\documentstyle[aps,prd,psfig,preprint,floats,tighten]{revtex}
\begin{document}
\draft
\preprint{OKHEP--99--04}

\title{Direct and Indirect Searches for Low-Mass Magnetic 
Monopoles\thanks{Paper contributed to Kurt Haller's Festschrift.}}

\author{Leonard Gamberg,\thanks{Electronic address: gamberg@mail.nhn.ou.edu}
George R. Kalbfleisch,\thanks{Electronic address: grk@mail.nhn.ou.edu} and
Kimball A. Milton\thanks{Electronic address: milton@mail.nhn.ou.edu}}
\address{Department of Physics and Astronomy, University of Oklahoma, 
Norman, OK
73019, USA}

\date{\today}
\maketitle

\begin{abstract}
Recently, there has been renewed interest in the search for low-mass
magnetic monopoles.  At the University of Oklahoma we are performing
an experiment (Fermilab E882)
 using material from the old D0 and CDF detectors to set
limits on the existence of Dirac monopoles of masses of the order of
500 GeV.  To set such limits, estimates must be made of the production
rate of such monopoles at the Tevatron collider, and of the binding strength
of any such produced monopoles to matter.  Here we sketch the still
primitive theory of such interactions, and indicate why we believe a
credible limit may still be obtained. 
On the other hand, there have been proposals that the classic Euler-Heisenberg
Lagrangian together with duality
 could be employed to set limits on magnetic monopoles having
masses less than 1 TeV, based on virtual, rather than real processes.
The D0 collaboration at Fermilab has used
such a proposal to set mass limits based on the nonobservation of pairs of
photons each with high transverse momentum.  We critique the 
underlying theory, by showing that the cross section 
violates unitarity at the quoted limits 
and is unstable with respect to radiative corrections. We therefore believe that
no significant limit can be obtained from the current experiments,
based on virtual monopole processes.
\end{abstract}
\pacs{PACS number(s): 14.80.Hv,13.85.Rm,12.20.Ds,11.30.Fs}

\section{Introduction}
The notion of magnetic charge has intrigued physicists since Dirac \cite{dirac}
showed that it was consistent with quantum mechanics provided a suitable 
quantization condition was satisfied: For a monopole of magnetic charge $g$ in 
the presence of an electric charge $e$, that quantization condition is
(in this paper we use rationalized units)
\begin{equation}
{eg\over4\pi}={N\over2}\hbar c,
\label{quant}
\end{equation}
where $N$ is an integer.  For a pair of dyons, that is, particles carrying
both electric and magnetic charge, the quantization condition is replaced
by \cite{schwinger,schwingera}
\begin{equation}
{e_1g_2-e_2g_1\over4\pi}={N\over 2}\hbar c,
\label{squant}
\end{equation}
where $(e_1,g_1)$ and $(e_2,g_2)$ are the charges of the two dyons.\footnote{An
additional factor of 2 appears on the right hand side of these conditions if
a symmetrical solution is adopted---see Eq.~(\ref{quant2}) below.}

With the advent of ``more unified'' non-Abelian theories, classical composite
monopole solutions were discovered \cite{nonabel}.  The mass of these monopoles
would be of the order of the relevant gauge-symmetry breaking scale,
which for grand unified theories is of order
$10^{16}$ GeV or higher.  But there are models where the electroweak symmetry
breaking can give rise to monopoles of mass $\sim 10$ TeV \cite{ewmono}.
 Even the latter are not yet accessible to accelerator experiments, 
 so limits on heavy monopoles depend either on cosmological considerations
\cite{cosmo}, or detection of cosmologically produced (relic) monopoles
impinging upon the earth or moon \cite{relic}.
However, {\it a priori}, there is no reason that Dirac/Schwinger monopoles
or dyons of arbitrary mass might not exist: In this respect, it 
is important to set limits below the 1 TeV scale.

Such an experiment is currently in progress at the University of Oklahoma
\cite{ou}, where we expect to be able to set limits on {\it direct\/} monopole
production at Fermilab up to several hundred GeV.  This will be a substantial
improvement over previous limits \cite{prev}.  But {\it indirect\/} 
searches have
been proposed and carried out as well.  De R\'ujula \cite{DeRujula} proposed 
looking at the three-photon decay of the $Z$ boson, where the process proceeds
through a virtual monopole loop. If we use his formula \cite{DeRujula} for the
branching ratio for the $Z\to3\gamma$ process, compared to the current
experimental upper limit \cite{exp}
for the branching ratio of $10^{-5}$, we can rule out
monopole masses lower than about 400 GeV, rather than the 600 GeV quoted in
Ref.~\cite{DeRujula}.
Similarly, Ginzburg and Panfil \cite{ginz1} and more recently Ginzburg and
Schiller \cite{ginz2} considered the production of two photons with
high transverse momenta by the collision of two photons produced either
from $e^+e^-$ or quark-(anti-)quark collisions. Again the final photons are 
produced through a virtual monopole loop.  Based on this theoretical scheme,
 an experimental limit has 
appeared by the D0 collaboration \cite{d0}, which sets the following bounds
on the monopole mass $M$:
\begin{equation}
{M\over N}>\left\{\begin{array}{cc}
610 \mbox{ GeV}&\mbox{ for } S=0\\
870 \mbox{ GeV}&\mbox{ for } S=1/2\\
1580 \mbox{ GeV}&\mbox{ for } S=1 \end{array}\right.,
\end{equation}
where $S$ is the spin of the monopole.
It is worth noting that a mass limit of 120 GeV for a Dirac monopole has been
set by Graf, Sch\"afer, and Greiner \cite{graf}, based on the monopole
contribution to the vacuum polarization correction to the muon anomalous
magnetic moment. (Actually, we believe that the correct limit, obtained
from the well-known textbook formula \cite{js1} for the $g$-factor correction
due to a massive Dirac particle is 60 GeV.)

The purpose of the present paper is to, first, describe the key theoretical
elements necessary for the establishment of a direct limit for production
of monopoles at Fermilab: An estimate of the production cross section for
monopole-antimonopole pairs at the collider must be made, and then an
estimate of the binding probability of such produced monopoles
with matter must be given, so that we can predict how many monopoles would
be bound to the detector elements that are run through our induction detector.
Such estimates were given in our proposal to Fermilab; the complete analysis
will be given in the experimental papers to follow.  Here the emphasis will
be on the elementary processes involved.

