite{vilenkin}, given the equation of state is that of the
Goto-Nambu string $U=T=$const., and the light deflection is
independent of the impact parameter~\cite{defl}.

\section{Local string genesis}

Forming cosmic strings during a phase transition is a very complicated
problem involving thermal and quantum phase
fluctuations~\cite{zurek}. As it is far from being clear how will $a$
and $\eta$ fluctuations be correlated (even though they presumably
will), one can consider to begin with the possibility that a network
of two different kinds of strings will be formed right after the phase
transition, call them $a-$strings and $\eta-$strings, with the meaning
that an $a-$string is generated whenever the axion field winds
(ordinary axion string) while an $\eta-$string appears when the Higgs
field $\Phi$ winds. Both kinds of strings are initially global since
for both of them, only part of the covariant derivatives can be made
to vanish. However, if one were to have two such strings separated by
a distance $D$, they would exert on each other an attractive force,
the system energy being essentially $\ln m_{_A} D$, and that would
be minimized by setting $D\sim m_{_A}^{-1}$ so that one would end up
with only one string, satisfying the localisation
condition~(\ref{a}). We therefore expect the string network to
consist, after some time, in only local strings.

Another way to see this is to consider the stability of an $\eta-$string
with
$a=0$. The conservation of ${\cal J}$ implies, as one
can fix $\partial_\mu A^\mu=0$, that $\Box a =0$, whose general
time-dependent solution is $a = a(|{\bf r}|\pm t)$. Given the
cylindrical symmetry, this solution can be further separated into
$a=f(r-t) \theta$. This means that having a winding of $a$ that sets
up propagating away from the string is among the solutions. As this
configuration ultimately would minimize the total energy, provided
$\lim _{t\to\infty}f=-\delta_{_{GS}}M_{_P}/2X$, this means that the
original string is unstable and will evolve into the stationary solution
that we derived in the previous section.

Let us now consider an axionic string with no Higgs winding. The
situation here is actually simpler: as $A^\mu\not= 0$, the vacuum
solution $\Phi =0$ is no longer a solution, and here again the axionic
string field configuration is unstable. As a result of
Eq.~(\ref{Boxphi}), the Higgs field amplitude tends to vanish in the
string's core. At this point, it becomes, near the core, topologically
possible for its phase to start winding around the string, which it
will do since, again, this minimizes the total energy. As in the
previous case, such a winding will propagate away from the string.

It should be remarked at this point that these time evolution can in
fact only be accelerated when one takes into account the coupling
between $a$ and $\eta$: if any one of them is winding, then the other
one will exhibit a tendency to also wind, in order to locally minimize
the energy density. Indeed, it is not even really clear whether the
string configurations we started with  would even be present
at the string forming phase transition. What is clear, however, is
that after some time, all the string network would consist of local
strings having no long distance interactions. This means in particular
that the relevant scale, if no inflationary period is to occur after
the string formations, should not exceed the GUT scale in order to
avoid cosmological contradictions.

\section*{Conclusions}

Spontaneous breaking of a pseudo-anomalous U(1) gauge symmetry leads
to the formation of cosmic strings whose energy per unit length is
localized around their cores, contrary to what the presence of the
axion field in these theories might have suggested. This happens in
the simple case we've considered here, namely that of a single scalar
field aquiring a VEV at the symmetry breaking. In order to be general,
this result should be generalized to the case where more than one
field gets a VEV; this we now prove.

The potential we consider is that given by Eq.~(\ref{pot}) which, in
full generality, can be rewritten in the form
\begin{equation} V(\Phi_i) = {g^2\over 2} (\Phi ^\dagger {\bf X} \Phi
+ \delta_{_{GS}} M_{_{P}}^2 )^2,\label{pot-gen}\end{equation} where
${\bf X}$ is an $N\times N$ hermitian matrix, and $\Phi$ takes values
in an $N-$dimensional vector space ${\cal V}$. We denote by $p$ the
number of negative eigenvalues of ${\bf X}$ and $\phi$ the restriction
of $\Phi$ to that subspace ${\cal V}_p\in {\cal V}$ spanned by the
eigenvectors of ${\bf X}$ with negative eigenvalues. Once
diagonalized, ${\bf X}$ can be written as
\begin{equation} {\bf X} = \left( \matrix{ {\bf M} & {\bf 0}\cr
{\bf 0} & {\bf P}}\right),\end{equation}
with ${\bf M}$ and ${\bf P}$ containing respectively the negative and
positive eigenvalues.

