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\title{Q Rings}
\author{M. Axenides$^a$, E. Floratos$^a$, S. Komineas$^b$, L. Perivolaropoulos$^a$,
}
\address{$^a$ Institute of Nuclear Physics, N.C.R.P.S. Demokritos, 153
10, Athens, Greece \\ $^b$ Physikalisches Institut, Universit\"at
Bayreuth, D-95440 Bayreuth, Germany.}
\date{\today}
\maketitle
\begin{abstract}
We show the existence of new stable ring-like localized scalar
field configurations whose stability is due to a combination of
topological and nontopological charges. In that sense these
defects may be called semitopological. These rings are Noether
charged and also carry Noether current (they are superconducting).
They are local minima of the energy in scalar field theories with
an unbroken U(1) global symmetry. We obtain numerical solutions of
the field configuration corresponding to large rings and derive
virial theorems demonstrating their stability. We also derive the
minimum energy field configurations in 3D and simulate the
evolution of a finite size Q ring on a three dimensional lattice
thus generalizing our demonstration of stability.
\end{abstract}
\narrowtext
Non-topological solitons (Q balls) have been studied
extensively in the literature in one, two and three
dimensions\cite{lp92}. They are localized time dependent field
configurations with a rotating internal phase and their stability
is due to the conservation of a Noether charge $Q$\cite{c85}. In
three dimensions, the only localized, stable configurations of
this type have been assumed to be of spherical symmetry hence the
name Q balls. The generalization of two dimensional (planar) Q
balls to three dimensional Q strings leads to loops which are
unstable due to tension. Closed strings of this type are naturally
produced during the collisions of spherical Q balls and have been
seen to be unstable towards collapse due to their
tension\cite{bs00,lpwww}.
There is a simple mechanism however that can stabilize these
closed loops. It is based on the introduction of an additional
phase on the scalar field that twists by $2\pi N$ as the length of
the loop is scanned. This phase introduces additional pressure
terms in the energy that can balance the tension and lead to a
stabilized configuration, the {\it Q ring}. This type of pressure
is analogous to the pressure of the superconducting string
loops\cite{Witten:1985eb} (also called `springs'\cite{hht88}). In
fact it will be shown that Q rings carry both Noether charge and
Noether current and in that sense they are also superconducting.
However they also differ in many ways from superconducting
strings. Q rings do not carry two topological invariants like
superconducting strings but only one: the winding $N$ of the phase
along the Q ring. Their stability is due to the conservation of
both the topological twist and the nontopological Noether charge.
Hence Q rings may be viewed as semitopological defects. In
contrast to Q balls, they are local minima of the Hamiltonian
separated from Q balls by a finite energy barrier. In what follows
we demonstrate the existence and metastability of Q rings in the
context of a simple model. We use the term 'metastability' instead
of `stability' because {\it finite size} fluctuations can lead to
violation of cylindrical symmetry and decay of a Q ring to one or
more Q ball as demonstrated by our numerical simulations.
Consider a complex scalar field whose dynamics is determined by
the Lagrangian \be \label{model} {\cal L}={1\over 2}
\partial_\mu \Phi^*
\partial^\mu \Phi - U(|\Phi |) \ee
The model has a global $U(1)$ symmetry and the associated
Noether current is \be \label{current} J_\mu = Im(\Phi^*
\partial_\mu \Phi) \ee with conserved Noether charge $ Q=\int
d^3 x \; J_0 $. Provided that the potential of (\ref{model})
satisfies certain conditions \cite{c85,lp92} the model accepts
stable Q ball solutions which are described by the ansatz $ \Phi =
f(r) e^{i \omega t}$. The energy density of this Q ball
configuration is localized and spherically symmetric. The
stability is due to the conserved charge $Q$.
In addition to the $Q$ ball there are other similar stable
configurations with cylindrical or planar symmetry but infinite,
not localized energy in three dimensions. For example an infinite
stable Q string that extends along the z axis is described by the
ansatz \be \label{stranz} \Phi = f(\rho) e^{i \omega t}\ee where
$\rho$ is the azimouthal radius. This configuration has also been
called `planar' or 'two dimensional' Q ball\cite{lp92}.
