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\begin{document}
\title  {Superconducting Knot Solitons in Weinberg-Salam Model}

\author{Y. M. Cho}
\email{ymcho@yongmin.snu.ac.kr}
\affiliation{School of Physics, College of Natural Sciences, 
Seoul National University,
Seoul 151-742, Korea  \\ }

\begin{abstract}
~~~~~We demonstrate the existence of stable knot solitons in the 
standard electroweak theory whose topological quantum number 
$\pi_3(S^3)$ is fixed by the Higgs doublet. The electroweak knots 
are made of the helical hypermagnetic flux tube which has a non-trivial 
dressing of the Higgs field, and carry a net supercurrent made
of W-boson. We estimate the mass of the lightest 
knot to be around 15 TeV, and propose that the knots could 
be an ideal candidate of superconducting cosmic string.
\end{abstract}
\pacs{12.15.-y, 14.80.-j, 11.27.+d, 13.90.+i}
\keywords{electroweak knot, superconducting knot, topological knot}
\maketitle

%\section{Introduction}

Ever since Dirac proposed his theory of monopoles the topological 
objects in physics have been the subject of intensive studies 
\cite{dirac,skyrme}. In particular the finite energy topological 
solitons have been widely studied in theoretical physics 
\cite{skyrme,thooft}. A remarkable type of solitons is the knots, 
a prototype of which was discovered in the Skyrme-Faddeev 
non-linear sigma model \cite{fadd1,ussr}. Consider 
\bea 
&{\cal 
L}_{SF} = - \dfrac{\mu^2}{2} (\partial_\mu \hat n)^2 - 
\dfrac{1}{4} (\partial_\mu \hat n \times \partial_\nu \hat n)^2, 
\eea 
where $\hat n$ (with $\hat n^2 =1$) is the non-linear sigma 
field. The Lagrangian has the following equation of motion 
which allows the knot solitons \cite{cho3,cho4}
\bea 
&\hn \times
\partial^2 \hn - \dfrac{g}{\mu^2} ( \partial_\mu H_{\mu\nu} )
\partial_\nu \hn = 0, \nn\\
&H_{\mu\nu} = -\dfrac{1}{g}\hn \cdot (\partial_\mu \hn \times
\partial_\nu \hn) = \partial_\mu C_\nu - \partial_\nu C_\mu.
\eea 
Notice that since $H_{\mu\nu}$ is closed, one can always 
introduce the potential $C_\mu$ for the field $H_{\mu\nu}$,
as far as $\hat n$ is smooth everywhere.  
The knots are topological, whose quantum number $\pi_3 (S^2)$
fixed by $\hat n$ is given by 
\bea 
&Q_k = 
\dfrac{g^2}{32\pi^2} \int \epsilon_{ijk} C_i H_{jk} d^3x. 
\eea 
Similar knots have been shown to exist in Skyrme theory and 
in the generalized Skyrme theory \cite{cho3}, and
in plasma physics \cite{fadd2}.

The interest on these solitons, however, has been confined by and 
large to theoretical physics community, because the so-called 
standard models do not seem to admit such topological objects. 
The only topological objects which have been known to exist in the standard 
models are the electroweak monopoles and dyons which has a 
non-trivial $W_\mu$ and $Z_\mu$ dressing \cite{cho97}. 
Unfortunately these objects carry an infinite energy, which makes 
it less interesting to experimental physics 
community. 

{\it The purpose of this Letter is to demonstrate the existence of the 
superconducting knots in the standard Weinberg-Salam model of 
electroweak theory, whose mass is estimated to start from  around
15 TeV. Just like the knots in the non-linear sigma model our knots 
are topological. Unlike these knots, however, ours are made of 
the hypermagnetic flux tube which has a non-trivial dressing of 
the Higgs field. More importantly our knots are physical, whose 
existence could be confirmed by high energy experiments (possibly 
with LHC at CERN)}. 

To see the existence of the knots in the Weinberg-Salam theory we 
need a better understanding of the non-trivial topology of the theory, 
which is an essential ingredient of non-Abelian gauge theory.   
