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\title{A model of glueballs} \author{Roman V. Buniy and Thomas
W. Kephart\\ \emph{Department of Physics and Astronomy}\\
\emph{Vanderbilt University, Nashville, TN 37235}}
%\email{roman.buniy@vanderbilt.edu}
%\author{Thomas W. Kephart}
%\email{kephartt@ctrvax.vanderbilt.edu}
%\affiliation{Department of Physics and Astronomy,\\
%Vanderbilt University, Nashville, TN 37325.}
\date{}
\maketitle



\begin{abstract}
We model the observed glueball mass spectrum in terms of energies for
tightly knotted and linked QCD flux tubes. The data is fit well with
one parameter. We predict additional glueball masses.
\end{abstract}

%\pacs{}



%\section{Introduction}


\emph{Introduction.---} The interpretation of non-$q\bar{q}$ states is
a puzzle with a long and controversial history~\cite{Swanson}. Many
experiments~\cite{PDG} report states that do not fit neatly into the
quark model. These states can be broadly classified as: (1)\ hybrids,
which are bound states of quarks and gluons like $q\bar{q}G$ with
quantum numbers $J^{PC}=0^{-+}$, $1^{-+}$, $1^{--}$, $2^{-+}$,
$\ldots$; (2)\ exotics, for example, four and six quark states such as
$qq\bar{q}\bar{q}$ and $qqq\bar{q}\bar{q}\bar{q}$ with quantum numbers
$J^{PC}=0^{--}$, $0^{+-}$, $1^{-+}$, $2^{+-},\ldots$; (3)\ glueballs
with pointlike or collective (strings \`{a} la
Nielsen--Olesen~\cite{Nielsen-Olesen}, or flux tubes) glue. Glueballs
do not contain valence quarks, but there could be sea/virtual quarks
within the glueball or in the currents that support the flux
tubes. From a bag model perspective one is led to suppose that the
lightest non--$q\bar{q}$ states are those with no constituent quarks,
i.e., the glueballs. Lattice calculations, QCD sum rules, electric
flux tube models, and constituent glue models leads to a consensus
that the lightest non--$q\bar{q}$ states are glueballs with quantum
numbers $J^{++}=0^{++}\ $and 2$^{++}$~\cite{West}. We will model all
$J^{++}$ states (i.e., all $f_{J}$ and $f'_J$ states listed by the
Particle Data Group (PDG)~\cite{PDG}), some of which will be
identified with rotational excitations, as knotted/linked color
magnetic flux tubes~\cite{comment1}.

Besides the fact they do not fit in the quark model, glueballs have
some other expected signatures, including: enhanced central production
in gluon rich channels, branching fractions incompatible with
$q\bar{q}$ decay, reduced $\gamma\gamma$ coupling, and OZI
suppresion. All the $J^{++}$ states we consider have some or all of
these properties. For instance, none have substantial branching
fractions to $\gamma\gamma$. However, mixing with $q\bar{q}$ isoscalar
states can obscure some of the properties. A number of candidates with
masses below $2.5\,GeV$ have been identified. Beyond their masses and
widths, and some of their branching ratios~\cite{PDG}, much remains to
be learned about these states.

Knotted magnetic fields (which we will treat as solitons) have been
suggested as candidates for a number of plasma phenomena in systems
ranging from astrophysical, to atmospheric~\cite{lightning}, to
Bose-Einstein condensates~\cite{Bose-Einstein}. The energies and
configurations of these solitons are difficult to quantify since they
depend on parameters of the plasma, including temperature, pressure,
density, ionic content, etc.; however, we will argue that in QCD, a
well defined soliton energy can be identified.

As has been shown in plasma physics, tight knots and links (defined
below) correspond to metastable minimum energy configurations. We will
argue by analogy that tight knots and links of chromomagnetic flux are
glueballs. (In what follows, we often use the term ``knots'' to mean
knots and/or links.)

Movement of fluids often exhibits topological properties (for a
mathematical review see e.g.~\cite{Arnold}). For conductive fluids,
interrelation between fluid motion and magnetic fields via
magnetohydrodynamics may cause magnetic fields, in their turn, to
exhibit topological properties. For example, for a perfectly
conducting fluid, the (Abelian) magnetic helicity $L_H=\int\d ^3
x\,\eps_{ijk}A_i\p_jA_k$ is an invariant of motion~\cite{Woltier}, and
this quantity can be interpreted in terms of knottedness of magnetic
flux lines~\cite{Moffatt:1969}.