The energy of this configuration can be made finite and localized
in three dimensions by considering closed Q strings. These
configurations which have been shown to be produced during
spherical Q ball collisions\cite{bs00,lpwww} are unstable towards
collapse due to their tension. In order to stabilize them we need
a pressure term that will balance the effects of tension. This
term appears if we substitute the string ansatz (\ref{stranz}) by
the ansatz of the form \be \label{qringanz}\Phi = f(\rho) e^{i
\omega t} e^{i \alpha(z)} \ee where $\alpha(z)$ is a phase that
varies uniformly along the z axis. This phase introduces a new
non-zero $J_z$ component to the conserved current density
(\ref{current}). The corresponding current is of the form \be
\label{jzcons} I_z= \int d z \;{{d \alpha}\over {d z}} \;
2\pi\int d\rho \; \rho \; f^2 \ee Consider now closing the
infinite Q string ansatz (\ref{qringanz}) to a finite (but large)
loop of size $L$. The energy of this configuration may be
approximated by \bea E &=& {{Q^2}\over {4 \pi L \int d \rho \;
\rho \; f^2}} + \pi \; L \; \int d \rho \; \rho \; f'^2 \nn
\\ &+& {{(2\pi N)^2 \pi}\over L}\int d \rho \; \rho \; f^2 + 2 \pi L \int d
\rho \; \rho U(f)\nn \\ &\equiv & I_1 + I_2 + I_3 + I_4 \eea where
we have assumed $\alpha (z) = {{2 \pi N}\over L} z $ and the terms
$I_i$ are all positive. Also $Q$ is the charge conserved in $3D$
defined as
\be
Q=\omega 2\pi L\int d\rho \; \rho \;f^2 \ee The winding $2 \pi
N=\int d z \;{{d \alpha}\over {d z}}$ is topologically conserved
and therefore the current (\ref{jzcons}) is very similar to the
current of superconducting strings.
After a rescaling $ \rho \longrightarrow \sqrt{\lambda_1} \rho$, $
z\longrightarrow \lambda_2 z $ the rescaled energy may be written
as \be E={1 \over {\lambda_1 \lambda_2}} I_1 + \lambda_2 I_2
+{\lambda_1 \over \lambda_2} I_3 + \lambda_1 \lambda_2 I_4 \ee
Derrick's theorem\cite{d64} can be satisfied and collapse in any
direction can be evaded due to the time
dependence\cite{k97,akpf00}. We extremize E with respect to
$\lambda_1, \lambda_2$ and set $\lambda_1\!=\!\lambda_2=1$ to obtain the
following virial conditions \ba I_3 + I_4 &=& I_1 \label{virial1}
\\ I_2 + I_4 &=& I_1 + I_3 \label{virial2} \ea
In order to check the validity of these conditions numerically we
must first solve the ode which $f$ obeys. This is of the form
\be
f'' +{1\over \rho} f' + (\omega^2 -(2\pi N)^2/L^2) f -U'(f) = 0
\label{fode} \ee
%where we have used the rescaled potential of Ref. \cite{}.
with boundary conditions $ f(\infty)=0$ and ${df\over d\rho}(0)=
0$. Equation (\ref{fode}) is identical with the corresponding
equation for 2D Qballs\cite{akpf00} (see ansatz (\ref{stranz}))
with the replacement of $\omega^2$ by \be \omega^2 -{{(2\pi
N)^2}\over {L^2}} \equiv \omega'^2\ee Solutions of (\ref{fode})
for various $\omega'$ and $U(f) = {1 \over 2}f^2 - {1 \over 3} f^3
+ {B \over 4}f^4$ with $B=4/9$ were obtained in Ref.
\cite{akpf00}. Now it is easy to see that the first virial
condition (\ref{virial1}) may be written as
\be
\omega'^2 \int d\rho \; \rho \; f^2 = 2 \int d\rho \; \rho U(f)
\label{virnew} \ee This is exactly the virial theorem for 2D
Qballs\cite{akpf00} (infinite Q strings) with $N=0$ and field
ansatz given by (\ref{stranz}) with $\omega$ replaced by
$\omega'$. The validity of such a virial condition has been
verified in Ref. \cite{akpf00}. This therefore is an effective
verification of our first virial condition (\ref{virial1}).
The second virial condition (\ref{virial2}) can be written (using the first virial (\ref{virial1})) as
\be
2 I_3 = I_2
\ee
which implies that
\be
{{2\pi N^2} \over L^2} = {{\int d\rho \; \rho f'^2}\over {\int
d\rho \; \rho f^2}} \label{vir2} \ee This can be viewed as a
relation determining the value of $L$ required for balancing the
tension.
