Fractionally charged extended objects and superselection rules
Narendra Sahu
Topological objects resulting from symmetry breakdown may be either stable or metastable depending on the pattern of symmetry breaking. However, if they trap zero-energy modes of fermions, and in the process acquire non-integer fermionic charge, the metastable configurations also get stabilized. In the case of Dirac fermions the spectrum of the number operator shifts by $1/2$. In the case of majorana fermions it becomes useful to assign negative values of fermion number to a finite number of states occupying the zero-energy level, constituting a majorana pond. We determine the parities of these states and prove a superselection rule. Thus decay of objects with half-integer fermion number is not possible in isolation or by scattering with ordinary particles. The result has important bearing on cosmology as well as condensed matter physics.
hep-th0208027.tex
%\documentclass[11pt,twocolumn]{article}
\documentclass[english,aps,floats,twocolumn,showpacs,prl,amsfonts]{revtex4}
%\documentclass[aps,floats,showpacs,preprint,amsfonts]{revtex4}

\usepackage{pslatex}
\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}

\makeatletter
\usepackage{babel}
\makeatother

%\setlength\textwidth{5.75 in}
\setlength\topmargin{1mm}
%\setlength\topmargin{5.mm}
%\setlength\textheight{8.5 in}
%\setlength\textheight{8.25in}
%\addtolength\evensidemargin{-1.cm}
%\addtolength\oddsidemargin{-1.cm}

\def\sI{{\cal I}}
\def\sP{{\cal P}}
\def\sC{{\cal C}}
\def\ssA{\scriptstyle A}
\def\ssB{\scriptstyle B}
\def\ssC{\scriptstyle C}
\def\ssI{\scriptstyle I}
\def\ssM{\scriptstyle M}
\def\ssS{\scriptstyle S}
\def\dsI{{\cal I}^\dagger}
\def\dsP{{\cal P}^\dagger}
\def\dsC{{\cal C}^\dagger}
\newcommand{\rarrow}{\rightarrow}
%\def\state{#}{|#\rangle }
\newcommand{\state}[1]{|#1\rangle} 
\newcommand{\brastate}[1]{\langle #1 |}

\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}


\begin{document}

\title{Fractionally charged extended objects and superselection rules}
\author{Narendra Sahu}
\email{narendra@phy.iitb.ac.in} 
\author{Urjit A. Yajnik}
\email{yajnik@phy.iitb.ac.in}
\affiliation{Department of Physics, Indian Institute of Technology,
Mumbai 400076 INDIA}

%\maketitle

\bigskip
\bigskip
\begin{abstract}
Topological objects resulting from symmetry breakdown 
may be either stable or metastable 
depending on the pattern of symmetry breaking. 
However, if they trap zero-energy modes of fermions, 
and in the process acquire non-integer fermionic 
charge,  the metastable configurations also get stabilized. 
In the case of Dirac fermions the spectrum of the number
operator shifts by $1/2$. In the case of majorana fermions 
it becomes useful to assign negative values  of fermion
number to a finite number of states occupying the 
zero-energy level, constituting a \textit{majorana pond}. 
We determine the parities of these states
and prove a superselection rule. Thus decay of objects
with half-integer fermion number is not possible  
in isolation or by scattering with ordinary particles.
The result has important bearing on cosmology as well as
condensed matter physics.
\end{abstract}

\pacs{11.27.+d, 11.30.Er, 11.30.Fs, 98.80.Cq}

\maketitle

\section{Introduction}
Solitons present the possibility of extended objects 
as stable states within Quantum Field Theory. Although
these solutions are obtained from semi-classical arguments
in weak coupling limit, their validity as quantal states 
is justified based on the associated topological conservation 
laws. 
A more  curious occurrence is that of fermionic zero-energy modes
trapped on such solutions. Their presence requires,
according to well known arguments\cite{JandR}\cite{Jrev}, an
assignment of half-integer fermion number to the
solitonic states. In the usual treatment, the back reaction 
of the fermion zero-modes on the soliton itself is ignored. 
However, the fractional values of the fermionic charge have 
interesting consequence for the fate of the soliton if 
the latter is not strictly stable.
The reason for this is that if the configuration
were to relax to trivial vacuum in isolation, there is 
no particle-like state available for carrying the fractional 
value of the fermionic charge. Dynamical stability of such objects
was pointed out in \cite{devega}, in cosmological context in \cite{stern},
\cite{steandyaj} and more recently
in \cite{Davis:1997bu}\cite{Davis:1999wq}\cite{DavKibetal}. 
Fractional fermion number phenomenon also occurs in condensed
matter systems and its wide ranging implications call for
a systematic understanding of the phenomenon.