A best way to take care of the topology is to introduce 
a topological field $\hn$ which selects the non-Abelian charge 
direction at each space-time point, and reparametrize the 
non-Abelian gauge potential into the restricted potential which 
makes $\n$ a covariant constant and the valence potential which 
forms a covariant vector field \cite{cho1,cho2}. To demonstrate 
this let $\vec A_\mu$ be the $SU(2)$ gauge potential of the 
Weinberg-Salam theory and let 
\bea 
& \vec{A}_\mu =A_\mu \n - 
\oneg \n\times\pro_\mu\n+\X_\mu\nonumber
         = \hat A_\mu + \X_\mu, \nn\\
&  (A_\mu = \n\cdot \vec A_\mu,~ \n^2 =1,~ 
\hat{n}\cdot\vec{X}_\mu=0), 
\eea 
where $ A_\mu$ is the 
``electric'' potential. Notice that the restricted potential 
$\hat A_\mu$ is precisely the connection which leaves $\n$ 
invariant under parallel transport, 
\bea 
\D_\mu \n = \pro_\mu \n 
+ g {\hat A}_\mu \times \n = 0. \eea Under the infinitesimal 
gauge transformation \bea \delta \n = - \vec \alpha \times \n  
\,,\,\,\,\, \delta \A_\mu = \oneg  D_\mu \vec \alpha, \eea one has 
\bea &&\delta A_\mu = \oneg \n \cdot \pro_\mu \valpha,\,\,\,\
\delta \hat A_\mu = \oneg \D_\mu \valpha  ,  \nn \\
&&\hspace{1.2cm}\delta \X_\mu = - \valpha \times \X_\mu  . 
\eea 
This tells that $\hat A_\mu$ by itself describes an $SU(2)$ 
connection which enjoys the full $SU(2)$ gauge degrees of 
freedom. Furthermore the valence potential $\vec X_\mu$ forms a 
gauge covariant vector field under the gauge transformation. But 
what is really remarkable is that the decomposition is 
gauge-independent. Once the gauge covariant topological field 
$\hat n$ is given, the decomposition follows automatically 
independent of the choice of a gauge \cite{cho1,cho2}. 

Notice that $\hat{A}_\mu$ retains the full topological 
characteristics of the original non-Abelian potential. Clearly, 
the isolated singularities of $\hat{n}$ define $\pi_2(S^2)$ which 
describes the non-Abelian monopoles.  Indeed, $\hat A_\mu$ with 
$A_\mu =0$ and $\hat n= \hat r$ describes precisely the Wu-Yang 
monopole \cite{wu,cho80}.  Besides, with the $S^3$ 
compactification of $R^3$, $\hat{n}$ characterizes the Hopf 
invariant $\pi_3(S^2)\simeq\pi_3(S^3)$ which describes the 
topologically distinct vacua \cite{cho4,cho79}. 

The restricted potential $\hat{A}_\mu$ has a dual 
structure,
\begin{eqnarray}
& \hat{F}_{\mu\nu} = (F_{\mu\nu}+ H_{\mu\nu})\hat{n}\mbox{,}\nonumber \\
& F_{\mu\nu} = \partial_\mu A_{\nu}-\partial_{\nu}A_\mu \mbox{,}\nonumber \\
& H_{\mu\nu} = -\dfrac{1}{g} \hat{n}\cdot(\partial_\mu
\hat{n}\times\partial_\nu\hat{n}) = \partial_\mu 
C_\nu-\partial_\nu C_\mu,
\end{eqnarray}
where $C_\mu$ is the ``magnetic'' potential \cite{cho1,cho2}. 
Notice that $H_{\mu\nu}$ here is exactly the same $H_{\mu\nu}$ 
appeared in (2). Thus, one can identify the non-Abelian magnetic 
potential by 
\bea 
\vec C_\mu= -\frac{1}{g}\hat n \times 
\partial_\mu\hat n , 
\eea 
in terms of which the magnetic field is 
expressed by 
\bea 
\vec H_{\mu\nu}=\partial_\mu \vec 
C_\nu-\partial_\nu \vec C_\mu+ g \vec C_\mu \times \vec C_\nu 
=H_{\mu\nu}\hat n. 