These virial conditions can be used to lead to a determination of
the energy as \be E=2(I_1+I_3) \ee In the thin wall limit where
$2 \pi \int d \rho \rho f^2 = A f_0^2$ ($A$ is the surface of a
cross section of the Q ring) this may be written as \be E \simeq
{Q^2 \over {2 L A f_0^2}} +{{(2\pi N)^2 A f_0^2}\over {2 L}}
\label{etwa} \ee and can be minimized with respect to $f_0^2$.
The value of $f_0$ that minimizes the energy in the thin wall
approximation is
\be
f_0=\sqrt{Q\over {2\pi N A}} \ee Substituting this value back on
the expression (\ref{etwa}) for the energy we obtain \be E={{2 \pi
N Q}\over L} \ee This is consistent with the corresponding
relation for spherical Q balls which in the thin wall
approximation lead to a linear increase of the energy with $Q$.
The above virial conditions demonstrate the persistance of the Q
ring configuration towards shrinking or expansion in the two
periodic directions of the Q ring torus for large radius. In order
to study Q rings of any size we must perform an energy
minimization in 3D and subsequently a numerical simulation of the
time evolution. It can be performed for any potential $U(\Phi)$ of
a polynomial or logarithmic form that admits Q balls. In what
follows we adopt the cubic potential \be \label{potential} U(\phi)
= {1 \over 2}|\Phi|^2 - {1 \over 3} |\Phi|^3 + {B \over 4}|\Phi|^4
\ee The ansatz that captures the above mentioned properties of the
Q ring is \be \label{eq:ansatz} \Phi = f(\rho,z)\; e^{i [\omega t
+ N\phi]} \ee where the center of the coordinate system now is in
the center of the torus that describes the Q ring and the ansatz
is valid for {\it any} radius of the Q ring.
The energy of this configuration is \ba E &=& {1 \over 2} {Q^2
\over \int f^2 dV}
+ {1 \over 2} \int \left[ \left({\partial f \over \partial \rho} \right)^2
+ {N^2 f^2 \over \rho^2} \right]\;dV \nn \\
&+& {1 \over 2} \int
\left[ \left({\partial f \over \partial z} \right)^2 \right]\;dV
+ \int U(f)\,dV \label{eq:energy2}
\ea The field equation for $\Phi$ is
\begin{equation}
\label{eq:equation} \ddot{\Phi} - \Delta \Phi + \Phi - |\Phi| \Phi
+ B |\Phi|^2 \Phi = 0
\end{equation} Substituting the ansatz (\ref{eq:ansatz}) we find that $f(\rho,z)$
should satisfy
\begin{equation}
\label{eq:ansatzequation}
{\partial^2 f \over \partial \rho^2}
+ {1 \over \rho}\,{\partial f \over \partial \rho}
- {N^2 f \over \rho^2} + {\partial^2 f \over \partial z^2}
+ (\omega^2-1) f + f^2 - B f^3 = 0
\end{equation}
In order to solve this equation we minimize the energy
(\ref{eq:energy2}) at fixed Q using a relaxation algorithm. In the
algorithm, we have used the initial ansatz:
\begin{equation}
f(\rho,z) \sim e^{(\rho-\rho_0)^2+z^2}
\end{equation} where $\rho_0$ is a fixed initial radius.
The energy minimization resulted to a non-trivial configuration
$f(\rho,z)$ for a given set of parameters $B, N, Q$ in the
expression for the energy. We then used
\begin{equation}
\label{omegdef}
\omega = {Q \over \int f^2\, dV}
\end{equation}
to calculate $\omega$ and constructed the full Q ring configuration
using (\ref{eq:ansatz}). As a further test to the stability of the
solution and the energy minimization algorithm we fed the
resulting field configuration as initial condition to a leapfrog
algorithm simulation the full field evolution in 3D by solving
equation (\ref{eq:equation}) in a $81^3$ lattice with reflective
boundary conditions where the second spatial derivative is set to
zero on the boundary. The relaxed configurations with B=4/9, n=1
and various Q's were evolved for about 50 internal Q ring periods
$T=2\pi /\omega$.
\begin{figure}[htbp]
\begin{center}
\psfig{file=fig1.ps,width=8truecm,bbllx=60bp,bblly=200bp,bburx=520bp,bbury=710bp}
\caption{Charge density contour plots of x-z plane cuts for evolved
3D profiles of relaxed Q rings ($N=1$). No significant change of
the configurations was observed during the full 3D evolution of
about $50$ internal rotation periods for any value of Q.}
\label{fig:1}
\end{center}
\end{figure}
\vspace{-20pt} The contour plots of the final frames of the
simulations on the $x-z$ plane are shown in Fig. 1. It was
verified that the Q ring configurations evolve with practically no
distortion and are metastable despite the long evolution.