The impossibility of connecting half-integer valued states 
to integer valued states suggests that a superselection 
rule\cite{www} \cite{Sweinberg} is operative. 
In a theory with a conserved charge (global or local), a 
superselection rule operates among sectors of distinct charge 
values because the conservation of charge is associated 
with the inobservability of rescaling operation $\Psi\rightarrow 
e^{iQ}\Psi$. 
For the case of the Dirac fermions, the gauge symmetry is broken, 
however the overall phase
corresponding to the fermion number continues to be a symmetry
of the effective theory. This permits independent rescaling
of the sectors with different values of the fermionic number;
thus superselecting bosonic from fermionic sectors. In the case
at hand, half-integer values occur, preventing such states from 
decaying in isolation to the trivial ground state \cite{devega}
\cite{stern}. 

For the case of majorana fermions where the number operator
is not conserved by interactions the validity of such results
is far from obvious. However, the occurrence of half-integral
values can be shown  to be intimately connected to the charge 
conjugation invariance\cite{sudyaj} of the theory. Here we 
show that in such a case it is possible to assign parities to
the lowest energy solitonic sector which are compatible
with parity assignment in the vacuum sector. A superselection
rule can then be proved for majorana fermions.

%Indeed, the example of \cite{GolWil} shows that arbitrary real
%values of trapped fermion number are possible in a
%theory that does not respect charge conjugation invariance.

In the following we begin with constructing explicit examples 
of cosmic strings which are metastable and have odd number of 
zero-modes, for both the cases of Dirac and  Majorana masses. 
The masses in each case are derived from spontaneous
symmetry breaking. We then discuss assignment of fermion number
to topological objects, including the need for a finite spectrum
of negative values for states of trapped majorana fermions.
%This helps in determining parities of the non-interacting states. 
Subsequently we derive the required superselection rule. The 
paper ends with summary and conclusion.

\section{Examples}\label{sec:toyexa}
Here we construct two examples in which the topological objects of 
a low energy theory are metastable due to the embedding of the 
low energy  symmetry group in a larger symmetry group at higher 
energy. Examples of this kind were considered in \cite{presvil}.
Borrowing the strategies for bosonic sector from there, we include 
appropriate fermionic content to ensure the zero-modes.

\subsection*{\textbf{A} Dirac fermions}
 
Consider first the theory of two scalars $\vec \Sigma$ 
and $\sigma$ respectively real triplet and complex 
doublet under a local $SU(2)$. 
The Lagrangian is taken to be 
%\cite{presvil}
\begin{eqnarray}
{\cal L} = &-&\frac{1}{4} F^{\mu\nu a}F^a_{\mu\nu}
+ \frac{1}{2} D_\mu {\vec \Sigma} \cdot D^\mu \vec\Sigma 
+ D_\mu\sigma^\dag D^\mu\sigma   \nonumber \\
&-&\lambda_1(\vec\Sigma\cdot\vec\Sigma - \eta_1^2)^2
-\lambda_2(\sigma^\dag\sigma - \eta_2^2)^2  \nonumber \\
&+&\lambda_{12}\eta_1 \vec\Sigma\cdot\sigma^\dag{\vec \tau}\sigma
\end{eqnarray}
where isovector notation is used and $\tau$ are the Pauli 
matrices. It is assumed that $\eta_1\gg\eta_2$ and that the 
coupling $\lambda_{12}>0$ satisfies $\lambda_{12}\eta_1^2\ll$
$\lambda_2\eta_2^2$