Eq.~(\ref{pot-gen}) admits an accidental U($N$) symmetry, of which
the anomalous U(1) is part; this is not a simple U($N$)$\times$U(1)
symmetry as each field component transforms differently under U(1) as
indicated on Eq.~(\ref{gauge}). The vacuum configuration is now given
by
\begin{equation} \langle \phi^\dagger {\bf M} \phi \rangle = -
\delta_{_{GS}}
M_{_{P}}^2 ,\end{equation} so it would seem that the remnant symmetry
would be U($N-p$), i.e. a scheme U($N)\to$U($N-p$), and a
topologically trivial vacuum manifold~\cite{kibble}. Hence, one would
naively not expect cosmic string formation in such a model. This
conclusion is in fact not correct, as only part of the original
U($N$) is gauged, namely the anomalous U(1)subgroup, leading to a
N\oe{t}her current
\begin{equation} J^\mu \propto ig [\phi^\dagger {\bf M} \partial^\mu
\phi
- (\partial ^\mu\phi^\dagger) {\bf M} \phi ] + 2g^2 A^\mu \phi^\dagger
{\bf M} \phi,\end{equation} which can be made nonzero by imposing a
phase variation for $\phi$ as $\sim \exp (in\theta)$. Once set to a
nonzero value, this current will remain so for topological
reasons~\cite{semi}, being called a semi-local or embedded defect.
All the previous discussions concerning the simple $p=1$ model hold
also for these vortices, including their coupling to the axion field.

The cosmological evolution of the network of strings formed in these
theories also leads to serious constraints on the Green-Schwarz
coefficient provided no domain wall form connecting the strings;
otherwise, the network is known to rapidely (i.e. in less than a
Hubble time) decay into massive radiation and the usual constraint
relative to the axion mass would hold~\cite{kibble,wall-string}. If
however the string network is considered essentially stable, then its
impact on the microwave background limits the symmetry breaking scale
$\delta _{_{GS}} M_{_{P}}$ through the observational requirement that
the temperature fluctuations be not too large~\cite{dT-T}, i.e.  $$GU
\alt 10^{-6},$$ with $G$ the Newton constant $G\sim
M_{_{P}}^{-2}$. Therefore, the cosmological constraint reads 
\begin{equation}
\delta_{_{GS}}  \alt 10^{-2}, \label{delcons}
\end{equation}
a very restrictive constraint indeed. It should be remarked at this
point that the usual domain wall formation leading to a rapid
evaporation of the network does not semm valid in the case at hand. In
fact, the axion $\hat a$, defined through Eq.~(\ref{hata}), as the
solution (\ref{a}) set up, vanishes everywhere. Therefore, when the
Peccei-Quinn symmetry is broken, the axion itself does not have to
wind around the string. Hence, it has no particular reason for taking
values in all its allowed vacuum manifold so that no domain wall will
form. 

The strings that we have discussed here might appear in connection with
a scenario of inflation. Indeed, the potential (\ref{pot}) is used
for inflation in the scenario known as $D$-term inflation \cite{Dterm}:
inflation takes place in a direction neutral under $U(1)$ and the
corresponding vacuum energy is simply given by:
\begin{equation}
V_0 = {1 \over 2} g^2 \delta^2_{_{GS}} M_{_P}^4.
\end{equation}
The $U(1)$-breaking minimum is reached after inflation, which leads to
cosmic strings formation. Such an inflation era cannot therefore dilute the
density of cosmic strings and one must study a mixed scenario 
\cite{jeannerot}.  It is interesting to note that, under the assumption
that microwave background anisotropies are predominantly produced by
inflation, the experimental data puts a constraint \cite{LR,KM-R.ERR}  
on the scale
$\xi \equiv \delta_{_{GS}}^{1/2} M_{_P}$ which is stronger than
(\ref{delcons}). Several ways have been proposed \cite{KM-R,ERR}
in order to lower this scale. They would at the same time ease 
the constraint (\ref{delcons}).


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\end{document}


