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\begin{center}{{\tenbf EXTENDED TECHNICOLOR MODEL BUILDING}
\vglue 1.0cm
{\tenrm Thomas Appelquist and John Terning\footnote{Talk presented by 
John Terning at the International Workshop for Electroweak Symmetry Breaking,
Nov. 12-15, 1991, Hiroshima.}\\}

\baselineskip=13pt
{\tenit Department of Physics,Yale University\\}
\baselineskip=12pt
{\tenit New Haven, CT 06511, U.S.A.\\}
\vglue 0.3cm
%{\tenrm and\\}
%\vglue 0.3cm
%{\tenrm SECOND AUTHOR'S NAME\\}
%{\tenit Group, Company, Address, City, State ZIP/Zone, Country\\}
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{\tenrm ABSTRACT}}
\end{center}
\vglue 0.3cm
{\rightskip=3pc
 \leftskip=3pc
 \tenrm\baselineskip=12pt
 \noindent
We discuss constraints on extended technicolor model building, and show how 
to rule out classes of models.  We also present some speculations on neutrino
masses and discuss a toy model of leptons. 
\vglue 0.6cm}
{\elevenbf\noindent 1. Introduction}
\vglue 0.4cm
\baselineskip=14pt
\elevenrm
The recent resurgence of technicolor (TC) theories has produced a body of 
work\cite{walking,strongETC,KMEN} that
demonstrates that there are no serious technical obstacles to describing 
the observed particle mass spectrum (while avoiding flavor changing neutral 
currents (FCNC's) and keeping $\rho$ close to 1) in terms of a TC model.  That is,
given the ability to choose an arbitrary TC gauge group with an arbitrary number
of technifermions, and representing the extended technicolor (ETC) interactions
by four-Fermi interactions with arbitrary mass scales and arbitrary
couplings for each of the ordinary fermions one can produce the entire observed
range of fermion masses up to a few hundred GeV for the t quark 
without any phenomenological disasters. Even the 
${\tw S}$ 
parameter may be held at bay\cite{S}.
Though such an exercise is interesting as a sort of existence proof, one worries
that since there are more parameters than observables it is impossible to tell 
whether this success is the result of having the qualitatively correct physics
or merely the power of parameter fitting.  

To do better we have to be more 
ambitious and construct models that explain the real world rather than just 
describe it, i.e. models with fewer parameters than the Standard Model.
Such models should make testable predictions, and have the potential to be 
ruled out by experiment. In principle an explicit ETC model would fit this bill,
since it would reduce all the parameters of the Higgs sector to  just one 
parameter for each ETC gauge group.  In reality it is unlikely that given the
``right" ETC model we would be able to accurately analyze the non-perturbative
dynamics involved, so we might be forced to parameterize 
at least some of our ignorance.  Even so, we could still hope to have fewer
parameters than observables, and hence a testable model.

In this paper we will discuss some constraints on model building, show how to
rule out some classes of models, give some speculations on neutrino
masses and finish with a discussion of a toy model of leptons.
%\vglue 0.2cm
%\elevenit \noindent 1.2. Section Headings}
%\vglue 0.1cm
\vglue 0.5cm
{\elevenbf\noindent 2. Constraints on Model Building}
\vglue 0.4cm
There are several constraints that one might like to impose on a realistic
ETC model.  First of all, we expect that there should be more than one ETC
scale.  The absence of FCNC's (inferred from the $K-\overline{K}$
mass splitting) requires that the ETC scale associated with the the s quark
(that is the mass of the gauge boson that connects the s quark to 
technifermions, and,
more importantly, the d) be at least 200 TeV.  On the other hand, to obtain a
t quark mass of a few hundred GeV (without excessive fine tuning) we expect
an ETC scale of order 5 TeV or lower.  Such arguments suggest that it is 
natural to have
at least three different ETC scales, one for each family.  

Another
constraint arises from trying to obtain a large t-b mass splitting, while
keeping $\rho$ within 0.5\% of 1.  To do this without fine tuning it is 
natural to require that the right-handed t and right-handed b be in different
representations of the ETC group ($SU(2)_L$ gauge invariance requires the
left-handed t and b to be in the same representation).  Having t$_R$ and b$_R$
in different representations allows for the possibility that the t and b are
associated with different ETC scales (This scenario has been exploited by
King and Mannon and by Einhorn and Nash\cite{KMEN}).  