The vacuum expectation value (VEV) 
$\vec\Sigma = (0\  0\ \eta_1)^T$ 
%\qquad  \sigma = \pmatrix{0\cr \eta_2} \ee
breaks the $SU(2)$ to the $U(1)$ generated by $\exp(i\tau^3
\alpha/2)$. The effective theory of the $\sigma$ can be
rewritten as the theory of two complex scalar $\sigma_u$
and $\sigma_d$ for the up and and the down components respectively.
The potential of the effective theory favors the minimum
\be
\sigma_u=\eta_2, \qquad \sigma_d=0
\ee 
In the effective theory $\sigma_u$ enjoys a $U(1)$ invariance 
$\sigma_u \rarrow e^{i\alpha}\sigma_u$ which is broken by the
above VEV to $\mathbb{Z}\equiv\{e^{2n\pi i}\}$, $n=1,2,\ldots$. 
This makes possible vortex solutions
with an ansatz in the lowest winding number sector
\be
\sigma_u(r, \theta) = \eta_2 f(r) e^{i\theta}
%; \quad \Sigma_3 = \eta_1
\ee
The vortex configuration is a local minimum, however it
can decay by spontaneous formation of a monopole-antimonopole
pair \cite{presvil}. These monopoles are permitted by the 
first breaking $SO(3) \rarrow SO(2)$ in the $\Sigma$ sector.
Paraphrasing  the discussion of \cite{presvil}, we have 
$SU(2)\rarrow$$U(1)\rarrow \mathbb{Z}$. 
The vortices are stable in the low energy theory because 
$\pi_1(U(1)/\mathbb{Z})$ is nontrivial. But in the $SU(2)$,
the $\mathbb{Z}$ lifts to $\{e^{4n\pi i\tau^3/2}\}=I$ 
making it possible to unwind the vortex by crossing an energy 
barrier.

Consider now the introduction of a doublet of fermion species
$\psi_L\equiv (N_L, E_L)$ assumed to be left handed and
a singlet right handed species $N_R$. The Yukawa coupling of these
to the $\sigma_u$ is given by $h{\overline{N_R}}\sigma^\dag\psi_L$,
which in the vortex sector reads
\be
{\cal L}_{\sigma-\psi} \sim h \eta_2 f(r)(e^{i\theta}
{\overline{N_R}} N_L + h.c. )
\ee
The lowest energy bound states resulting from this coupling
are characterized by a topological index, \cite{Eweinberg}
\( \mathcal{I} \equiv n_L - n_R\) where 
$n_L$ and $n_R$ are the zero modes of the left handed and 
the right handed fermions respectively. This index can be computed
using the formula \cite{Eweinberg}\cite{GanLaz}
\be
\mathcal{I} = \frac{1}{2\pi i}(\ln \textrm{det}M)\vert^{2\pi}_{\theta=0}
\label{eq:index}
\ee
where  $M$ is the position 
dependent effective mass matrix for the fermions. 
In the present case this gives rise to a single zero-energy mode 
for the  fermions of species $N$. According to well known reasoning 
\cite{JandR} to be recapitulated 
%in sec. \ref{sec:ferno} 
below, this  requires the assignment of either of the values 
$\pm1/2$ to the fermion number of this configuration. We shall show 
below that this induces stability of the  
vortex against spontaneous  decay in isolation. 

\subsection*{\textbf{B} Majorana fermions}
The example above can be extended to the case where the $N$ is a
majorana fermion. Being a singlet $N$ admits a mass term
$M_{\ssM} {\overline{N_R^{\ssC}}}N_R$, $M_{\ssM}$ signifying majorana mass. 
This could also be a spontaneously 
generated mass due to the presence of a
neutral scalar $\chi$ with coupling terms 
$h_{\ssM}{\overline{N_R^{\ssC}}}N_R\chi + h.c.$. 
If this $\chi$ acquires a VEV at energies higher than the 
$\Sigma$, the $N$ particles possess a majorana mass. 
Thus fermions that are asymptotically seen as majorana
particles can possess odd number of zero mode on the vortex.

Finally we present the case where majorana mass is spontaneously
generated at the same scale at which the vortex forms. Consider
an $SU(3)$ theory (global or local) which is broken to identity
by two scalars, $\Phi$ an octet acquiring a VEV $\eta_1\lambda_3$ 
($\lambda_3$ here being the third Gell-Mann matrix) and 
$\phi$, a ${\bar 3}$, acquiring the VEV $\langle\phi^k\rangle=\eta_2
\delta^{k 2}$, with $\eta_2\ll \eta_1$. Thus
\be
SU(3){\begin{array}{cc} 
8 \\ \vspace{3pt}
\longrightarrow \end{array} }
U(1) {\begin{array}{cc} \overline{3}\\ \vspace{3pt}  
{\longrightarrow} \end{array}} I 
\ee
It can be checked that this pattern of VEVs can be generically
obtained from the quartic scalar potential of the above Higgses. 
The first phase of breaking gives rise to monopoles, while the 
second gives rise  to cosmic strings. The strings however are 
unstable because the vacuum manifold of the full theory is simply 
connected.