A secondary purpose of this paper is to critique the theory of 
Refs.~\cite{DeRujula},
\cite{ginz1}, \cite{ginz2}, and \cite{graf}, and thereby demonstrate
that experimental limits, such as that of Ref.~\cite{d0}, based on virtual
processes are unreliable. We will show that the theory is based on 
a naive application of electromagnetic duality; the resulting
cross section cannot be valid because
unitarity is violated for monopole masses as low as the quoted limits, and
the process is subject to enormous, uncontrollable radiative corrections.
It is not correct, in any sense, as Refs. \cite{ginz2} and \cite{d0}
state, that the effective expansion parameter is $g\omega/M$, where $\omega$
is some external photon energy; rather, the factors of $\omega/M$ emerge
kinematically from the requirements of gauge invariance at the one-loop
level.  If, in fact, a correct calculation introduced such 
additional factors of $\omega/M$, arising from the complicated coupling
of magnetic charge to photons, we argue that no limit could be deduced
for monopole masses from the current experiments. It may even be the
case, based on preliminary field-theoretic calculations, that processes
involving the production of real photons vanish.

\section{Eikonal Approximation for Electron- (or Quark-) Monopole Scattering}

It is envisaged that if monopoles are sufficiently light, they would be
produced by a Drell-Yan type of process occurring in $p\overline p$ collisions
at the Tevatron.  The difficulty is to make a believable estimate of the
elementary process $q\overline q\to\gamma^*\to M\overline M$, where $q$
stands for quark and $M$ for magnetic monopole.   It is not known how
to calculate such a process using perturbation theory; indeed, perturbation
theory is inapplicable to monopole processes because of the quantization
condition (\ref{quant}).  It is only because of that consistency condition
that the Dirac string, for example, disappears from the result.  

Only formally has it been shown that the quantum field theory of electric
and magnetic charges is independent of the string orientation, or, more
generally, is gauge and rotationally invariant \cite{schwinger,zwanziger}.
It has not yet proved possible to develop generally consistent schemes for
calculating processes involving real or virtual magnetically charged particles.
Partly this is because a sufficiently general field theoretic formulation
has not yet been given; this defect will be addressed elsewhere \cite{gm}.
However, the nonrelativistic scattering of magnetically charged particles
is well understood\footnote{Contrary to the statement in the second
reference in Ref.~\cite{ginz2}, the nonrelativistic calculation is
exact, employs the quantization condition, and uses no ``unjustified
extra prescription.''}
 \cite{dyondyon,dm}.  Thus it should not be surprising that
an eikonal approximation gives a string-independent result for electron-monopole
scattering provided the condition (\ref{quant}) is satisfied.  More than
two decades ago Schwinger proposed \cite{schwingera} and
Urrutia carried out such a calculation \cite{urrutia}.  
Indeed, this subject has had arrested development. Since this is the
only successful field-theoretic calculation yet presented, it may be useful
to review it here. (A more detailed discussion will be presented in
Ref.~\cite{gm}.)

The interaction between electric ($J^\mu$) and magnetic (${}^*J^\mu$) 
currents is given by
\begin{equation}
W^{(eg)}=
-\epsilon_{\mu\nu\sigma\tau}\int(dx)(dx')(dx'')J^\mu(x)\partial^\sigma
D_+(x-x')f^\tau(x'-x''){}^*J^\nu(x'').
\label{weg}
\end{equation}
Here $D_+$ is the usual photon propagator, and
the arbitrary ``string'' function $f_\mu(x-x')$ satisfies
\begin{equation}
\partial_\mu f^\mu(x-x')=\delta(x-x').
\end{equation}
It turns out to be convenient for this
calculation to choose a symmetrical string, which
satisfies
\begin{equation}
f^\mu(x)=-f^\mu(-x).
\end{equation}
In the following we choose a string lying along the straight line $n^\mu$, in
which case the function may be written as a Fourier transform
\begin{equation}
f_\mu(x)={n_\mu\over 2i}\int{(dk)\over(2\pi)^4}e^{ikx}\left({1\over n\cdot k
-i\epsilon}+{1\over n\cdot k+i\epsilon}\right).
\label{string}
\end{equation}

In the high-energy, low-momentum-transfer regime, the scattering amplitude
between electron and monopole is obtained from Eq.~(\ref{weg}) by inserting
the classical currents,
\begin{mathletters}
\begin{eqnarray}
J^\mu(x)&=&e\int_{-\infty}^\infty d\lambda \,{p_2^\mu\over m}\,
\delta\left(x-{p_2\over m}\lambda\right),\\
{}^*J^\mu(x)&=&e\int_{-\infty}^\infty d\lambda'\,
{p_2^{\prime\mu}\over M}\,\delta\left(x+b-{p'_2\over M}\lambda'\right),
\end{eqnarray}
\end{mathletters}
where $m$ and $M$ are the masses of the electron and monopole, respectively.
Let us choose a coordinate system such that
the incident momenta of the two particles have spatial components
along the $z$-axis:
\begin{equation}
p_2=(p,0,0,p),\quad p_2'=(p,0,0,-p),
\end{equation}
and the impact parameter lies in the $xy$ plane:
\begin{equation}
b=(0,{\bf b},0).
\end{equation}
Apart from kinematical factors, the scattering amplitude is simply the
transverse Fourier transform of the eikonal phase,
\begin{equation}
I({\bf q})=\int d^2b\, e^{-i{\bf b\cdot q}}\left(e^{i\chi}-1\right),
\label{eikonal}
\end{equation}
where $\chi$ is simply $W^{(eg)}$ with the classical currents substituted,
and $\bf q$ is the momentum transfer.

First we calculate $\chi$; it is immediately seen to be, if $n^\mu$ has
no time component,
\begin{equation}
\chi={eg\over2}\int{d^2k_\perp
\over(2\pi)^2}{{\bf \hat z\cdot(\hat n\times k_\perp})\over k_\perp^2-i\epsilon}
e^{i{\bf k_\perp\cdot b}}\left({1\over {\bf \hat n\cdot k_\perp}-i\epsilon}
+{1\over {\bf \hat n\cdot k_\perp}+i\epsilon}\right),
\label{chi}
\end{equation}
where $\bf k_\perp$ is the component of the photon momentum perpendicular
to the $z$ axis.
From this expression we see that the result is independent of the angle $\bf n$
makes with the $z$ axis.  We next use proper-time representations for the 
denominators in Eq.~(\ref{chi}),
\begin{mathletters}
\begin{eqnarray}
{1\over k_\perp^2}&=&\int_0^\infty ds\, e^{-sk_\perp^2},\\
{1\over {\bf \hat n\cdot k_\perp}-i\epsilon}+
{1\over {\bf \hat n\cdot k_\perp}+i\epsilon}&=&{1\over i}\left[\int_0^\infty
d\lambda-\int_{-\infty}^0 d\lambda\right]e^{i\lambda {\bf \hat n\cdot k_\perp}}
e^{-|\lambda|\epsilon}.
\end{eqnarray}
\end{mathletters}
We then complete the square in the exponential and perform the Gaussian
integration to obtain
\begin{equation}
\chi={eg\over4\pi}{\bf \hat z\cdot(\hat n\times b)}\int d\lambda{1\over
(\lambda+{\bf b\cdot \hat n})^2+b^2-({\bf b\cdot\hat n})^2},
\end{equation}
or
\begin{equation}
\chi={eg\over2\pi}\tan^{-1}\left({\bf \hat n\cdot b}\over{\bf \hat z\cdot(b
\times \hat n})\right).
\end{equation}
Because $e^{i\chi}$ must be continuous when $\bf \hat n$ and $\bf b$ lie in the
same direction, we must have the Schwinger quantization condition for
an infinite string, 
\begin{equation}
eg=4\pi N,
\label{quant2}
\end{equation}
where $N$ is an integer.