\eea 
With the decomposition (1), one has 
\bea 
\vec{F}_{\mu\nu}&=&\hat F_{\mu \nu} + \D _\mu \X_\nu - \D_\nu 
\X_\mu + g\X_\mu \times \X_\nu, 
\eea 
so that the Yang-Mills 
Lagrangian is expressed as 
\bea 
&{\cal L} =-\dfrac{1}{4}
{\hat F}_{\mu\nu}^2 -\dfrac{1}{4}(\D_\mu\X_\nu-\D_\nu\X_\mu)^2  \nn \\
&-\dfrac{g}{2} {\hat F}_{\mu\nu} \cdot (\X_\mu \times 
\X_\nu)-\dfrac{g^2}{4} (\X_\mu \times \X_\nu)^2. 
\eea 
This shows 
that the Yang-Mills theory can be viewed as a restricted gauge 
theory made of the restricted potential, which has the additional 
gauge covariant valence potential as its source \cite{cho1,cho2}.

The decomposition (1) reveals the deep connection between the 
non-Abelian gauge theory and the Skyrme-Faddeev theory. In fact 
from the decomposition we have 
\bea 
{\cal L}_{SF} = - 
\dfrac{1}{4} \vec H_{\mu\nu}^2 - \dfrac{\mu^2} {2} \vec C_\mu^2. 
\eea 
This tells that the Skyrme-Faddeev theory can be interpreted 
as a massive Yang-Mills theory where the gauge potential has the 
special form (9), which indicates the existence of the 
electroweak knots in the Weinberg-Salam theory. Our decomposition 
(1), which has recently become known as the ``Cho decomposition'' 
\cite{faddeev2} or the ``Cho-Faddeev-Niemi decomposition'' 
\cite{lang}, was introduced long time ago in an attempt to 
demonstrate the monopole condensation in QCD \cite{cho1,cho2}. 
But only recently the importance of the decomposition in 
clarifying the non-Abelian dynamics has become appreciated by 
many authors \cite{faddeev2,lang}. Indeed it is this 
decomposition which has played a crucial role to establish the  
``Abelian dominance'' in Wilson loops in QCD \cite{cho00}, and 
the possible connection between the Skyrme-Faddeev action and the 
effective action of QCD in the infra-red limit \cite{cho4,cho01} 

With these preliminaries we now demonstrate the existence of the 
electroweak knots in Weinberg-Salam theory. Consider the 
Lagrangian 
\bea 
&{\cal L} =-\dfrac{1}{4} {\vec 
F}_{\mu\nu}^2-\dfrac{1}{4}
G_{\mu\nu}^2 -|\tilde D_\mu \phi|^2 \nn\\
&+ m^2\phi^{\dagger}\phi - \dfrac{\lambda}{2} (\phi^{\dagger} \phi)^2, 
\eea 
where $\vec F_{\mu\nu}$ and $G_{\mu\nu}$ 
are the field strengths of the $SU(2)$ and $U(1)$ gauge potential 
$\vec A_\mu$ and $B_\mu$, $\phi$ is the Higgs doublet, and 
\bea 
\tilde D_\mu \phi = ( \partial_\mu + \dfrac{g}{2i} \vec \sigma
\cdot \vec A_\mu +\dfrac{g'}{2i} B_\mu) \phi 
=(D_\mu +\dfrac{g'}{2i} B_\mu ) \phi. \nn 
\eea 
The equation of 
motion of the Lagrangian is given by 
\bea 
&{\tilde D}^2\phi 
=\lambda(\phi^{\dagger} \phi
-\dfrac{m^2}{\lambda})\phi, \nn\\
&D_\mu \vec F_{\mu \nu} = g \Big[(\tilde D_\nu \phi)^{\dagger} \dfrac{
\vec\sigma}{2i} \phi - \phi ^{\dagger} \dfrac{ \vec\sigma}{2i}(\tilde
D_\nu \phi) \Big], \nn\\
&\partial_\mu G_{\mu \nu} = \dfrac{g'}{2i} \Big[(\tilde D_\nu
\phi)^{\dagger}\phi - \phi ^{\dagger}(\tilde D_\nu \phi) \Big]. 
