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\title{On the existence of finite-energy classical glueballs\\ in
gauge theories with sources}

\author{Roman V. Buniy}
\email{roman.buniy@vanderbilt.edu}
\affiliation{Vanderbilt University, Nashville, TN 37235}
\author{Thomas W. Kephart}
\email{kephartt@ctrvax.vanderbilt.edu}
\affiliation{Vanderbilt University, Nashville, TN 37235}
\date{March 21, 2003}

\begin{abstract}
We show that for classical gauge theories with sources, an initial
lump of gauge and matter fields evolves into either a lump of matter
fields with no gluonic content or a configuration where the gauge and
matter fields are periodic in time.
\end{abstract}

\pacs{}

\maketitle


Constraints imposed by requiring non-radiating asymptotic behavior
restrict the form of classical solutions (solitons or lumps) in gauge
theories with sources. While these solitons are expected to be highly
complicated objects, we will still be able to find general results by
placing mild requirements on the fields.

Despite their non-linearity pure Yang-Mills fields do not hold
themselves together to give finite-energy solutions that are either
time-independent\cite{Coleman,Deser:1976wq} or periodic in time
\cite{Pagels:1977ck}. An even stronger result was proved: the only
finite-energy non-singular non-radiating solutions with arbitrary time
dependence are vacuum solutions~\cite{Coleman:1977hd}. This ``no-go''
theorem forbids the existence of classical glueballs in a pure
Yang-Mills system. The presence of sources changes this situation. In
this note we investigate conditions which are imposed on the
energy-momentum of gauge fields by the existence of such classical
lumps. The result is applicable to a wide class of gauge theories. Our
particular interest~\cite{Buniy:2002yx} is in glueballs in QCD, where
the $SU_C(3)$ gauge fields couple to quarks and confinement is
involved.  This work is a zeroth order analysis towards understanding
this complete physical situation.

We consider classical field theory with the Lagrangian density \ba
{\cal L}=-\fr{1}{4}F^a_{\mu\nu}F_a^{\mu\nu}+{\cal L}_m\ea which
includes gauge fields $A^a_\mu$ via field strengths $F^a_{\mu\nu}$
together with the matter fields via ${\cal L}_m$~\footnote{In QCD or
the Standard Model ${\cal L}_m$ depends on vectors, scalars, and
spin-$\fr{1}{2}$ fermions in such a way that ${\cal L}$ is
renormalizable. However, the results given here are classical, so we
need not require renormalizability.}; our metric is $(+,-,-,-)$.  The
explicit form of the functional ${\cal L}_m$ will not be needed.

We will be interested in the part of the energy-momentum tensor for
the gauge fields \emph{alone}. In terms of electric $E^a_i=F^a_{i0}$
and magnetic $H^a_i=\fr{1}{2}\epsilon_{ijk}F^a_{jk}$ fields, the
components of this symmetric tensor are
\ba&&\theta_A^{00}=\fr{1}{2}(E^a_i E^a_i+H^a_i H^a_i),\ \ \ \ \
\theta_A^{0i}=\epsilon^{ijk}E^a_j
H^a_k,\nn\\&&\theta_A^{ij}=\fr{1}{2}\delta^{ij}\left(E^a_k E^a_k+H^a_k
H^a_k\right)-E^{ai} E^{aj}-H^{ai} H^{aj}.\label{theta}\ea $\theta_A$
is traceless, as in the sourceless case, but it is not conserved,
$\p_\mu\theta_A^{\mu\nu}=J^a_{A\mu}F^{\mu\nu}_a$, when the matter
currents $J^a_{A\mu}=-(\p{\cal L}_m/\p A^\mu_a)$ are present.

It is useful to study the time and radial dependence of the quantities
\ba G^\mu(t,R)=\int_{r\le R}\d^3x\,f(r)\theta_A^{0\mu},\label{G}\ea
where $r=|{\mathbf x}|$. Taking the time derivative of (\ref{G}), we
find \ba\p_0 G^\mu(t,R)=\int_{r\le R}\d^3x\left(f J^a_{A\nu}
F^{\nu\mu}_a+\theta_A^{i\mu}\p_i f\right)-\int_{r=R}\d^2 S_i
f\theta_A^{i\mu}.\label{dG}\ea The only requirements imposed on the
time-independent scalar function $f(r)$ is that the integral in
Eq.~(\ref{G}) converges and that the surface term in Eq.~(\ref{dG})
can be ignored~\footnote{Instead of the scalar function $f(r)$ we
could have used higher tensor moments $x^{i_1}\ldots x^{i_n}g(r)$ with
a similar result.}.