To carry out the integration in Eq.~(\ref{eikonal}), choose $\bf b$ to
make an angle $\psi$ with $\bf q$, and the projection of $\bf \hat n$ in the
$xy$ plane to make an angle $\phi$ with $\bf q$; then
\begin{equation}
\chi={eg\over2\pi}(\psi-\phi-\pi/2).
\end{equation}
To avoid the appearance of a Bessel function, we first integrate over
$b=|{\bf b}|$, and then over $\psi$:
\begin{eqnarray}
I({\bf q})&=&\int_0^{2\pi}d\psi\int_0^\infty b\,db\,e^{-ibq(\cos\psi-i\epsilon)}
e^{2iN(\psi-\phi-\pi/2)}\nonumber\\
&=&{4\over i}{e^{-2iN(\phi+\pi/2)}\over q^2}
\oint_C{dz\,z^{2N-1}\over(z+1/z-i\epsilon)^2}
\nonumber\\
&=&-{4\pi N\over q^2}e^{-2iN\phi},
\end{eqnarray}
where $C$ is a unit circle about the origin, and
where again the quantization condition (\ref{quant2}) has been used.
Squaring this and putting in the kinematical factors we obtain Urrutia's
result \cite{urrutia}
\begin{equation}
{d\sigma\over dt}={(eg)^2\over 4\pi}{1\over t^2}, \quad t=q^2,
\end{equation}
which is exactly the same as the nonrelativistic, small angle result
found, for example, in Ref.~\cite{dyondyon}.
This calculation, however, points the way toward a proper relativistic
treatment, and will be extended to the crossed process, quark-antiquark
production of monopole-antimonopole pairs elsewhere.

\section{Binding of monopoles to matter}
Once the monopoles are produced in a collision at the Tevatron, they travel
through the detector, losing energy in a well-known manner (see, e.g., 
Ref.~\cite{ce}), presumably ranging out, and
eventually binding to matter in the detector (Be, Al, Pb,
for example).  The purpose of this section is to review the theory of the
binding of magnetic charges to matter.

We consider the binding of a monopole of magnetic charge $g$
to a nucleus of charge $Ze$, mass ${\cal M}=A m_p$, and magnetic moment
\begin{equation}
\mbox{\boldmath{$\mu$}}={e\over m_p}\gamma{\bf S},
\end{equation}
$\bf S$ being the spin of the nucleus.  (We will assume here that the
monopole mass $\gg {\cal M}$, which restriction could be easily removed.)
Other notations for the magnetic moment are
\begin{equation}
\gamma=1+\kappa={g_S\over2}.
\end{equation}
The charge quantization condition is given by Eq.~(\ref{quant}).
Because the nuclear charge is $Ze$, the relevant angular momentum
quantum number is [recall $N$ is the magnetic charge quantization number
in Eq.~(\ref{quant})]
\begin{equation}
l={NZ\over2}.
\end{equation}
We do not address the issue of dyons \cite{Wu}, which for the correct sign
of the electric charge will always bind electrically to nuclei.

\subsection{Nonrelativistic binding for $S=1/2$}
In this subsection we follow the early work of Malkus \cite{Malkus} and
the more recent paper of Bracci and Fiorentini \cite{Bracci}.
(There are also the results given in Ref.~\cite{Sivers}, but 
this reference seems to contain errors.)

The neutron ($Z=0$) is a special case.  Binding will occur in the
lowest angular momentum state, $J=1/2$, if 
\begin{equation}
|\gamma|>{3\over2N}
\end{equation}
Since $\gamma_n=-1.91$, this condition is satisfied for all $N$.

In general, it is convenient to define a reduced gyromagnetic ratio,
\begin{equation}
\hat\gamma={A\over Z}\gamma,\quad \hat\kappa=\hat\gamma-1.
\end{equation}
This expresses the magnetic moment in terms of the mass and charge of
the nucleus.
Binding will occur in the special lowest angular momentum state
$J=l-{1\over2}$ if 
\begin{equation}
\hat \gamma>1+{1\over4l}.
\label{lowest}
\end{equation}
Thus binding can occur here only if the anomalous magnetic moment
 $\hat\kappa>1/4l$.  The proton, with $\kappa=1.79$, will bind.

Binding can occur in higher angular momentum states $J$ if and only if
\begin{equation}
|\hat\kappa|>\kappa_c={1\over l}\left|J^2+J-l^2\right|.
\label{kappac}
\end{equation}
For example, for $J=l+{1\over2}$, $\kappa_c=2+3/4l$, and for
$J=l+{3\over2}$, $\kappa_c=4+15/4l$.  Thus ${}^3_2$He, which is spin 1/2,
will bind in the first excited angular momentum state because $\hat\kappa
=-4.2$.

Unfortunately, to calculate the binding energy, one must regulate
the potential at $r=0$.  The results shown in Table 1 assume a hard
core.

\subsection{Nonrelativistic binding for general $S$}
The reference here is \cite{Olaussen}.  The assumption made here is that
$l\ge S$.  (There are only 3 exceptions, apparently: ${}^2$H,
${}^8$Li, and ${}^{10}$B.)