\eea 
Now, let 
\bea 
&\phi =\rho \xi,~~~\xi^{\dagger}\xi =1, 
~~~\hat n = \xi^{\dagger} \vec
\sigma\xi, \nn\\
&A_\mu = \hat n \cdot \vec A_\mu ,~~~C_\mu = \dfrac{2}{g}i \xi 
^{\dagger} 
\partial _\mu \xi, 
\eea 
and choose the ansatz
\bea
&\vec X_\mu = f_1 \partial_\mu \hat n +f_2 \hat n \times
\partial_\mu \hat n,~~~~~\chi = f_1 + i f_2,\nn\\
&A_\mu = 0, ~~~ \chi^* \chi = \dfrac{1}{g^2}. 
\eea 
With the ansatz we find that 
the second equation of (15) is automatically satisfied, so that 
the equation of motion is reduced to 
\bea 
&\partial ^2 \rho - \dfrac{1}{4} 
\Big[(\partial _\mu \hat n)^2 + (g'B_\mu + g C_\mu )^2 \Big] \rho = \lambda
(\rho^2-\dfrac{m^2}{\lambda})\rho, \nn\\
&\Big\{ (\partial_\mu \hat n)^2+ \Big[ \partial^2 \hat n \nn\\
&+\big(2 \dfrac{\partial_\mu \rho}{\rho} - i ( g' B_\mu + g C_\mu)\big) 
\partial_\mu \hat n \Big]\cdot \vec \sigma \Big\} \xi =0, \nn\\
&\partial_\mu G_{\mu\nu} = \dfrac {g'}{2} \rho^2 ( g' B_\mu + g 
C_\mu). 
\eea 
The second equation is a matrix equation, which 
requires the following condition to allow a solution 
\bea 
&{\rm 
Det} \Big\{(\partial _\mu \hat n)^2 + \Big[\partial^2 \hat n \nn\\ 
&+ \big(2 \dfrac{ \partial_\mu \rho}{\rho} 
- i ( g' B_\mu + g C_\mu) \big)  
\partial_\mu \hat n \Big] \cdot \vec \sigma \Big\} = 0.
\eea 
This, together with the last equation of (18), can be 
expressed as 
\bea 
\hat n \times \partial ^2 \hat n + 2 
\dfrac{\partial_\mu \rho}{ \rho} \hat n \times \partial_\mu \hat 
n \pm \dfrac{ 2}{ g'\rho^2} (\partial_\mu G_{\mu\nu}) 
\partial_\nu \hat n = 0. 
\eea 
But notice that this is nothing but 
a knot equation. Indeed, in the absence of the Higgs field, this 
becomes exactly the knot equation (2), provided that 
\bea 
B_\mu 
=\mp \dfrac{gg'}{2} C_\mu,~~~~G_{\mu\nu} =\mp \dfrac{gg'}{2} 
H_{\mu\nu}. 
\eea 
This means that (20) describes the Faddeev-Niemi 
knot coupled to the Higgs field $\rho$ and the $U(1)$ gauge field 
$B_\mu$. Furthermore, it is evident from (18) that the equation 
for $B_\mu$ is very much like the London equation in 
superconductors. Indeed with the ansatz (21) the equation of 
motion (18) is reduced to 
\bea 
&\partial ^2 \rho -  
\Big[\dfrac{1}{4}(\partial _\mu \hat n)^2+ \dfrac{1}{g'^2}(
\dfrac{g'^2}{2} \mp 1)^2 B_\mu ^2 \Big] \rho 
= \lambda
(\rho^2-\dfrac{m^2}{\lambda})\rho, \nn\\
&\hat n \times \partial ^2 \hat n + 2 \dfrac{\partial_\mu \rho}{
\rho} \hat n \times \partial_\mu \hat n - \dfrac{g}{ \rho^2}
(\partial_\mu H_{\mu\nu}) \partial_\nu \hat n = 0, \nn\\
&\partial_\mu G_{\mu\nu}=(\dfrac{g'^2}{2}\mp 1)\rho^2 B_\mu. 
\eea 
This guarantees that $B_\mu$ displays the Meissner effect which 
forms hypermagnetic flux tubes, which together with the Higgs 
field forms the electroweak knots. The fact that the knots exist 
in the absence of the Higgs field, together with the 
manifestation of the Meissner effect of the $U(1)$ gauge field, 
guarantees the existence of the knot solutions described by (22). 