A powerful constraint on ETC model building was originally 
elucidated by Eichten and Lane\cite{EL}, 
who showed that the absence of a visible axion implies a limit on
the number of irreducible representations (irreps) of the ETC gauge group.
The number of exactly massless Nambu-Goldstone bosons is given by the number 
of spontaneously broken global U(1)'s minus the number of global anomalies.  
For 
vector-like ETC theories of the form ETC${\twelverm\otimes SU(2)_L \otimes 
U(1)}$ this implies that there
can be at most 3 conjugate pairs of irreps, which implies that quarks and 
leptons 
should not be in seperate irreps.  Such considerations lead one to consider 
grand unified technicolor theories (GUTT's) where all interactions 
(including TC) are unified in a single gauge group.  This is the simplest and,
 in principle, most predictive type of ETC model.
We are allowed at most one irrep of the GUTT gauge group, again in order to 
avoid axions\cite{EL}.
\vglue 0.5cm
{\elevenbf \noindent 3. Ruling Out Classes of Models}
\vglue 0.4cm
Combining the limit on the number of irreps discussed previously with the
requirement of three families, we can rule out classes of GUTT's.  As we have
seen we are allowed only one irrep of the GUTT gauge group.  In order to 
protect ordinary fermions from gaining GUTT scale masses, this single irrep 
must be complex.  Having a complex irrep also allows us to imagine that the
GUTT gauge symmetry can be broken down with fermion bilinears, that is without
using any scalar fields.  However our GUTT must be free of gauge anomalies, so
the search can be limited to $E_6$ and $O(4N+2)$ since these are the only
anomaly safe groups with complex representations.  The {\bf 27} of $E_6$ and the
{\bf 64} of $O(14)$ are too small to contain three families, so the smallest 
possible
GUTT is $O(18)$ with all fermions in the 256 dimensional spinor.  

This type of
theory has been previously discussed\cite{Bag} as an ordinary GUT, and it may
be helpful to review that scenario.  If we imagine $O(18)$ breaking to
$O(10)\otimes O(8)$ we find that the {\bf 256} decomposes as 
$({\bf 16},{\bf 8}')+({\bf \overline{16}},{\bf 8}'')$. The {\bf 16}
of $O(10)$ contains one ordinary family (when $O(10)$ breaks to the Standard
Model in the usual fashion) with an extra right handed neutrino,
while the ${\tw {\bf \overline{16}}}$ is a family with left and right 
interchanged.
We will refer to the ${\tw {\bf \overline{16}}}$ as a right-family and the 
{\bf 16} as a left-family.  At this
stage the 8 left-families and the 8 right-families are protected from 
getting masses by gauge symmetries.
However, if the $O(8)$ gauge group breaks entirely, then nothing prevents the
left-families from  getting masses with the right-families, and there will be no
light families left over to make up the observed three families.  

We
can now see what must happen in a GUTT scenario.  If the $O(8)$ breaks
down to a TC gauge group such that more of the right-families are technicolored
than the left-families, then there will be some excess left-families which
are protected from getting masses until $SU(2)_L$ is broken.  Thus we can 
proceed straightforwardly by simply listing all the possible TC groups, 
embedding them (along with the Standard Model) in all possible ways 
in $O(18)$ and then counting the 
number of extra techni-singlet left-families\footnote{The tables of ref. 
\citenum{group} are invaluable in this analysis.}.  
(This family counting proceedure
was originally suggested by Georgi\cite{Georgi} for ordinary GUT's.)  
In fact we can imagine breaking patterns that do not automatically generate a
family structure; in these cases we can simply count quarks and antiquarks.
It should be noted that this procedure does not rely on knowing the actual pattern of
symmetry breaking between the GUTT scale and the TC scale; different patterns
of breaking can result in the same unbroken TC group.  The
result of this counting procedure is that there is an essentially 
unique TC group that produces three
families from the $O(18)$ GUTT:  $Sp(4)_{TC}$.  (There exists an embedding of 
$SU(2)$ into ${\tw Sp(4)}$ where ${\tw {\bf 4}\rightarrow {\bf 4}}$, and 
${\tw {\bf 5}\rightarrow {\bf 5}}$, but
all of our further considerations will apply to this degenerate case as well.)  
This pattern of breaking was originally discovered by Gell-Mann, Ramond, and 
Slansky\cite{GMRS} and by Farhi and Susskind\cite{FS}.  What we have shown is 
that there are no other possibilities for $O(18)$.
In the $Sp(4)_{TC}$ scenario
the ${\bf 8}' \rightarrow {\bf 5}\oplus{\bf 1}\oplus{\bf 1}\oplus{\bf 1}$
and the 
${\bf 8}'' \rightarrow {\bf 4}\oplus{\bf \overline{4}}$.  