Binding in the lowest angular momentum state $J=l-S$ is 
given by the same criterion (\ref{lowest}) as in spin 1/2.
Binding in the next state, with $J=l-S+1$ occurs if
$\lambda_{\pm}>{1\over4}$ where
\begin{equation}
\lambda_{\pm}=\left(S-{1\over2}\right){\hat\gamma\over S}l
-2l-1\pm\sqrt{(1+l)^2+(2S-1-l){\hat\gamma\over S}l+
{1\over4}l^2\left({\hat\gamma\over S}\right)^2}.
\end{equation}
 The previous result for $S=1/2$ is recovered, of course.
$S=1$ is a special case: Then $\lambda_-$ is always negative, while
$\lambda_+>{1\over4}$ if $\hat\gamma>\gamma_c$, where
\begin{equation}
\gamma_c={3\over4l}{(3+16l+16l^2)\over9+4l}.
\end{equation}

For higher spins, both $\lambda_\pm$ can exceed $1/4$:  
\begin{eqnarray}
\lambda_+>{1\over4} &\mbox{ for }& \hat\gamma>\gamma_{c-}\\
\lambda_->{1\over4} &\mbox{ for }& \hat\gamma>\gamma_{c+}
\end{eqnarray}
where for $S={3\over2}$
\begin{equation}
(\gamma_c)_\mp={3\over4l}(6+4l\mp\sqrt{33+32l}).
\end{equation}
For ${}_4^9$Be, for which $\hat\gamma=-2.66$, we cannot have binding
because $3>\gamma_{c-}>1.557$, $3<\gamma_{c+}<8.943$, where the
ranges come from considering different values of $N$ from 1 to $\infty$.
For $S={5\over2}$,
\begin{equation}
(\gamma_{c})_\mp={36+28l\mp\sqrt{1161+1296l+64l^2}\over12l}.
\end{equation}  So ${}^{27}_{13}$Al will bind in either of
these states, or the lowest angular momentum state, because
$\hat\gamma=7.56$, and 
$1.67>\gamma_{c-}>1.374$, $1.67<\gamma_{c+}<4.216$.


\subsection{Relativistic spin-1/2}
Kazama and Yang treated the Dirac
equation \cite{Kazama}.  See also \cite{oooo} and \cite{Wu}.


In addition to the bound states found nonrelativistically, deeply
bound states, with $E_{\rm binding}=M$ are found. 
These states always exist for $J\ge l+1/2$.   For $J=l-1/2$, these
 (relativistic) $E=0$ bound
states exist only if $\kappa>0$. 
 Thus (modulo the question of
form factors) Kazama and Yang \cite{Kazama}
expect that electrons can bind to monopoles.
(We suspect that one must take the existence of these deeply bound
states with a fair degree of skepticism.  See also \cite{Walsh}.)

As expected, for $J=l-1/2$
we have weakly bound states only for $\kappa>1/4l$, which is the same
as the nonrelativistic condition (\ref{lowest}), and for $J\ge l+1/2$, only if
$|\hat\kappa|>\kappa_c$, where $\kappa_c$ is given in Eq.~(\ref{kappac}).



\subsection{Relativistic spin-1}
Olsen, Osland, and Wu considered this situation \cite{Olsen}.

In this case, no bound states exist, unless an additional interaction
is introduced  (this is similar to what happens nonrelativistically,
because of the bad behavior of the Hamiltonian at the origin).
Bound states are found if an ``induced magnetization'' interaction
(quadratic in the magnetic field)
is introduced.  Binding is then found for the lowest angular momentum
state $J=l-1$ again if $\hat\kappa>1/4l$.  For the higher angular
momentum states, the situation is more complicated:
\begin{itemize}
\item for $J=l$: bound states require $l\ge 16$, and
\item for $J\ge l+1$: bound states require $J(J+1)-l^2\ge 25$.
\end{itemize}
But these results are probably highly dependent on the form of the
additional interaction.  The binding energies found are inversely
proportional to the strength $\lambda$ of this extra interaction.

\begin{table}[ht]
\centering
\begin{tabular}{||c|ccccccc||}\hline
Nucleus&Spin&$\gamma$&$\hat\gamma$&$J$&$E_b$&Notes&Ref\\ \hline
$n$&${1\over2}$&$-1.91$&&${1\over2}$&350 keV&NR,hc&\cite{Sivers}\\
${}_1^1$H&${1\over2}$&2.79&2.79&$l-{1\over2}=0$&15.1 keV
&NR,hc&\cite{Bracci}\\
&&&&&320 keV
&NR,hc&\cite{Sivers}\\
&&&&&50--1000 keV&NR,FF&
\cite{Olaussen}\\
&&&&&263 keV&R&
\cite{oooo}\\
${}_1^2$H&$1$&$0.857$&1.71&$l-1=0$ ($N=2$)&${130\over\lambda}$ keV&R,IM
&\cite{Olsen}\\
${}_2^3$He&${1\over2}$&$-2.13$&$-3.20$&$l+{1\over2}={3\over2}$&
13.4 keV&NR,hc&\cite{Bracci}\\
${}_{13}^{27}$Al&${5\over2}$&3.63&7.56&$l-{5\over2}=4$
&2.6 MeV&NR,FF&\cite{Olaussen}\\
${}_{13}^{27}$Al&${5\over2}$&3.63&7.56&$l-{5\over2}=4$
&560 keV&NR,hc&\cite{Goebbel}\\
${}_{48}^{113}$Cd&${1\over2}$&$-0.62$&$-1.46$&$l+{1\over2}={49\over2}$&
6.3 keV&NR,hc&\cite{Bracci}\\
\hline
\end{tabular}
\caption{Weakly bound states of nuclei to a magnetic monopole.
The angular momentum quantum number $J$ of the lowest bound state
is indicated.  In Notes, NR means nonrelativistic and R relativistic
calculations;
hc indicates an additional hard core interaction is assumed, while
FF signifies use of a form factor. IM=induced magnetization, the
additional interaction employed for the relativistic spin-1 calculation.
We use  $N=1$ except for the deuteron, where $N=2$ is required for
binding.}
\end{table}

\subsection{Remarks on binding}
Clearly, this summary indicates that the theory of monopole binding
to nuclear magnetic dipole moments is rather primitive.  The angular
momentum criteria for binding is straightforward; but in general
(except for relativistic spin 1/2) additional interactions have to
be inserted by hand to regulate the potential at $r=0$.  The results
for binding energies
clearly are very sensitive to the nature of that additional interaction.
It cannot even be certain that binding occurs in the allowed states.
In fact, however, it seems nearly certain that monopoles will bind
to all nuclei, even, for example, Be, because the magnetic field in
the vicinity of the monopole is so strong that the monopole will
disrupt the nucleus and will bind
to the nuclear, or even the subnuclear, constituents.