Now add a multiplet of left-handed fermions belonging to $\overline{15}$. 
Its mass terms arise from the following coupling to the $\overline{3}$
\be
{\cal L}_{\textrm{Majorana}} = h_M \overline{\psi^{\ssC}}^{\{ij\}}_{k}
\psi^{\{lm\}}_{n}\phi^{r}
(\epsilon_{ilr}\delta^{n}_{j}\delta^{k}_{m})
\ee
The indices symmetric under exchange have been indicated by curly brackets. 
No mass terms result from the $8$ because it cannot provide a singlet
from tensor product with $\overline{15}\otimes \overline{15}$ 
\cite{Slansky:yr}. 
After substituting the $\phi$ VEV a systematic 
enumeration shows that all but the two components $\psi^{\{22\}}_{1}$ 
and $\psi^{\{22\}}_{3}$ acquire
majorana masses at the second stage of the breaking. Specifically 
we find the majorana mass matrix to be indeed rank $13$. Thus, using 
either of  the results \cite{JandRossi} or \cite{GanLaz} i.e.,\  eq. 
(\ref{eq:index})
we can see that there will be $13$   zero modes present in the lowest 
winding sector of the cosmic string.

\section{Assignment of fermion number}\label{sec:ferno}
We now recapitulate the reasoning behind the assignment of
fractional fermion number. We focus on the Majorana fermion
case, which is more nettlesome, while the treatment of the
Dirac case is standard \cite{JandR}\cite{Jrev}. In the prime example 
in $3+1$ dimensions 
of a single left-handed fermion species coupled to an abelian
Higgs model according to 
\be
\mathcal{L}_\psi = i\overline{\psi}\gamma^\mu D_\mu \psi
-( h \phi \overline{\psi^C}\psi + h.c.)
\ee
the following result has been obtained\cite{JandRossi}.
For a vortex oriented along 
the $z$-axis, and in the winding number sector $n$, 
the fermion zero-modes are of the form
\be
\psi_0({\bf x})=\pmatrix{1\cr 0}\left[U(r)e^{il\theta}
+V^*(r)e^{i(n-1-l)\theta}\right]g_l(z+t)
\ee
In the presence of the vortex, $\tau^3$ (here representing
Lorentz transformations on spinors) acts as the
matrix which exchanges solutions of positive frequency
with those of negative frequency. It is therefore identified
as the ``particle conjugation'' operator.
In the above ansatz, the $\psi$ in the zero-frequency 
sector are charge self-conjugates, $\tau^3\psi=\psi$,
and have an associated left moving zero mode along the
vortex. The functions satisfying $\tau^3\psi=-\psi$
are not normalizable. The situation is reversed 
when the winding sense of the scalar field is reversed, ie, for
$\sigma_u\sim$$e^{-in\theta}$. In the winding number sector $n$, 
regular normalizable solutions\cite{JandRossi} exist for
for $0\leq l\leq n-1$. The lowest energy sector of the vortex is 
now \(2^n\)-fold degenerate, and each zero-energy mode needs to be 
interpreted as contributing a value $\pm 1/2$ to the total 
fermion number of the individual states\cite{JandR}. This conclusion
is difficult to circumvent if the particle spectrum is to reflect
the charge conjugation symmetry of the theory \cite{sudyaj}. 
The lowest possible value of the induced number in this sector
is $-n/2$. Any general state of the system is built from one
of these states by additional integer number of fermions.
All the states in the system therefore possess half-integral
values for the fermion number if $n$ is odd. 