The dynamics of the magnetic fields follows the dynamics of the liquid
(magnetic flux lines are ``frozen'' into the fluid), and one finds
that a perfectly conducting, viscous and incompressible fluid relaxes
to a state of magnetic equilibrium without a change in
topology~\cite{Moffatt:1985}. As a result, for topologically
non-trivial plasma flows (with knotted streamlines), the ``freezing''
condition forces topological restrictions on possible changes in field
configurations. For linked non-intersecting loops $C_a$ with magnetic
fluxes $\Phi_a$, the helicity becomes~\cite{Moffatt:1969}
$L_H=\frac{k}{8\pi}\sum_{a\not=b}L(C_a,C_b)\Phi_a\Phi_b$. Here
$L(C_a,C_b)$ is the Gauss linking number.~\cite{helicity_comment} By
its topological nature, the helicity can be one of the quantum numbers
characterizing glueballs. However, there is another invariant called
the knot energy that is less obvious but as important in the
classification of solitonic knots.



%\section{Knot energies}


\emph{Knot energies.---} Consider a hadronic collision that produces
some number of baryons and mesons plus a gluonic state in the form of
a closed flux tube (or a set of tubes). From an initial state, the
fields in the flux tubes quickly relax to an equilibrium
configuration, which is topologically equivalent to the initial
state. (We assume topological quantum numbers are conserved during
this rapid process.) The relaxation proceeds through minimization of
the magnetic energy. Flux conservation and energy minimization force
the fields to be homogeneous across the tube cross sections. This
process occurs via shrinking the tube length, and halts to form a
``tight'' knot or link. The radial scale will be set by $\Lambda
_{\textrm{QCD}}^{-1}$. The energy of the final state depends only on
the topology of the initial state and can be estimated as follows. An
arbitrarily knotted tube of radius $r$ and length $l$ has the volume
$\pi r^2 l$. Using conservation of flux, the energy becomes $\propto
l(\tr\Phi^2)/(\pi r^2)$. Fixing the radius of the tube (to be
proportional to $\Lambda_{\textrm{QCD}}^{-1}$), we find that the
energy is proportional to the length $l$. The dimensionless ratio
$\ve(K)=l/(2r)$ is a topological invariant and the simplest definition
of the ``knot energy''~\cite{knot_energy}.

Many knot energies have been calculated by Monte Carlo
methods~\cite{knot-energies} and certain types can be calculated
exactly (see below), while for other cases simple estimates can be
made (see Table~\ref{table}). For example, the knot energy of the
connected product of two knots $K_1$ and $K_2$ satisfies \ba
\ve(K_1\#K_2)<\ve(K_1)+\ve(K_2)\ea A rule of thumb is \ba
\ve(K_1\#K_2)\approx \ve(K_1)+\ve(K_2)-(2\pi-4),\ea which results from
removing two half tori, one from each knot, and replacing these with
two connecting cylinders of lengths $r$. This, for example, gives
$\ve(3_1\#3_1)$ and $\ve(3_1\#3_1^*)$ to about $5\%$.

Most of the knot energies in Table 1 have been taken from
\cite{knot-energies}, but we have independently calculated the energy
of $2^2_{1}$ and $4^3_1$ exactly and the energy for several other
knots and links approximately. We find $\ve(2^2_{1})=4\pi\approx
12.57$, to be compared with the Monte Carlo value $12.6$. We also find
$\ve(4^3_{1})=6\pi+2$, where there is no Monte Carlo comparison
available.



%\section{Model}


\emph{Model.---} In our model, the chromomagnetic
fields~\cite{comment3} $F_{ij}$ are confined to knotted tubes, each
carrying one quantum of conserved flux~\cite{flux}~\cite{soliton}. We
consider a static Lagrangian density \ba{\cL}=\frac{1}{2}\tr
F_{ij}F^{ij}-V,\label{L}\ea where, similar to the MIT bag
model~\cite{MIT-bag}, we included the possibility of a constant energy
density $V$. To account for conservation of the magnetic flux $\Phi$,
we add to (\ref{L}) the term \ba\tr\lambda\{\Phi/(\pi
r^2)-\frac{1}{2}\eps^{ijk}n_iF_{jk}\},\nn\ea where $n_i$ is the normal
vector to a section of the tube and $\lambda$ is a Lagrange
multiplier. Varying the full Lagrangian with respect to $A_i$, we find
\ba D^j(F_{ij}-\frac{1}{2}\eps_{ijk}n^k\lambda)=0,\nn\ea which has the
constant field \ba F_{ij}=(\Phi/\pi r^2)\eps_{ijk}n^k\label{F}\ea as
its solution. With this solution, the energy is positive and
proportional to $l$ and thus the minimum of the energy is achieved by
shortening $l$, i.e. tightening the knot.