\subsection{Binding of monopole-nucleus complex to material lattice}

Now the question arises: Is the bound complex of nucleus and monopole
rigidly attached to the crystalline lattice of the material?
This is a simple tunneling situation.  The decay rate is estimated
by the WKB formula
\begin{equation}
\Gamma\sim {1\over a}e^{-2\int_a^b\sqrt{2{\cal M}(V-E)}},
\end{equation}
where the potential is crudely
\begin{equation}
V=-{\mu g\over4\pi r^2}-gBr,
\end{equation}
$\cal M$ is the nuclear mass $\ll$ monopole mass,
and the inner and outer turning points, $a$ and $b$ are the zeroes of
$E-V$.  Provided the following equality holds,
\begin{equation}
(-E)^{3}\gg {g^3\mu B^2\over4\pi},
\end{equation}
which should be very well satisfied, since the right hand side equals
$10^{-20} N^3$ MeV$^3$, we can write the decay rate as
\begin{equation}
\Gamma\sim N^{-1/2} 10^{23} \mbox{s}^{-1}
 \exp\left[-{8\sqrt{2}\over 3 \cdot 137}
\left({-E\over m_e}\right)^{3/2}{B_0\over NB}A^{1/2}\left(
{m_p\over m_e}\right)^{1/2}\right],
\end{equation}
where the characteristic field, defined by $eB_0=m^2_e$, is $4\times 10^9$ T.
If we put in $B=1.5$T, and $A=27$, $-E=2.6$MeV, appropriate for 
${}_{13}^{27}$Al, we have for the exponent, for $N=1$, $-2\times 10^{11}$,
corresponding to a rather long time!  To get a 10 yr lifetime, the 
binding energy would have to be only of the order of 1eV.  Monopoles
bound with kilovolt or more energies will stay around forever.

Then the issue is whether  the entire Al atom can be extracted
with the 1.5 T magnetic field present in CDF.  The answer seems to be
unequivocally NO.  The point is that the atoms are rigidly bound in a
lattice, with no nearby site into which they can jump.  A major disruption
of the lattice would be required to dislodge the atoms, which would
probably require kilovolts of energy \cite{Furneaux}.
Some such disruption was made by the monopole when it came to rest and
was bound in the material, but that disruption would be very unlikely to
be in the direction of the accelerating magnetic field.  Again, a simple
Boltzmann argument shows that any effective binding slightly bigger than 1 eV
will result in monopole trapping ``forever.''  This argument applies equally
well to binding of monopoles in ferromagnets.  If monopoles bind strongly
to nuclei there, they will not be extracted by 5 T fields, contrary to
the arguments of Goto et al. \cite{goto} 
The corresponding limits on monopoles from
ferromagnetic samples  of Carrigan et al. \cite{carrigan} are suspect.



\section{Duality and the Euler-Heisenberg Lagrangian}

Finally, let us consider the process contemplated in Refs.~\cite{ginz2}
and \cite{d0}, that is
\begin{equation}
\left(\begin{array}{ccc}
qq&\to& qq\\
\bar qq&\to&\bar qq
\end{array}\right)+\gamma\gamma,\quad \gamma\gamma\to\gamma\gamma,
\end{equation}
where the photon scattering process is given by the one-loop light-by-light
scattering graph shown in Fig.\ \ref{fig1}.  
\begin{figure}
\centerline{\psfig{figure=lbl.ps,height=3in,width=3in,angle=270}}
\caption{The light-by-light scattering graph for either an electron
or a monopole loop.}
\label{fig1}
\end{figure}
If the particle in the loop
is an ordinary electrically charged electron, this process is 
well-known \cite{js,js1,ll}. If, further, the photons involved are of very low
momentum compared the the mass of the electron, then the result may be
simply derived from the well-known Euler-Heisenberg Lagrangian \cite{eh}, which
for a spin 1/2 charged-particle loop in the presence of homogeneous 
electric and magnetic fields is\footnote{We emphasize that 
Eq.~(\ref{ehlagrangian}) is only valid when $\partial_\alpha F_{\mu\nu}=0$.}
\begin{equation}
{\cal L}=-{\cal F}-{1\over8\pi^2}\int_0^\infty {ds\over s^3}e^{-m^2 s}
\left[(es)^2{\cal G}{\mbox{Re}\cosh esX\over\mbox{Im}\cosh esX}-1-{2\over3}
(es)^2{\cal F}\right].
\label{ehlagrangian}
\end{equation}
Here the invariant field strength combinations are
\begin{equation}
{\cal F}={1\over 4}F^2={1\over2}({\bf H}^2-{\bf E}^2),\quad
{\cal G}={1\over 4}F  {}^*F={\bf E\cdot H},
\end{equation}
${}^*F_{\mu\nu}={1\over2}\epsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}$ being
the dual field strength tensor, and the argument of the hyperbolic cosine
in Eq.~(\ref{ehlagrangian}) is given in terms of
\begin{equation}
X=[2({\cal F}+i{\cal G})]^{1/2}=[({\bf H}+i{\bf E})^2]^{1/2}.
\end{equation}
If we pick out those terms quadratic, quartic and 
sextic in the field strengths,
we obtain\footnote{Incidentally, note that the coefficient of the last term is 
36 times larger than that given in Ref.~\cite{DeRujula}.}
\begin{eqnarray}
{\cal L}&=&-{1\over4}F^2+{\alpha^2\over360}{1\over m^4}
[4(F^2)^2+7(F  {}^*F)^2]\nonumber\\
&&\mbox{}-{\pi\alpha^3\over630}{1\over m^8}F^2[8(F^2)^2+13 (F {}^*F)^2]+\dots.
\label{ehlag}
\end{eqnarray}
The Lagrangian for a spin-0 and spin-1 charged particle in the loop 
is given by similar formulas which are derived in Ref.~\cite{js,js1}
and (implicitly) in Ref.~\cite{spin1}, respectively.

Given this homogeneous-field effective Lagrangian, it is a simple matter to
derive the cross section for the $\gamma\gamma\to\gamma\gamma$ process in the
low energy limit. (These results can, of course, be directly calculated
from the corresponding one-loop Feynman graph with on-mass-shell photons.
See Refs.~\cite{js1,ll}.)
Explicit results for the differential cross section are given 
by Ref.~\cite{ll}:
\begin{equation}
{d\sigma\over d\Omega}={139\over32400\pi^2}\alpha^4{\omega^6\over m^8}
(3+\cos^2\theta)^2,
\end{equation}
and the total cross section for a spin-1/2 charged particle in the loop 
is\footnote{The numerical coefficient in the total cross section for a spin-0
and spin-1 charged particle in the loop is $119/20250\pi$ and $2751/250\pi$,
respectively.  Numerically the coefficients are $0.00187$, $0.0306$, and
$3.50$ for spin 0, spin 1/2, and spin 1, respectively.}
\begin{equation}
\sigma={973\over10125\pi}\alpha^4{\omega^6\over m^8}.
\label{llcs}
\end{equation}
Here, $\omega$ is the energy of the photon in the center of mass frame,
$s=4\omega^2$. This result is valid provided $\omega/m\ll 1$.
The dependence on $m$ and $\omega$ is evident from the 
Lagrangian (\ref{ehlag}),
the $\omega$ dependence coming from the field strength tensor.
Further note that perturbative quantum corrections are small, 
because they are of relative order $3\alpha\sim10^{-2}$ \cite{dicus}.  
Processes in which
four final-state photons are produced, which may be easily calculated from
the last displayed term in Eq.~(\ref{ehlag}), are even smaller, being
of relative order $\sim\alpha^2 (\omega/m)^8$.  So light-by-light scattering,
which has been indirectly observed through its contribution to the
anomalous magnetic moment of the electron \cite{anmm}, is completely under
control for electron loops.