Since
all of the right-families are technicolored, we expect no super-heavy fermions,
that is, except for right-handed Majorana neutrinos, the largest fermion masses
are of the order of hundreds of GeV. 
Above 1 TeV, there is the equivalent of 16 families contributing to beta 
functions, so asymptotic freedom for QCD and $Sp(4)_{TC}$ is lost ($ \alpha_s$ will blow up
around 35 TeV).  For this theory to make sense, it may be
necessary to rely on some non-perturbative dynamics, but computations then
become difficult. (For an attempt along these lines, assuming the the existence
of nontrivial ultraviolet fixed points, see ref.~\citenum{Raby}.)
One could also consider larger GUTT's like $O(22)$.  It is easy to
find TC groups which, when embedded in ${\tw O(22)}$, give four families, 
but a systematic search for three family models has not been made.

Since the smallest GUTT models seem beyond our computational abilities, it is
natural to consider the more general possibility of partial unification,  that
is where color is unified with TC so that the gauge group is of the form 
$SU(N)_{ETC}\otimes SU(2)_L \otimes U(1)$.  We will consider a fermion content of the
form $({\bf R_1},{\bf 2}) \oplus ({\bf R_2},{\bf 1}) \oplus 
({\bf R_3},{\bf 1})$.  We will also assume that 
$SU(3)_C$ is embedded in $SU(N)_{ETC}$ is the simplest possible way (i.e. the
{\bf N} decomposes to a {\bf 3} and ${\tw N-3}$ color singlets) and that 
${\bf R_1}$, ${\bf R_2}$, and ${\bf R_3}$ 
are antisymmetric irreps.  These assumptions ensure that the model will only
contain {\bf 3}'s and ${\bf \overline{3}}$'s of color\cite{Georgi}, that is, 
we eschew quixes,
queights, etc.  We also assume that ${\bf R_1}$, ${\bf R_2}$, and ${\bf R_3}$ 
are not all identical, while requiring that the $SU(N)_{ETC}$ gauge anomaly 
vanishes.  
Finally, by counting the number of quarks and antiquarks and 
requiring three families we can
show that such models are ruled out for ${\tw N < 10}$.

Since such general group theoretical arguments do not get us very far towards
a realistic model, we will take a more phenomenological approach in the 
remainder of the paper.

\vglue 0.5cm
{\elevenbf \noindent 4. Speculations on Neutrino Masses \hfil}
\vglue 0.4cm
The fact that only extremely light left-handed neutrinos are seen poses special
problems in
TC theories.  Fortunately there is a simple explanation available in
the usual seesaw mechanism\cite{GMRS,models}.  The idea is that right-handed 
neutrinos
get large Majorana masses, so that the left-handed neutrinos end up with masses
given by a Dirac mass squared divided by the Majorana mass.  Presumably, the
neutrino Dirac masses are of the same order as their charged leptonic partners'
masses.  The remaining problem is to estimate the Majorana masses.  One
possibility is that the Majorana masses are much larger than any ETC 
scales, then
all the left-handed neutrinos are light, typically with a mass less than 
$10^{-4}$ eV for $\nu_{e}$.  This assumes that although there are right-handed
techni-neutrinos with ETC interactions, the right-handed neutrinos have no 
ETC interactions or are exclusively in real representations of the ETC group.
  