One puzzle immediately arises, what is the meaning of
negative values for the fermion number operator for
\textit{Majorana} fermions? In the trivial vacuum,
particles are identified with anti-particles according to
the Majorana condition
\be
\sC \psi \dsC\  =\  \pm \psi 
\label{majcond}
\ee
making negative values for the number meaningless.
Here $\sC$ is the charge conjugation operator and the choice 
between $+$ and $-$ on the right hand side has to be determined 
from experiment. We shall first verify that in the zero-mode 
sector we must indeed assign negative values to the number 
operator. It is sufficient to treat 
the case of a single zero-mode, which generalizes easily
to any larger number of zero-modes.
The number operator possesses the properties
 \be
[ N, \psi ]\ =\ -\psi\qquad{\rm and} 
\qquad [ N, \psi^\dagger ]\ =\ \psi^\dagger
\label{psiN}
\ee
\be
\sC N \dsC\ =\ N
\label{ccN}
\ee
Had it been the Dirac case, there should be a minus sign 
on the right hand side of eq. (\ref{ccN}). This is absent due 
to the Majorana condition. The fermion field operator is now 
expanded as
\be
\psi\ =\ c\psi_0 + \left\{ \sum_{{\bf \kappa},s} 
a_{{\bf \kappa},s}\chi_{{\bf \kappa},s}(x)\ \\
+\ \sum_{{\bf k},s} b_{{\bf k},s}u_{{\bf k},s}(x)\ 
+\ h.c. \right\}
\label{eq:expansion}
\ee
where the first summation is over all the possible bound states of 
non-zero frequency with real space-dependence of the form $\sim
e^{-{\bf \kappa\cdot x_\perp}}$ in the transverse space directions
${\bf x}_\perp$, and the second summation is over all 
unbound states, which are asymptotically plane waves. Note that
hermitian conjugates of the operators $a_{{\bf \kappa},s}$ and 
$b_{{\bf k},s}$ are  included above in the ``$h.c.$'' but not 
that of the solitary mode operator $c$. Then
the Majorana condition (\ref{majcond}) requires that we demand
\be
\sC\ c\ \dsC\ =\ \pm c \qquad{\rm and} 
\qquad \sC\  c^\dagger\  \dsC\ =\ \pm c^\dagger
\label{ccc}
\ee
Unlike the Dirac case, the $c$ and $c^\dagger$ are not
exchanged under charge conjugation. 
%and consistency with  standard anti-commutation relations 
%for the $c$ and $c^\dagger$ is retained.
The only non-trivial irreducible realization of this
algebra is to require the existence of a doubly degenerate
ground state with states $\state{-}$ and $\state{+}$ satisfying
\be
c\state{-}\ =\ \state{+}\qquad {\rm and}\qquad 
c^\dagger\state{+}\ =\ \state{-}
\label{cstates}
\ee
with the simplest choice of phases. Now we find
\begin{eqnarray}
\sC\  c\  \dsC \sC \state{-}\ &=\ \sC\state{+}\\ \vspace{3mm}
\Rightarrow \hspace{2mm} \pm c (\sC \state{-})\ &=\ (\sC\state{+})
\label{Ctransform}
\end{eqnarray}
This relation has the simplest non-trivial solution
\be
\sC\state{-}\ =\ \eta^-_{\ssC} \state{-}\qquad
{\rm and}\qquad \sC\state{+}\ =\ \eta^+_{\ssC} \state{+}
\label{Cproperty}
\ee
where, for the consistency of  (\ref{cstates}) and (\ref{Ctransform})
$\eta^-_{\ssC}$ and $\eta^+_{\ssC}$ must satisfy
\be
\pm (\eta^-_{\ssC})^{-1}\eta^+_{\ssC}\ =\ 1
\ee
Finally we verify that we indeed get values $\pm 1/2$ for $N$.
The standard fermion number operator
\be
N_F = \frac{1}{2}[ \psi^\dag \psi - \psi \psi^\dag ]
\ee
%\be
%N_{\mathrm{vac}}\ =\ {1\over 2}(\psi^\dagger \psi\ -\ \psi\psi^\dagger) 
%\label{Noperator} 
%\ee
acting on these two states gives,
\be
{1\over 2}(c\ c^\dagger\  -\  c^\dagger\  c)\ \state{\pm}\ =\ 
\pm{1 \over 2}\state{\pm}
\ee
The number operator indeed lifts the degeneracy of the
two states. For $s$ number of zero modes, the ground
state becomes $2^s$-fold degenerate, and the fermion number
takes values in integer steps ranging from $-s/2$ to $+s/2$.
For $s$ odd the values are therefore half-integral.
%As we shall see below, additionally, separate
%parities must be assigned to the two possible states,
%thus identifying physically observable states unambiguously.
Although uncanny, these conclusions accord with some known
facts. They can be understood as spontaneous symmetry breaking 
for fermions\cite{mcwilczek}. The negative values of the number thus
implied occur only in the zero-energy sector and do not 
continue indefinitely to $-\infty$. Instead of an unfathomable 
\textit{Dirac sea} we have a small \textit{Majorana pond} 
at the threshold.