How is this applicable to photon scattering through a monopole loop? At first
blush this calculation seems formidable.  The interaction of a magnetically
charged particle with a photon involves a ``string,'' as described by
the function $f_\mu$ given in Eq.~(\ref{string}).
The interaction between electric and magnetic charges is given by
the complicated expression (\ref{weg}).
This coupling is equivalent to the interaction between the magnetic
current ${}^*J^\mu$ and the electromagnetic field,
\begin{eqnarray}
W_{\rm int}=\int(dx)(dx') {}^*F_{\mu\nu}(x')f^\nu(x'-x){}^*J^\mu(x).
\label{mmint}
\end{eqnarray}
From Eqs.~(\ref{weg}) and (\ref{string}) 
one obtains the relevant string-dependent 
monopole-photon coupling vertex in momentum space,
\begin{equation}
\Gamma_\mu(q)=ig{\epsilon_{\mu\nu\sigma\tau} n^\nu q^\sigma\gamma^\tau\over
n\cdot q-i\epsilon},
\label{mmvertex}
\end{equation}
where we have, for variety's sake, chosen a semi-infinite string.
As we have noted,
the choice of the string is arbitrary; reorienting the string is a kind
of gauge transformation.  In fact, it is this requirement that leads to
the quantization conditions (\ref{quant}) and (\ref{squant}). 
% The consistency
%of magnetic charge has been demonstrated in quantum mechanics (for
%example, see Refs.~\cite{dyondyon} and \cite{dm}), 
%but never completely in quantum field theory.\footnote{Arguments have been
%given to demonstrate the relativistic invariance of the theory, and the
%string independence of the action for classical particle currents
%\cite{schwinger}.  See also
%Ref.~\cite{brandt}. This consistency is a consequence of the quantization
%condition (\ref{quant}) or (\ref{squant}).  We should also bear in mind
%Schwinger's warning, in the first reference in Ref.~\cite{schwinger}:
%``Relativistic invariance will appear to be violated
%in any treatment based on a perturbation expansion.  Field theory is
%more than a set of `Feynman's rules.'~'' } The use of the string-dependent
%vertex (\ref{mmvertex}) directly is not meaningful.  In this regard,
%the ``remedy'' proposed by Deans \cite{deans} and cited as a solution
%to the gauge-string dependence of Drell-Yan processes is implausible.

The authors of Refs.~\cite{DeRujula}, \cite{ginz1}, and \cite{ginz2} do not
attempt a calculation of the ``box'' diagram with the 
interaction (\ref{mmint}).
Rather, they (explicitly or implicitly) appeal to duality, that is, the
symmetry that the introduction of magnetic charge brings to Maxwell's 
equations:
\begin{equation}
{\bf E}\to {\bf H}, \quad {\bf H}\to -{\bf E},
\label{duality}
\end{equation}
and similarly for charges and currents.  Thus the argument is that 
for low energy photon processes
it suffices to compute the fermion loop graph in the presence of
zero-energy photons, that is, in the presence of static, constant fields.
The box diagram shown in Fig.~\ref{fig1} with a spin-1/2 
monopole running around the loop in the presence of a homogeneous $\bf E, H$ 
field is then obtained from that analogous process with an electron in the 
loop in the presence of a homogeneous $\bf H, -E$ field, with the substitution 
$e\to g$.  Since the Euler-Heisenberg Lagrangian (\ref{ehlag}) is invariant 
under the substitution (\ref{duality}) on the fields alone, 
this means we obtain the low energy 
cross section $\sigma_{\gamma\gamma\to\gamma\gamma}$ through the monopole
loop from Eq.~(\ref{llcs}) by the substitution $e\to g$, or
\begin{equation}
\alpha\to\alpha_g={137\over 4}N^2,\quad N=1,2,3,\dots.
\label{subs}
\end{equation}

\subsection{Inconsistency of the Duality Approximation}

It is critical to emphasize that the Euler-Heisenberg Lagrangian is 
an effective
Lagrangian for calculations at the {\it one fermion loop level\/} for low
energy, i.e., $\omega/M\ll1$. It is commonly asserted that the Euler-Heisenberg
Lagrangian is an {\it effective Lagrangian\/} in the sense used in chiral
perturbation theory \cite{weinberg,halter}.  This is not true.  The QED
expansion generates derivative terms which do not arise in the effective
Lagrangian expansion of the Euler-Heisenberg Lagrangian \cite{dicus}.
One can only say that the Euler-Heisenberg Lagrangian is a good approximation
for light-by-light scattering (without monopoles)
at low energy because radiative corrections are down by factors of 
$\alpha$.  However, it becomes unreliable if radiative corrections are
large.\footnote{The same has been noted in another context by Bordag et al.
\cite{bordag}.}

In this regard, both the Ginzburg \cite{ginz1,ginz2} and the De R\'ujula
\cite{DeRujula} articles, particularly Ref.~\cite{ginz2}, are rather misleading
as to the validity of the approximation sketched in the previous section.  
They state that the expansion parameter is not $g$ but $g\omega/M$, $M$ being 
the monopole mass, so that the perturbation expansion may be valid for large 
$g$ if $\omega$ is small enough.  But this is an invalid argument.  
It is only when external photon lines are attached that 
extra factors of $\omega/M$ occur, due to the appearance of the field
strength tensor in the Euler-Heisenberg Lagrangian.  Moreover, the powers
of $g$ and $\omega/M$ are the same only for the $F^4$ process.
The expansion parameter
is $\alpha_g$, which is huge.  Instead of radiative corrections being of the
order of $\alpha$ for the electron-loop process, these corrections will
be of order $\alpha_g$, which implies an uncontrollable sequence of
corrections.  For example, the internal radiative correction to the box
diagram in Fig.~\ref{fig1} have been computed by Ritus \cite{ritus} and by
Reuter, Schmidt, and Schubert \cite{reuter} in QED.  In the $O(\alpha^2)$
term in Eq.~(\ref{ehlag}) the coefficients of the $(F^2)^2$ and the 
$(F\tilde F)^2$ terms are multiplied by $\left(1+{40\over9}{\alpha\over\pi}
+O(\alpha^2)\right)$ and $\left(1+{1315\over252}{\alpha\over\pi}+O(\alpha^2)
\right)$,
respectively.  The corrections become meaningless when we {\it replace\/}
$\alpha\to\alpha_g$.

 This would seem to be a
devastating objection to the results quoted in Ref.~\cite{ginz2} and used
in Ref.~\cite{d0}.  But even if one closes one's eyes to higher order effects,
it seems clear that the mass limits quoted are inconsistent.