This completes the demonstration of the existence of new knots 
made of the hypermagnetic flux tube and the Higgs field in the 
standard electroweak theory. 

Notice that the ansatz (17) practically describes a vacuum for 
the $SU(2)$ gauge potential $\vec A_\mu$. Indeed, with $f_1 = 0$ 
and $f_2 = 1/g$, we have $\vec A_\mu = 0$. This tells that our 
knots are essentially the knots of the $U(1)$ gauge theory 
coupled to the Higgs doublet. Nevertheless we like to emphasize 
that it is the non-Abelian structure of the Weinberg-Salam model 
that plays the crucial role for the existence of the knots. To 
understand this it is important to realize that, even though 
$\vec A_\mu = 0$ according to the ansatz (17), our knots
have a nontrivial component of W-boson. This is because $\vec X_\mu$
in our decomposition becomes nothing but $\vec W_\mu$ in the
physical gauge where $\xi$ becomes trivial \cite{cho3,cho97}.
Furthermore in this physical gauge the $SU(2)$ gauge 
potential acquires a non-trivial $U(1)$ component $-C_\mu$.  
This is because 
$C_\mu$ defined in (16) describes precisely the magnetic 
potential of (8), which becomes the $U(1)$ part of $\vec A_\mu$ 
(up to the signature) in the physical gauge  
\cite{cho4,cho79}. This confirms that the existence of our knots 
is really due to the non-Abelian structure of the electroweak 
theory.  

Just like the Faddeev-Niemi knots our knots are topological. But 
there is an important difference between the two knots. Unlike 
the Faddeev-Niemi knots in which the non-trivial homotopy  is 
provided by $\pi_3(S^2)$ of the non-linear sigma field $\hn$, 
here it is the non-Abelian topology $\pi_3(S^3)$ of the Higgs 
doublet $\phi$ which provides the non-trivial homotopy. Indeed 
our knot quantum number $\pi_3(S^3)$ is given by 
\bea 
Q_k = - 
\dfrac {1}{4\pi^2} \int \epsilon_{ijk} \xi^{\dagger} 
\partial_i \xi ( \partial_j \xi^{\dagger} 
\partial_k \xi ) d^3 x. 
\eea 
Of course one can easily show that
(3) and (23) produce the same quantum number, due to the Hopf 
fibring $\pi_3(S^3) \simeq \pi_3(S^2)$. But certainly 
$\pi_3(S^2)$ and  $\pi_3(S^3)$ are different. Furthermore, in 
the physical gauge where $\xi$ becomes trivial, 
the knot quantum number can be 
shown to describe nothing but the vacuum number of the $SU(2)$ 
gauge potential \cite{cho4,cho79}. This reassures the fact 
that our electroweak knots are fundamentally non-Abelian.
  
To understand the physics behind the electroweak knots, notice 
that  with (17) and (21) the Lagrangian (14) is reduced to
\bea 
&{\cal L}= -\dfrac{g'^2}{16}(
\partial_\mu \hn \times\partial_\nu \hn)^2
- \dfrac{\rho^2}{4} \Big[g^2 \vec W_\mu^2 + (g^2 + g'^2)  
Z_\mu^2 \Big] \nn\\
&- (\partial_\mu \rho)^2 +m^2 \rho^2 - \dfrac{\lambda}{2}\rho^4, 
\eea 
where $\vec W_\mu$ and $Z_\mu$ are the weak bosons given by
\bea 
&\vec W_\mu = f_1 \partial_\mu \hn + f_2 \hn \times \partial_\mu \hn, \nn\\
&Z_\mu 
=\dfrac{-1}{\sqrt{g^2 + g'^2}} (g'B_\mu + g C_\mu). \nn 
\eea 
This tells that the 
Weinberg-Salam theory can be reduced to a generalized 
Skyrme-Faddeev theory, in which the non-linear sigma field $\hn$ 
interacts with the weak bosons $\vec W_\mu$, $Z_\mu$, 
and the Higgs field $\rho$. 
Furthermore this confirms that it is exactly the 
spontaneous symmetry breaking of $SU(2) \times U(1)$ to 
$U(1)_{em}$ which induces the Meissner effect to 
the hypermagnetic flux of our knots.