\section{Induced stability}\label{sec:indsta}

As a step towards deriving our superselection rule,
we determine the parities of the zero-energy states. 
Majorana fermions can be assigned a unique parity \cite{Sweinberg}, 
either of the values $\pm i$.
Let us choose $i$ to be the parity of the free single fermion
states in the trivial vacuum. The fermion spectrum should
look the same as trivial vacuum far away from the vortex 
\footnote{This should be understood to be the clustering 
property of local field theory being obeyed by the vortex
ground state.}. In turn the parities of the latter states should
be taken to be the same as those of the trivial ground state.
Next, any of these asymptotic free fermions is capable of
being absorbed by the vortex (see for instance \cite{Davis:1999ec}). 
In the zero energy sector this absorption would cause a 
transition from \( \state{-1/2} \) to \( \state{1/2} \)
and cause a change in parity by $i$.
Thus the level carrying fermion number \(+1/2\) should be
assigned a parity $e^{i\pi/2}$ relative to the \(-1/2\) state. 
Symmetry between the two states suggests that we assign parity 
$e^{i\pi/4}$ to the $N_F=1/2$ and $e^{-i\pi/4}$ to the $N_F=-1/2$ 
states. 

We now show the inappropriateness of superposing states of
half-integer valued fermion number and integer valued fermion
number \cite{www}. The operation $\sP^{\displaystyle4}$, parity 
transformation performed four times  must return the system 
to the original  state, upto a phase. Consider forming the 
state $\Psi_{\ssS}=$$\frac{1}{\sqrt{2}}(\state{1/2}+\state{1})$ 
from states of half-integer and integer value for the fermion number. 
But 
\be
\sP^4\ \Psi_{\ssS}\ =\ {1\over \sqrt{2}}(-\state{1/2}+\state{1})\ 
%\equiv\ \psi_{\ssA}
\ee
Thus this operation identifies a state with another orthogonal
to it. Similarly, application of $\sP^{\displaystyle2}$ which 
should also leave the physical content of a state unchanged
results in yet another linearly independent state, 
$\frac{1}{\sqrt{2}}(i\state{1/2}-\state{1})$.
Thus the space of superposed states collapses to a trivial
vector space. The conclusion therefore is that it is not
possible to superpose such sectors. In turn there can be
no meaningful operator  possessing non-trivial matrix
elements between the two spaces. This completes our proof
of the theorem.

\section{Conclusion}

Metastable classical lumps, also referred to as embedded defects
can be found in several theories.
The conditions on the geometry of the vacuum manifold
that give rise to such defects were spelt out in \cite{presvil}.
We have studied the related question of fermion zero-energy 
modes on such objects. It is possible to construct examples
of cosmic strings in which the presence of zero-modes signals 
a fractional fermion number both for Dirac
and Majorana masses. 
It then follows that such a cosmic string cannot decay in
isolation because it belongs to a distinct superselected 
Quantum Mechanical sector. Thus a potentially metastable object 
can enjoy induced stability due to its bound state with fermions.

Although decay is not permitted in isolation, it certainly
becomes possible when more than one such objects come
together in appropriate numbers. In the early Universe such 
objects could have formed depending on the unifying group 
and its breaking pattern. Their disappearance would be
slow because it can only proceed through encounters between
objects with complementary fermion numbers adding up to an 
integer. Another mode of decay is permitted by change in
the ground state in the course of a phase transition.
When additional Higgs fields acquire a vacuum expectation 
value, in turn altering the boundary conditions for the
Dirac or Majorana equation, the number of induced zero
modes may change from being odd to even\footnote{Such a possibility,
though for topologically stable string can be found for realistic
unification models in \cite{steandyaj} \cite{Davis:1997bu} 
\cite{Davis:1999ec} \cite{widyajetal}.}, thus imparting
the loops an integer fermion number. The decay can then proceed
at the rates calculated in \cite{presvil}. 

In condensed matter systems the best known example is of
charge carriers of charge half the electronic charge in
polyacetylene \cite{Jackiw:wc} where the underlying
solitons have been identified as stable kinks. It would be 
interesting to find intrinsically metastable objects stabilized
by this mechanism.