\begin{figure*}
\centering \includegraphics[angle=0]{knot1}
\caption{\label{figure1} The shortest knot/link solitonic flux
configuration has the topology of two linked tori, which in knot
theory notation is $2^2_1$. This corresponds to the lightest glueball
candidate $f_0(600)$.}
\end{figure*}

\begin{figure*}
\centering \includegraphics[angle=0]{knot2}
\caption{\label{figure2} The second shortest solitonic flux
configuration is the trefoil knot $3_1$ corresponding to the second
lightest glueball candidate $f_0(980)$.}
\end{figure*}


We proceed to identify knotted and linked QCD flux tubes with
glueballs, where we include all $f_J$ and $f'_J$ states. The lightest
candidate is the $f_0(600)$, which we identify with the shortest
knot/link, i.e., the $2^2_1$ link (see Figure~\ref{figure1}); the
$f_0(980)$ is identified with the next shortest knot, the $3_{1}$
trefoil knot (see Figure~\ref{figure2}), and so forth. All knot and
link energies have been calculated for states with energies less then
$1700\,\textrm{MeV}$. Above $1700\,\textrm{MeV}$ the number of knots
and links grows rapidly, and few of their energies have been
calculated. However, we do find knot energies corresponding to known
$f_J$ and $f'_J$ states, and so can make preliminary identifications
in this region. (We focus on $f_J$ and $f'_J$ states from the PDG
summary tables. The experimental errors are also quoted from
PDG. There are a number of additional states reported in the extended
tables, but some of this data is either conflicting or inconclusive.)

Our detailed results are collected in Table \ref{table}, where we list
$f_J$ and $f'_J$ masses, widths, and our identifications of these
states with knots, together with the corresponding knot energies.

\begin{figure*}
\centering \includegraphics[angle=0]{plot}
\caption{\label{figure}Relationship between the glueball spectrum
$E(G)$ and knot energies $\ve(K)$. Each point in this figure
represents a glueball identified with a knot or link. The straight
line is our model and is drawn for the fit (\ref{fit1}).}
\end{figure*}


In Figure~\ref{figure} we compare the mass spectrum of $f$ states with
the identified knot and link energies. Since errors for the knot
energies in~\cite{knot-energies} were not reported, we conservatively
assumed the error to be $1\%$. A least squares fit to the most
reliable data gives \ba E(G)=(23.4\pm 46.1)+(59.1\pm 2.1)\ve(K)\ \ \
[\textrm{MeV}],\label{fit1}\ea with $\chi^2$ is $9.1$. The data used
in this fit is the first seven $f_J$ states (filled circles in
Figure~\ref{figure}) in the PDG summary tables. Inclusion of the
remaining seven (non-excitation) states (unfilled circles in
Figure~\ref{figure}) in Table~\ref{table}, where either the glueball
or knot energies are less reliable, does not significantly alter the
fit and leads to \ba E(G)=(26.9\pm 24.9)+(58.9\pm 1.0)\ve(K)\ \ \
[\textrm{MeV}],\label{fit2}\ea with $\chi^2=10.1$. The fit
(\ref{fit1}) is a good self-consistency check~\cite{comment2} of our
model, in which $E(G)$ is proportional to $\ve(K)$. Better HEP data
and the calculation of more knot energies will provide further tests
of the model and improve the high mass identification.

In terms of the bag model~\cite{MIT-bag}, the interior of
tight knots correspond to the interior of the bag. The flux through
the knot is supported by current sheets on the bag boundary (surface
of the tube). Knot complexity can be reduced (or increased) by
unknotting (knotting) operations~\cite{Rolfsen,Kauffman}. In terms of
flux tubes, these moves are equivalent to reconnection
events~\cite{reconnection}. Hence, a metastable glueball decays via
reconnection. Once all topological charge is lost, metastability is
lost, and the decay proceeds to completion.

We have assumed one fluxoid per tube. There may be states with more
than one fluxoid, but these would presumably have somewhat fatter flux
tubes with higher flux densities and higher energies. For example, the
two fluxoid trefoil knot $3_1$ would certainly have
$\ve(K)>2\,\ve(3_1)$ and a fairly reliable estimate gives $\ve(K)\approx
2\sqrt{2}\,\ve(3_1)$. Hence most multifluxoid states would be above the
mass range of known glueballs.