{\it But what is most interesting is that our knots are 
superconducting. To see this consider the unknot (with $Q_k = 1$)
which forms a toroidal ring made of the hypermagnetic flux.
Clearly the topology makes the magnetic flux helical.
But this helical magnetic flux inevitably requires
a helical supercurrent.
So the vortex ring should carry a net supercurrent along
the toroidal ring, in addition to the well-known
supercurrent which confines the magnetic flux.
Furthermore this supercurrent is hypercurrent, made of
W-boson as well as Z-boson. This makes the electroweak 
knot superconducting}. 
Clearly this makes our knot an excellent example of
superconducting cosmic string \cite{witten}, 
which suggests that our electroweak knot could have a deep
impact not only in high enregy physics but also in cosmology.

To discuss the physical 
implications of the knots we must know the energy of the knots. 
Clearly the energy of our knots (22) is given by 
\bea 
&E = 
\dfrac{}{} \int \Big[ \dfrac{g'^2}{16}(\partial_i \hn \times 
\partial_j \hn)^2 + \dfrac{m_Z^2}{2} 
\Big(\dfrac{1}{g^2+g'^2}(\partial_i \hn)^2 \nn\\ &+ Z_i^2 \Big) 
(\dfrac{\rho}{\rho_0})^2 + (\partial_i \rho)^2 + V(\rho) \Big] d^3 x, 
\eea 
where $\rho_0 = \sqrt{m^2/\lambda} \simeq 174~GeV$.
To estimate the energy it is important to remember the 
energy of the knots described by the Skyrme-Faddeev Lagrangian (1) 
is given by \cite{ussr} 
\bea 
E_n \geq 16 \pi^2 3^{3/8} ~|n|^{3/4} 
\mu \simeq 238 ~|n|^{3/4}  \mu, 
\eea 
where n is the knot 
quantum number given by (3). From this we can estimate the energy 
of our knots. In the absence of $\rho$ and $Z_\mu$, (25) implies that 
\bea 
&E_n \geq 16 \pi^2 
3^{3/8} ~|n|^{3/4}\dfrac{g'}{\sqrt{g^2 + g'^2}}~\dfrac{m_Z}{2}\nn\\
& \simeq 119~|n|^{3/4} ~\sin \theta~m_Z,
\eea 
where $m_Z$ and  
$\theta$ are the mass of the $Z_\mu$ boson
and the Weinberg 
angle. So, with $m_Z \simeq 90 ~GeV$ 
and $\sin^2 \theta \simeq 
0.23$, we arrive at the following lower bound for the energy of 
the electroweak knots, 
\bea 
E_n > 5.13~|n|^{3/4} ~TeV. 
\eea  
This (with the equipartition of energy)
suggests that the mass of the lightest electroweak knot is around 
$15 ~TeV$.  

Although we have not been able to obtain the analytic solutions 
of the knots yet, we believe that our analysis has established 
the existence of the stable electroweak knots 
without any doubt. A challenging task now would be 
to confirm the existence of the electroweak knots by 
high energy experiments. 

Note Added: One might worry about the stability of our knots. 
Obviously the knots have the topological stability. 
But we emphasize that they also enjoy  
the dynamical stability. To see this 
consider the unknot again. Here  
the net supercurrent necessarily creates an angular
momentum which provides the centrifugal force to 
forbid the collapse of the knot. So the topological supercurrent
provides the dynamical stability to the unknot.
It is this interplay between topology and dynamics 
which guarantees the stability of the electroweak knot.
Similar superconducting knots have also been shown to exist
in two-component Bose-Einstein condensates 
and two-gap superconductors \cite{cho5}. The details 
of the electroweak knot, together with the cosmological
implications, will be
published elsewhere \cite{cho6}.


{\bf ACKNOWLEDGEMENT} 

~~~We thank E. Babaev, L. Faddeev, H. W. Lee, A. Niemi, 
D. G. Pak, M. Walker, and Professor C. N. Yang  
for the illuminating 
discussions. The work is supported in part by a Korea Research 
Foundation Grant (KRF-2001-015-BP0085), and by the BK21 project 
of the Ministry of Education.

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