Finally it is interesting to conjecture about the fate of 
loops of cosmic strings. The zero modes on closed string loops 
have not been studied in detail, though dynamical stability due 
to superconductivity has been identified. If loops carry 
fractional fermion number and get stabilized by this mechanism 
that also would present very interesting possibilities for Cosmology.


\begin{acknowledgments}
We thank thank Sumit Das and P. Ramadevi 
for their for critical comments on the manuscript.
\end{acknowledgments}

%\bigskip
%\hrule


%\section*{References}
\begin{thebibliography}{99}
\bibitem{JandR} R. Jackiw and C. Rebbi, Phys Rev. {\bf D13}, 3398 (1976)

\bibitem{Jrev} R. Jackiw, Rev. Mod. Phys. 49, 681(1977) 

%\bibitem{GolWil} J. Goldstone and F. Wilczek, Phys. Rev. Lett.
%{\bf 47}, 986 (1981)

\bibitem{devega} H. de Vega, Phys. Rev. {\bf D18}, 2932 (1978)

\bibitem{stern} A. Stern; Phys. Rev. Lett.{\bf 52}, 2118(1983)

\bibitem{steandyaj} A. Stern and U. A. Yajnik, Nucl. Phys. {\bf B267}, 158 
(1986)

\bibitem{Davis:1997bu}
S.~C.~Davis, A.~C.~Davis and W.~B.~Perkins,
%``Cosmic string zero modes and multiple phase transitions,''
Phys.\ Lett.\ B {\bf 408}, 81 (1997) 

\bibitem{Davis:1999wq}
A.~C.~Davis, S.~C.~Davis and W.~B.~Perkins, \textsl{Proceedings of
COSMO99}, 
%``The fate of cosmic string zero modes,''
arXiv:hep-ph/0005091

\bibitem{DavKibetal}
A. C. Davis, T. W. B. Kibble, M. Pickles
and D. A. Steer, Phys. Rev. {\bf D62} 083516 (2000)

\bibitem{www} G. C. Wick, A. S. Wightman and E. P. Wigner, Phys. Rev. 
{\bf 88}, 101  (1952)

\bibitem{Sweinberg} S. Weinberg, {\it The Quantum Theory of fields}, Vol. I, 
Cambridge University Press, 1996

\bibitem{sudyaj} E. C. G. Sudarshan and U. A. Yajnik, Phys. Rev. {\bf D33}, 
1830 (1986)

\bibitem{Eweinberg} E. J. Weinberg, Phys. Rev. {\bf D24}, 2669 (1981)

\bibitem{GanLaz} N. Ganoulis and G. Lazarides, Phys. Rev.
{\bf D38} 547 (1988)

%

\bibitem{presvil} J. Preskill and A. Vilenkin, Phys. Rev. {\bf D47}, 
2324 (1993)

%\cite{Shankar:1976un}
%\bibitem{Shankar:1976un}
%R.~Shankar,
%``The SO(3) Monopole Catalog,''
%Phys.\ Rev.\ D {\bf 14}, 1107 (1976).

\bibitem{Slansky:yr}
R.~Slansky,
%``Group Theory For Unified Model Building,''
Phys.\ Rept.\  {\bf 79}, 1 (1981).

\bibitem{JandRossi} R. Jackiw and P. Rossi, Nucl. Phys.{\bf B190}, 681 (1981)

\bibitem{mcwilczek}
R.~MacKenzie and F.~Wilczek,
%``Illustrations Of Vacuum Polarization By Solitons,''
Phys.\ Rev.\ D {\bf 30}, 2194 (1984)

%\bibitem{siekib} X. A. Siemens and T. W. B. Kibble, Nucl. Phys.{\bf B438}, 
%307 (1995)

\bibitem{Davis:1999ec}
S.~C.~Davis, W.~B.~Perkins and A.~C.~Davis,
%``Cosmic string current stability,''
Phys.\ Rev.\ D {\bf 62}, 043503 (2000)

%\bibitem{das} S. R. Das, Nucl. Phys. {\bf B227}, 462 (1983)

\bibitem{widyajetal} U.~A.~Yajnik, H.~Widyan, D.~Choudhari, S.~Mahajan 
and A.~Mukherjee, Phys.\ Rev.\ D {\bf 59}, 103508 (1999)


\bibitem{Jackiw:wc}
R.~Jackiw and J.~R.~Schrieffer,
%``Solitons With Fermion Number 1/2 In Condensed Matter And 
%Relativistic Field Theories,''
Nucl.\ Phys.\ B {\bf 190} (1981) 253.