If we take the cross section given by Eq.~(\ref{llcs}) and make the
substitution (\ref{subs}), we obtain for the low energy light-by-light
scattering cross section in the presence of a monopole loop
\begin{equation}
\sigma_{\gamma\gamma\to\gamma\gamma}\approx {973\\pi}
{N^8\over\alpha^4}{\omega^6\over M^8}=4.2\times 10^4 \,N^8{1\over M^2}\left(
\omega\over M\right)^6.
\label{monocs}
\end{equation}
If the cross section were dominated by a single partial wave of angular
momentum $J$, the cross section would be bounded by
\begin{equation}
\sigma\le{\pi(2J+1)\over s}\sim {3\pi\over s},
\end{equation}
if we take $J=1$ as a typical partial wave. Comparing this with the
cross section given in Eq.~(\ref{monocs}), we obtain the following
inequality for the cross section to be consistent with unitarity,
\begin{equation}
{M\over\omega}\gtrsim 3 N.
\label{unitbd}
\end{equation}  
But the limits quoted \cite{d0} for the monopole mass are less than this:
\begin{equation}
{M\over N}>870 \mbox{ GeV}, \quad \mbox{spin } 1/2,
\end{equation}
because, at best, a minimum $\langle\omega\rangle\sim 300$ GeV; 
the theory cannot sensibly be applied below a monopole mass of about 1 TeV.
  (Note that changing the
value of $J$ in the unitarity limits has very little effect on the bound
(\ref{unitbd}) since an 8th root is taken: replacing $J$ by 50 reduces
the limit (\ref{unitbd}) only by 50\%.)

Similar remarks can be directed toward the De R\'ujula limits \cite{DeRujula}. 
That author, however, notes the ``perilous use of a perturbative expansion
in $g$.''  However, although he writes down the correct vertex, 
Eq.~(\ref{mmvertex}), he does not, in fact, use it,
instead appealing to duality, and even so he admittedly omits
enormous radiative corrections of $O(\alpha_g)$ without any justification
other than what we believe is a specious reference to the use of effective
Lagrangian techniques for these processes.

\subsection{Proposed Remedies}

Apparently, then, the formal small $\omega$ result obtained from the 
Euler-Heisenberg Lagrangian cannot be valid beyond a photon energy 
$\omega/M\gtrsim0.1$. The reader might ask why one cannot use
duality to convert the monopole coupling with an arbitrary photon to
the ordinary vector coupling.  The answer is that little is thereby
gained, because the coupling of the photon to ordinary charged particles
is then converted into a complicated form analogous to Eq.~(\ref{mmint}).
This point is stated and then ignored in Ref.~\cite{DeRujula} in the
calculation of $Z\to3\gamma$.
There is, in general, no way of avoiding the complication of including
the string.

We are currently undertaking realistic calculations of virtual (monopole
loop) and real (monopole production) magnetic monopole processes.  
These calculations are, as the reader may infer, somewhat difficult
and involve subtle issues of principle involving the string, and it will
be some time before we have results to present.  Therefore, here we wish
to offer plausible qualitative considerations, which we believe 
suggest bounds that call into question
the results of Ginzburg et al. \cite{ginz1,ginz2}.

Our point is very simple.  The interaction (\ref{mmint}) couples the magnetic
current to the dual field strength.  This corresponds to
the velocity suppression in the interaction 
of magnetic fields with electrically charged particles,
or to the velocity suppression in the interaction 
 of electric fields with magnetically charged
particles, as most simply seen in the magnetic analog of the Lorentz force,
\begin{equation}
{\bf F}=g({\bf B}-{{\bf v}\over c}\times {\bf E}).
\end{equation}
That is, the force between an electric charge $e$ and magnetic charge $g$,
moving with relative velocity $\bf v$ and with relative separation $\bf r$
is
\begin{equation}
{\bf F}=-{eg\over c}{{\bf v\times r}\over4\pi r^3}.
\end{equation}
This velocity suppression is reflected in nonrelativistic calculations.
For example, the energy loss in matter
of a magnetically charge particle is approximately obtained from that
of a particle with charge $Ze$ by the substitution \cite{ce}
\begin{equation}
{Ze\over v}\to{g\over c}.
\end{equation}
And the classical nonrelativistic dyon-dyon scattering cross section near
the forward direction is \cite{dyondyon}
\begin{equation}
{d\sigma\over d\Omega}\approx{1\over(2\mu v)^2}\left[\left(e_1g_2-e_2g_1
\over4\pi c\right)^2+\left(e_1e_2+g_1g_2\over4\pi v\right)^2\right]
{1\over(\theta/2)^4},\quad \theta\ll1,
\end{equation}
the expected generalization of the Rutherford scattering cross section at
small angles.  

Of course, the true structure of the magnetic interaction and the resulting 
scattering cross section is much more complicated.  For example, classical 
electron-monopole or dyon-dyon scattering exhibits rainbows and glories, 
and the quantum scattering exhibits
a complicated oscillatory behavior in the backward direction \cite{dyondyon}.  
These reflect the complexities of the magnetic interaction between 
electrically and magnetically charged particles, which can be represented as a 
kind of angular momentum \cite{dm,gold}.  Nevertheless, for the purpose of
extracting qualitative information, the naive substitution,
\begin{equation}
e\to {v\over c}g,
\end{equation}
seems a reasonable first step.\footnote{This, and the extension of this
idea to virtual processes, leaves aside the troublesome issue
of radiative corrections.  The hope is that an effective Lagrangian
can be found by approximately integrating over the fermions 
which incorporates these effects.  A first estimate of the effect of
incorporating radiative correction, however, may be made by
applying Pad\'e summation of the leading corrections found in 
Refs.~\cite{ritus,reuter}: 
\begin{equation}
\sigma\sim{\sigma^{\rm PT}\over(1-\alpha_g)^2}\sim{\sigma^{\rm PT}\over
1000N^4},
\end{equation}
which reduces the quoted mass limits of Ref.~\cite{d0} a bit.}
 Indeed, such a substitution was used in the
proposal \cite{ou} to estimate production rates of monopoles at Fermilab.

The situation is somewhat less clear for the virtual processes
considered here.  Nevertheless, the interaction (\ref{mmint}) suggests
there should, in general, be a softening of the vertex. In the current
absence of a valid calculational scheme, we will merely suggest two
plausible alternatives to the mere replacement procedure adopted in
Refs.~\cite{DeRujula,ginz1,ginz2,graf}.