%\section{Discussions and conclusions}


\emph{Discussions and conclusions.---} In principle, lattice
calculations can find any tame knot (knot without an infinite number
of crossings or other pathology \cite{Rolfsen}) configuration, since
there is always a contour through the lattice that represents the
knotted path by some specific Wilson loop. However, since one is
constrained by the rigidity of the lattice, energy minimization is
difficult and requires a very fine-grained lattice.  Thus we expect
shape-evolving Monte Carlo techniques \cite{knot-energies} to be much
more efficient and accurate for this purpose.




Now we must discuss the details of identifications made in
Table~\ref{table}. The four (unconfirmed) glueball states with masses
less then $1700\,\textrm{MeV}$ from the extended PDG tables are
identified as follows: (1) the $4^2_1$ link with $E(G)=1289$ and the
$4_1$ knot with $E(G)=1277$ are nearly degenerate, and the $f_1(1285)$
could actually be a pair of nearly degenerate states with identical
quantum numbers associated with these knots; this is a possible
interpretation of the $f_1(1285)$ mass measurements summarized on page
481 of Ref.~\cite{PDG}; (2) the $f_2(1430)$ is treated as a rotational
excitation of the $f_1(1420)$ and identified with the $5_1$ knot; the
energy difference between these two states, $\delta'$, is a few MeV,
but not well determined; this difference is of the order of what one
would expect for rotational excitations; [We approximate
$E(f_J)=E(f_0)+\frac{1}{2}J(J+1)\delta$.] (3) we treat the $f_1(1510)$
as the first and the $f'_2(1525)$ as the second rotational excitation
of the $f_0(1500)$, which we identify with the $5_2$ knot; now the
energy step size is $\delta\approx 5\textrm{MeV}$ which agrees with a
simple estimate; (4) we assign the $f_2(1565)$ and the $f_2(1640)$ to
the $5^2_1$ and the $6^3_3$ links respectively.

Further details of knot excitations would be interesting to
investigate, as would quantum and curvature corrections. At present we
do not have a reliable way to estimate all these effects, nor do we
have a good way to calculate glueball decays. However, we do expect
high mass glueball production to be suppressed because more complicated
non-trivial topological field configurations are statistically
disfavored.

Finally, knot solitons may also be able to survive within a
quark-gluon plasma (e.g., in the interior of a RHIC event, quark star,
or in the early universe). Complications will certainly arise in these
cases due to additional parameters describing the media, as with
knotted and linked electromagnetic plasma solitons; but if one holds
the parameters constant throughout the region of interest, the energy
spectrum will be universal for any such system up to a scaling.


\emph{Acknowledgments.---} We thank Med Webster, Kevin Stenson, Eric
Vaandering, Will Johns, Tom Weiler, and Jack Ng for useful comments
and discussions. This work was supported by U.S. DoE grant number
DE-FG05-85ER40226.





%\bibliographystyle{unsrt}
%\bibliography{general,knots,mhd}



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\bibitem{comment3} It has been argued that the confinement of color
magnetic flux tubes requires light quarks in the spectrum of the theory
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the full QCD Lagrangian for our model of tightly knotted glueballs. Other
possibilities are knotted color electric flux tubes, or some type of
knottedness in disoriented chiral condensates. Neither of the
possibilities seem as compelling as color magnetic flux tubes.
Specifically, the electric fields require a quark at one end and an
anti-quark at the other. If we try to knot this flux tube and make it
close by having the $q\bar{q}$ pair annihilate, we are left with an
electric flux tube with no current support (a color magnetic monopole
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through the boundary $S$ of $V$, in general, does not vanish:
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\bibitem{comment2} Note that $\chi ^{2}=251$ for a fit where the first
glueball is missed out, and $\chi ^{2}=355$ for a fit where the first
knot/link is missed out. This is strong evidence that our
identification is appropriate.

\bibitem{Rolfsen}
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\bibitem{Kauffman}
L.~H. Kauffman,
\newblock {\em Knots and physics},
\newblock World Scientific, 2001.

\bibitem{reconnection}
E.~Priest and T.~Forbes,
\newblock {\em Magnetic Reconnection: {MHD} Theory and Applications},
\newblock Cambridge, 2000.