\end{thebibliography}

\end{document}



hep-th0208027.extracted_bib
\begin{thebibliography}{99}
\bibitem{JandR} R. Jackiw and C. Rebbi, Phys Rev. {\bf D13}, 3398 (1976)

\bibitem{Jrev} R. Jackiw, Rev. Mod. Phys. 49, 681(1977) 

%\bibitem{GolWil} J. Goldstone and F. Wilczek, Phys. Rev. Lett.
%{\bf 47}, 986 (1981)

\bibitem{devega} H. de Vega, Phys. Rev. {\bf D18}, 2932 (1978)

\bibitem{stern} A. Stern; Phys. Rev. Lett.{\bf 52}, 2118(1983)

\bibitem{steandyaj} A. Stern and U. A. Yajnik, Nucl. Phys. {\bf B267}, 158 
(1986)

\bibitem{Davis:1997bu}
S.~C.~Davis, A.~C.~Davis and W.~B.~Perkins,
%``Cosmic string zero modes and multiple phase transitions,''
Phys.\ Lett.\ B {\bf 408}, 81 (1997) 

\bibitem{Davis:1999wq}
A.~C.~Davis, S.~C.~Davis and W.~B.~Perkins, \textsl{Proceedings of
COSMO99}, 
%``The fate of cosmic string zero modes,''
arXiv:hep-ph/0005091

\bibitem{DavKibetal}
A. C. Davis, T. W. B. Kibble, M. Pickles
and D. A. Steer, Phys. Rev. {\bf D62} 083516 (2000)

\bibitem{www} G. C. Wick, A. S. Wightman and E. P. Wigner, Phys. Rev. 
{\bf 88}, 101  (1952)

\bibitem{Sweinberg} S. Weinberg, {\it The Quantum Theory of fields}, Vol. I, 
Cambridge University Press, 1996

\bibitem{sudyaj} E. C. G. Sudarshan and U. A. Yajnik, Phys. Rev. {\bf D33}, 
1830 (1986)

\bibitem{Eweinberg} E. J. Weinberg, Phys. Rev. {\bf D24}, 2669 (1981)

\bibitem{GanLaz} N. Ganoulis and G. Lazarides, Phys. Rev.
{\bf D38} 547 (1988)

%

\bibitem{presvil} J. Preskill and A. Vilenkin, Phys. Rev. {\bf D47}, 
2324 (1993)

%\cite{Shankar:1976un}
%\bibitem{Shankar:1976un}
%R.~Shankar,
%``The SO(3) Monopole Catalog,''
%Phys.\ Rev.\ D {\bf 14}, 1107 (1976).

\bibitem{Slansky:yr}
R.~Slansky,
%``Group Theory For Unified Model Building,''
Phys.\ Rept.\  {\bf 79}, 1 (1981).

\bibitem{JandRossi} R. Jackiw and P. Rossi, Nucl. Phys.{\bf B190}, 681 (1981)

\bibitem{mcwilczek}
R.~MacKenzie and F.~Wilczek,
%``Illustrations Of Vacuum Polarization By Solitons,''
Phys.\ Rev.\ D {\bf 30}, 2194 (1984)

%\bibitem{siekib} X. A. Siemens and T. W. B. Kibble, Nucl. Phys.{\bf B438}, 
%307 (1995)

\bibitem{Davis:1999ec}
S.~C.~Davis, W.~B.~Perkins and A.~C.~Davis,
%``Cosmic string current stability,''
Phys.\ Rev.\ D {\bf 62}, 043503 (2000)

%\bibitem{das} S. R. Das, Nucl. Phys. {\bf B227}, 462 (1983)

\bibitem{widyajetal} U.~A.~Yajnik, H.~Widyan, D.~Choudhari, S.~Mahajan 
and A.~Mukherjee, Phys.\ Rev.\ D {\bf 59}, 103508 (1999)


\bibitem{Jackiw:wc}
R.~Jackiw and J.~R.~Schrieffer,
%``Solitons With Fermion Number 1/2 In Condensed Matter And 
%Relativistic Field Theories,''
Nucl.\ Phys.\ B {\bf 190} (1981) 253.


\end{thebibliography}