Subsequent evolution of the configurations shown in Fig. 1
involved non-symmetric fluctuations emerging from the cubic
boundaries. These finite size fluctuations can cause local
vanishing of the scalar field and corresponding violation of the
topological charge $N$ conservation. Thus they were
found\cite{lpwww1} to lead to a break up and eventual decay of the
Q ring to one or more Q balls (depending on the the number of
zeros developed by the scalar field) after more than $100$
internal rotation periods. Thus a Q ring is a metastable as
opposed to a stable configuration.
We have repeated the same numerical experiment with $f(\rho,z)$
corresponding to a ring of large radius with $N=0$ in the ansatz
(\ref{eq:ansatz}) used as initial condition in the evolution
algorithm. The result of the evolution is shown in the frames of
Fig. 2 which shows a collapsing Q ring. It collapses due to the
lack of pressure support ($N = 0$) within less than $10$ internal
rotation periods in contrast to the metastable cases of Fig. 1.
The unstable collapsing ring produces a pair of Q balls that
propagate in opposite directions along the z-axis. This unstable
loop is similar to the one previously seen in Q ball simulations
of scattering in 3D \cite{bs00,lpwww}.
\begin{figure}[htbp]
\begin{center}
\psfig{file=fig2.ps,width=8truecm,bbllx=60bp,bblly=200bp,bburx=520bp,bbury=710bp}
\caption{The 3D evolution (x-z plane cut) of an unstable Q loop with winding
$N=0$ significantly less time than the simulations of Fig.1
(about 10 internal field rotation periods).
The ring rapidly collapses and forms a pair of Q balls propagating
along the z axis. Cylindrical symmetry is implied.}
\label{fig:2}
\end{center}
\end{figure}
\vspace{-20pt}
The metastable Q ring configuration we have discovered is the simplest
metastable ring-like defect known so far. Previous attempts to
construct metastable ring-like configurations were based on pure
topological arguments (Hopf maps) and required gauge fields to
evade Derrick's theorem due to their static
nature\cite{Faddeev:1997zj,Perivolaropoulos:2000gn}. Metastable
moving non-relativistic field configurations have also been
constructed\cite{pap}. These attempts resulted in complicated
models that were difficult to study analytically or even
numerically. Q rings require only a single complex scalar field
and they appear in all theories that admit stable Q balls
including the minimal supersymmetric standard model (MSSM). The
simplicity of the theory despite the non-trivial geometry of the
field configuration is due to the combination of topological with
non-topological charges that combine to secure metastability
without added field complications.
The derivation of metastability of this configuration opens up
several interesting issues that deserve detailed investigation.
Here we outline some of these issues. \begin{itemize} \item {\bf
Formation of Q Rings:} Q rings can form in principle by variations
of the Kibble mechanism, the Affleck-Dine mechanism\cite{kk99} or
by collision of Q balls with non-trivial relative phases. The
effectiveness of these mechanisms for Q ring formation needs
careful numerical and analytical study.
\item {\bf Current Quenching:} The increase of the winding $N$
which implies increase of the current along the Q ring can not be
arbitrary. In a way similar to superconducting string loops there
is a maximum current and winding $N_{crit}$ beyond which the
gradient of the phase becomes high enough to favor a zero value of
the field inside the ring. The investigation of the dependence of
$N_{crit}$ on Q and other parameters of the theory is a
interesting issue.
%\item {\bf Embedding in realistic models:} What
%are the properties of Q rings when gauge fields are introduced and
%they are embedded in realistic models?
\item {\bf Cylindrical
Walls:} A stabilizing phase can also be introduced in closed
cylindrical Q walls whose tension can thus be balanced by the
pressure of the winding phase. What are the properties of these
objects which are essentially a simpler version of the Q rings
discussed here?
\end{itemize}
These and other interesting issues emerge as a consequence of our
results and await further investigation.
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