We first suggest, as seemingly Ref.~\cite{ginz2} does, 
that the approximate effective vertex incorporates 
an additional factor of $\omega/M$.
Thus we propose the following estimate for the $\gamma\gamma$ cross
section in place of Eq.~(\ref{monocs}),
\begin{equation}
\sigma_{\gamma\gamma\to\gamma\gamma}\sim10^4 N^8{1\over M^2}\left(\omega\over
M\right)^{14},
\label{goodcs}
\end{equation}
since there are four suppression factors in the amplitude.
Now a considerably larger value of $\omega$ is consistent with unitarity,
\begin{equation}
{M\over\omega}\gtrsim\sqrt{3N},
\end{equation}
if we take $J=1$ again.  We now must re-examine the $\sigma_{pp\to\gamma
\gamma X}$ cross section.

In the model
given in Ref.~\cite{ginz2}, where the photon energy distribution is
given in terms of the functions $f(y)$, $y=\omega/E$, 
the physical cross section is given by
\begin{equation}
\sigma_{pp\to\gamma\gamma X}=\left(\alpha\over\pi\right)^2\int{dy_1\over y_1}
{dy_2\over y_2}f(y_1)f(y_2)\sigma_{\gamma\gamma\to\gamma\gamma}=
\int dy_1 dy_2{d\sigma\over dy_1\,dy_2},
\end{equation}
where now ({\it cf.} Eq.~(25) of in the first reference in Ref.~\cite{ginz2})
\begin{equation}
{d\sigma\over dy_1\,dy_2}=\left(\alpha\over\pi\right)^2R E^6\left(E\over M
\right)^8y_1^6f(y_1)y_2^6f(y_2),
\label{newdsigma}
\end{equation}
where, for spin 1/2,  (up to factors of order unity)
\begin{equation}
R\sim{10^{-4}\over\alpha^4}\left( N\over M\right)^8.
\end{equation}
The result in (\ref{newdsigma}) differs from that in Ref.~\cite{ginz2}
by a factor of $(E/M)^8y_1^4y_2^4$.  The photon distribution 
function $y^2f(y)$ used is rather strongly
peaked at $y\sim 0.3$.  (This peaking is necessary to have any chance
of satisfying the low-frequency criterion.)  When we multiply by $y^4$, the
amplitude is greatly reduced and the peak is shifted above $y=1/2$, violating
even the naive criterion for the validity of perturbation theory.  
Nevertheless, the integral of the distribution function is reduced by two 
orders of magnitude, that is,
\begin{equation}
{\int_0^1 dy\,y^6f(y)\over\int_0^1dy\,y^2 f(y)}\sim 10^{-2}.
\end{equation}
This reduces the mass limit quoted in \cite{d0} by a factor of $1/\sqrt{3}$,
to about 500 GeV, where $\langle\omega\rangle/M\approx 0.9$.  This
dubious result makes us conclude that
it is impossible to derive any limit for the monopole mass from the present
data.

As for the De R\'ujula limit\footnote{We note that De R\'ujula also considers
the monopole vacuum polarization correction to $g_V/g_A$, $g_A$, and $m_W/m_Z$,
proportional to $(m_Z/M)^2$ in each case, once again ignoring both the
string and the radiative correction problem.  He assumes that the
monopole is a heavy vector-like fermion, and obtains a limit of
$M/N>8m_Z$.  Our ansatz changes $(m_Z/M)^2$ to $(m_Z/M)^4$, so that
$M/\sqrt{N}>\sqrt{8}m_Z\approx250$ GeV, a substantial reduction.}
from the $Z\to3\gamma$ process, if we
insert a suppression factor of $\omega/M$ at each vertex and integrate over
the final state photon distributions, given by Eq.~(18) 
of Ref.~\cite{DeRujula},
the mass limit is reduced to $M/\sqrt{N}\gtrsim1.4m_Z\sim120$
GeV, again grossly violating the low
energy criterion. And the limit deduced from the vacuum polarization
correction to the anomalous magnetic moment of the muon due to virtual
monopole pairs \cite{graf} is reduced to about 2 GeV.

The reader might object that this $\omega/M$ softening of the vertex has
little field-theoretic basis.  Therefore, we propose a second possibility that
does have such a basis.  The vertex (\ref{mmvertex}) suggests, and detailed
calculation supports (based on the tensor structure of the photon
amplitudes\footnote{For example, the naive monopole loop contribution to vacuum
polarization differs from that of an electron loop (apart from charge and
mass replacements) entirely by the replacement in the latter of
$(g_{\mu\nu}-q_\mu q_\nu/q^2)\to({\bf q}^2/q_0^2)(\delta_{ij}-q_iq_j/{\bf q}^2)
$, when $n^\mu$ points in the time direction. 
Apart from this different tensor structure, the vacuum polarization
is given by exactly the usual formula, found, for example in Ref.~\cite{js1}.
Details of this and related
calculations will be given elsewhere.}) the introduction of the 
string-dependent factor $\sqrt{q^2/(n\cdot q)^2}$ at each vertex, where $q$ is 
the photon momentum. Such a factor is devastating to the indirect monopole 
searches---for any process involving a real photon, such as that of the D0 
experiment \cite{d0} or for $Z\to3\gamma$ discussed in \cite{DeRujula}, the 
amplitude vanishes. Because such factors can and do appear in full monopole 
calculations, it is clearly premature to claim any limits based on virtual 
processes involving real final-state photons. 

\section{Conclusions}

The field theory of magnetic charge is still in a rather primitive state.
Indeed, it has been in an arrested state of development for the past two
decades.  With serious limits now being announced based on laboratory
measurements, it is crucial that the theory be raised to a useful level.

At the present time, we believe that theoretical estimates for the production
of real monopoles are more reliable than are those for virtual processes.
This is because, in effect, the former are dominated by tree-level processes.
We have indicated why the indirect limits cannot be taken seriously
at present; and of course only the real production processes offer the
potential of discovery.  Perhaps the arguments here will stimulate readers
to contribute to the further development of the theory, for it remains
an embarrassment that there is no well-defined quantum field theory of
magnetic charge.
%We do not take our reduced limits on monopole masses very seriously.  Rather,
%we believe they demonstrate our point that given the dual difficulties
%of theoretically treating monopoles, that is, incorporating the string
%and dealing with enormously strong coupling,\footnote{None of the papers
%dealing with virtual monopole effects, \cite{DeRujula,ginz1,ginz2,graf},
%in fact, incorporate these nontrivial effects.} it is premature to attempt
%to set any limits on monopole masses based on virtual effects.  A direct
%search stills seems much less problematic.




\section* {ACKNOWLEDGMENTS}

We are very pleased to dedicate this paper to Kurt Haller on the occasion of
his 70th birthday, in view of his many contributions to non-Abelian
gauge theory, of which dual QED is a disguised variant.
We thank Igor Solovtsov for helpful conversations and we are
grateful to the U.S.~Department of Energy for financial support.

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\end{references}

\end{document}