\end{thebibliography}





\newpage

\begin{center}
\begin{table}[htb]
\begin{center}
\caption{\label{table}Comparison between the glueball mass spectrum
and knot energies.}
\begin{tabular}{cccccc}\hline\hline
{\rule[-3mm]{0mm}{8mm} State} & Mass & Width & $K$~\footnotemark[1] &
$\ve(K)$~\footnotemark[2] & $E(G)$~\footnotemark[3] \\ \hline
{\rule[1mm]{0mm}{3mm} $f_0(600)$} & $400-1200$ & $600-1000$ & $2^2_1$
& $12.6\ [4\pi]$ & $768\ [766]$\\ $f_0(980)$ & $980\pm 10$ & $40-100$
& $3_1$ & $16.4$ & $993$\\ $f_2(1270)$ & $1275.4\pm 1.2$ &
$185.1^{+3.4}_{-2.6}$ & $4^3_1$ & $[6\pi+2]$ & $[1256]$\\ $f_1(1285)$
& $1281.9\pm 0.6$ & $24.0\pm 1.2$ & $4_1$ & $21.2$ & $1277$\\ & & &
$4^2_1$ & $(21.4)$ & $(1289)$\\ $f_1(1420)$ & $1426.3\pm 1.1$ &
$55.5\pm 2.9$ & $5_1$ & $24.2$ & $1454$\\$\{f_2(1430)$ & $\approx
1430\}$~\footnotemark[4] & & $5_1$ & $24.2$ & $1454+\delta'$\\
$f_0(1370)$ & $1200-1500$ & $200-500$ & $3_1*0_1$ & $(24.7)$ &
$(1484)$\\$f_0(1500)$ & $1507\pm 5$ & $109\pm 7$ & $5_2$ & $24.9$ &
$1496$\\ $\{f_1(1510)$ & $1518\pm 5$ & $73\pm 25\}$ & $5_2$ & $24.9$ &
$1496+\delta$\\$f'_2(1525)$ & $1525\pm 5$ & $76\pm 10$ & $5_2$ &
$24.9$ & $1496+3\delta$\\ $\{f_2(1565)$ & $1546\pm 12$ & $126\pm 12\}$
& $5^2_1$ & $(25.9)$ & $(1555)$\\ $\{f_2(1640)$ & $1638\pm 6$ &
$99^{+28}_{-24}\}$ & $6^3_3$ & $((27.3))$ & $((1638))$\\
\multicolumn{6}{c}{.\dotfill.}\\$f_0(1710)$ & $1713\pm 6$ & $125\pm
10$ & $6^3_2$ & $((28.6))$ & $((1714))$\\ & & & $3_1\#3_1^*$ & $28.9\
(30.5)$ & $1732\ (1827)$\\ & & & $3_1\#3_1$ & $29.1\ (30.5)$ & $1744\
(1827)$\\ & & & $6_2$ & $29.2$ & $1750$\\ & & & $6_1$ & $29.3$ &
$1756$\\ & & & $6_3$ & $30.5$ & $1827$\\ & & & $7_1$ & $30.9$ &
$1850$\\ & & & $8_{19}$ & $31.0$ & $1856$\\ & & & $8_{20}$ & $32.7$ &
$1957$\\ $f_2(2010)$ & $2011^{+60}_{-80}$ & $202\pm 60$ & $7_2$ &
$33.2$ & $1986$\\ $f_4(2050)$ & $2025\pm 8$ & $194\pm 13$ & $8_{21}$ &
$33.9$ & $2028$\\ & & & $8_1$ & $37.0$ & $2211$\\ & & & $10_{161,162}$
& $37.6$ & $2247$\\ $f_2(2300)$ & $2297\pm 28$ & $149\pm 40$ &
$8_{18}$, $9_1$ & $38.3$ & $2288$\\ $f_2(2340)$ & $2339\pm 60$ &
$319^{+80}_{-70}$ & $9_2$ & $40.0$ & $2389$\\ & & & $10_1$ & $44.8$ &
$2672$\\ {\rule[-3mm]{0mm}{3mm} } & & & $11_1$ & $47.0$ &
$2802$\\\hline\hline
\end{tabular}
\end{center}
\end{table}
\end{center}

\footnotetext[1]{Notation $n^l_k$ means a link of $l$ components with
$n$ crossings, and occurring in the standard table of links (see
e.g. \protect\cite{Rolfsen}) on the $k^\textrm{th}$ place. $K\#K'$
stands for the knot product (connected sum) of knots $K$ and $K'$ and
$K*K'$ is the link of the knots $K$ and $K'$.} \footnotetext[2]{Values
are from \protect\cite{knot-energies} except for our exact
calculations of $2^2_1$ and $4^3_1$ in square brackets, our analytic
estimates given in parentheses, and our rough estimates given in
double parentheses.}  \footnotetext[3]{$E(G)$ is obtained from
$\ve(K)$ using the fit (\ref{fit1}).} \footnotetext[4]{States in
braces are not in the PDG summary tables.}







\end{document}



