For now we restrict our attention to lumps at rest; we can choose a
frame such that this is the case for any lump moving slower than the
speed of light. To be localized, non-radiating fields of lumps at rest
must have an asymptotic behavior \ba
\lim_{r\to\infty}r^{3/2+\delta}F^a_{\mu\nu}(t,{\mathbf
x})=0,\label{limitF}\ea which is uniform in time~\footnote{For a lump
at rest, outside of a sphere with the radius \emph{independent} of
time, fields approach their asymptotic values with a given accuracy
for \emph{all} times. This guarantees absence of the outgoing
radiation~\cite{Coleman:1977hd}. For a lucid discussion of uniform
convergence see R.~Courant, \emph{Differential and Integral Calculus}
(Interscience, New York, 1937).}; here
$0<\delta<\frac{1}{2}$. Likewise, matter fields do not radiate their
energy away when their currents satisfy a similar limit \ba
\lim_{r\to\infty}r^{5/2+\delta} J^a_{A\mu}(t,{\mathbf
x})=0,\label{limitJ}\ea which also must be uniform in time.

As a result of these assumptions, for a sphere of large radius $R$,
the surface term vanishes and the right-hand side of Eq.~(\ref{dG})
goes uniformly in time to \ba K^\mu(t)=\int\d^3x\left(f J^a_{A\nu}
F^{\nu\mu}_a+\theta_A^{i\mu}\p_i f\right).\label{K}\ea This means that
for any positive $\epsilon$ there exists a \emph{time-independent}
$R_0$ such that for any $R>R_0$ we have \ba\left|\p_0
G^\mu(t,R)-K^\mu(t)\right|<\epsilon\ea for all $t\ge 0$. For such $R$,
\ba\left|G^\mu(t,R)-G^\mu(0,R)-\int_0^t\d
t'\,K^\mu(t')\right|<\epsilon t.\label{ineq1}\ea

On the other hand, from the properties of the energy-momentum tensor,
Eq.~(\ref{theta}), we deduce $|\theta_A^{0i}|\le\theta_A^{00}$ and so
$G^\mu(t,R)$ is bounded, \ba|G^\mu(t,R)|\le\max_{r\le
R}|f(r)|\max_{t\ge 0}E_A(t,R)=\Delta(R);\label{ineq2}\ea $E_A(t,R)$
being the energy of the gauge fields inside the sphere of radius
$R$. $\Delta(R)$ is a time-independent and bounded function of $R$.

Without a detailed form for the function $K^\mu(t)$ we cannot conclude
whether the bounds imposed on the function $G^\mu(t,R)$ are consistent
or inconsistent. There are, however, simple cases where inconsistency
is obvious: \begin{enumerate}
\item When $|K^\mu(t)|$ diverges as $t$ tends to infinity,
Eqs.~(\ref{ineq1}) and (\ref{ineq2}) cannot possibly be true
simultaneously.
\item If for large $t$, $K^\mu(t)$ approaches a nonzero constant
$K^\mu(\infty)$, the integral in Eq.~(\ref{ineq1}) asymptotically
approaches $K^\mu(\infty)t$. If we now choose
$\epsilon<\left|K^\mu(\infty)\right|$, then bounds (\ref{ineq1}) and
(\ref{ineq2}) still cannot be consistent for all $t\ge 0$ since
$\Delta(R)$ is finite (see Figure~\ref{figure}). It follows that the
only resolution is to set $K^\mu(\infty)=0$. Since the function $f(r)$
in Eq.~(\ref{K}) is general (up to satisfying the requirements
described above), we conclude that the gauge fields must
asymptotically approach zero as $t$ increases.  From the equations of
motion it then follows that the matter currents approach zero as
well. Note, since the matter currents $J_A$ vanish for the pure matter
Lagrangian, our conclusion does not limit the form of pure matter
solitons. For example, $J_\phi\not =0$ is not required.
\item If for large $t$, the function $K^\mu(t)$ approaches a periodic
function with oscillations around a constant $K^\mu(\infty)$, we
similarly conclude that $K^\mu(\infty)=0$, otherwise (\ref{ineq1}) and
(\ref{ineq2}) are again violated, and it follows that for large $t$
the gauge fields and the matter currents must periodically oscillate
around zero. Consequently, the matter fields becomes periodic in time
for large $t$.\end{enumerate} As a result, an initial lump of gauge
and matter fields can evolve either into a lump of matter fields with
no gluonic content (this means, in particular, that there cannot be
finite-energy time-independent classical glueballs, even if we include
matter), or into a time-periodic configuration of gauge and matter
fields.  For pure Yang-Mills theory, the role of $K^\mu(t)$ is played
by the conserved total energy, and our result reproduces Coleman's
conclusion in Ref.~\cite{Coleman:1977hd}.



\begin{figure}
\includegraphics{glueballsBK}
\caption{For $t>T$ the bound (\ref{ineq1}) is valid and (\ref{ineq2})
is not.\label{figure}}
\end{figure}


We now turn to the case of lumps moving with the speed of light and
modify the argument in Ref.~\cite{Coleman:1977hd} to include
sources. By choosing the 3-axis in the direction of the momentum, we
make the fields transverse with their components related by
$E_\alpha=\epsilon_{\alpha\beta}H_\beta$. In terms of light-cone
variables $x^{\pm}=x^0\pm x^3$, the only non-vanishing components of
the field-strength are $F_{+\alpha}=-2E_\alpha$. Since $F_{12}=0$, we
can perform a gauge transformation depending on $x^1$ and $x^2$ to set
$A_1$ and $A_2$ to zero. From $F_{-\alpha}=0$ it now follows that
$A_-$ is independent of $x^1$ and $x^2$, so we make a gauge
transformation depending on $x^-$ to set $A_-=0$. Next, from the
equations of motion we find $J_{A-}=0$ and \ba&&\p^\alpha\p_\alpha
A_+=J_{A+},\label{eq1}\\&&\p^+\p_\alpha A_++[A^+,\p_\alpha
A_+]=-J_{A\alpha}.\label{eq2}\ea Eq.~(\ref{eq2}) follows from
Eq.~(\ref{eq1}) by differentiation and using covariant conservation of
the current. We are left with only Eq.~(\ref{eq1}) to solve and its
general solution is \ba A_+(x^+,x^1,x^2)=\tilde{A}_+(x^+,x^1,x^2)
+\frac{1}{4\pi}\int\d\xi^1\d\xi^2\,J_{A+}(x^+,\xi^1,\xi^2)
\log\left[{(x^1-\xi^1)}^2+{(x^2-\xi^2)}^2\right],
\label{Poisson}\ea where $\tilde{A}_+$ is a solution 
to the Laplace equation $(\p^1\p_1+\p^2\p_2)\tilde{A}_+=0$. It is well
known that the only non-singular solution to this equation is a
function $\tilde{A}_+(x^+)$ of $x^+$ alone. For a non-singular
current, the second term in Eq.~(\ref{Poisson}) can have a singularity
only at infinity. For large $r$, the second term in
Eq.~(\ref{Poisson}) is asymptotically $\frac{1}{2\pi}I\log{r}$, where
\ba
I=\int\d\xi^1\d\xi^2\,J_{A+}=\oint_C(F_{+2}\d\xi^1-F_{+1}\d\xi^2),\ea
and $C$ is an infinite contour in the $(\xi^1,\xi^2)$ plane, which
encloses the sources. Observe that the contour integral form for $I$
is the gauge field flux in the transverse plane. Fields are
non-singular only when the transverse flux vanishes, $I=0$. The
asymptotic behavior in Eq.~(\ref{limitF}) ensures this. Thus we cannot
eliminate, but can constrain the form of lumps moving with the speed
of light.

To summarize, only specific types of localized solutions can exist in
classical gauge theories with sources: either they are periodic in
time or only matter fields are present for large $t$. Also, we have
found only mild restrictions on massless solutions. These conclusions
were reached for classical systems and they do not restrict forms of
possible quantum lumps; in fact, periodic solutions are just what one
would expect for stationary quantum states.




\bibliography{glueballsBK}



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