A more interesting possibility\cite{models} is that the
Majorana masses are of the same order as some ETC scale (or scales).  This
is natural if condensates of bilinears of right-handed neutrinos (Majorana
condensates) are involved 
in the dynamical breaking of the ETC gauge symmetry.  First we imagine a 
hierarchy of ETC scales (1000 TeV, 200 TeV, 25 TeV) in order to naturally arrange for a 
hierarchy of charged lepton masses ($m_e$, $m_\mu$, $m_\tau$).  The 25 TeV
scale is a natural scale for producing the mass of the $\tau$, and may also
produce the mass of the b when color enhancement effects are included\cite{color}.
Now we can estimate the
masses of the assocated neutrinos in such a scheme:
\begin{equation}
m_{\nu_{e}} \approx {(0.5 {\rm MeV})^2 \over {1000 {\rm TeV}}} \approx 
3 \times 10^{-4} {\rm eV}
\nonumber
\end{equation}   
\begin{equation}
m_{\nu_{\mu}} \approx {(100 {\rm MeV})^2 \over {200 {\rm TeV}}} \approx 
50  {\rm eV}
\nonumber
\end{equation}   
\begin{equation}
m_{\nu_{\tau}} \approx {(1 {\rm GeV})^2 \over {25 {\rm TeV}}} \approx 
40 {\rm keV} .
\end{equation}   
From these estimates we see three interesting mass scales arising.  The
$\nu_{e}$ mass is in the right range to impliment the MSW mechanism, provided
that there exists another light neutrino for  $\nu_e$ to mix with.  
A $\nu_{\mu}$ with a mass of the order of 10 eV is interesting as 
a dark matter candidate.  The $\nu_{\tau}$ mass is also in
an interesting mass range\cite{Simpson}.  However if such a heavy Majorana $\nu_{\tau}$
has a substantial mixing with the $\nu_e$ then another heavy 
neutrino\cite{Nelson} is required in order to stay below the experimental 
limit on 
neutrinoless double $\beta$ decay. 
For example, if a 17 keV $\nu_{\tau}$ has
a 1\% mixing with the $\nu_e$, and we use the $\nu_\mu$ as the second heavy
neutrino, then the $\nu_\mu$ must have either a 17 keV mass, or a mass in 
the range 150 - 250 keV for consistency with a variety of 
experiments\cite{Nelson}.
\vglue 0.5cm
{\elevenbf \noindent 5. A Toy Model \hfil}
\vglue 0.4cm
Finally, we will present a toy model of leptons 
which produces some of 
the neutrino masses discussed above.  As seen in section 2, quarks and 
leptons should come from the same irrep in order to avoid massless
visible axions.  Here we postpone quark lepton unification to some scale 
higher than the ETC scale, so we will be able to discuss leptons seperately. 
The ETC gauge group in our model
is $SU(5)$, which is assumed to commute with the standard model gauge 
interactions.  $SU(5)$ was chosen so that we can break it down to the smallest
TC group, $SU(2)_{TC}$, and produce three families.  
The fermion content of the model is:
\begin{equation}{\nu_\sigma}_L ~~
\left( \begin{array}{lr}
N & E \\ 
N & E \\ 
\nu_\tau & \tau \\ 
\nu_\mu & \mu \\
\nu_e & e
\end{array} \right) _L ~~
\left( \begin{array}{lcccr}
0 & \nu_e & n & n & N \\
- \nu_e & 0 & n & n & N \\
- n & - n & 0 & \nu_\sigma & \nu_\tau \\
- n & - n & - \nu_\sigma & 0 & \nu_\mu \\
- N & - N & - \nu_\tau & - \nu_\mu & 0
\end{array} \right) _R ~~
\left( \begin{array}{l} E \\ E \\  \tau \\  \mu \\ e
\end{array} \right) _R ~~.
\end{equation}

Note that two extra sets of right-handed techni-neutrinos (the $n$'s) and
an extra singlet neutrino (${\nu_\sigma}_R$) are needed to fill out the 
{\bf 10}.  In general we can expect that the breaking of gauge symmetries at a
higher unification scale may leave some sterile particles.  Here  
we assume that the ${\nu_\sigma}_R$ has a corresponding (but sterile) 
${\nu_\sigma}_L$. 
The most attractive channel for condensation in this model is for
the {\bf 10} to condense with itself, breaking $SU(5)$ down to $SU(4)$.  Each of the
{\bf 5}'s splits into a {\bf 4} and a singlet; these new singlets being the leptons of the
first generation.  The
{\bf 10} splits into a {\bf 4} and a {\bf 6}:
\begin{equation}
\left( \begin{array}{l} 
 N \\  N\\  \nu_\tau \\ \nu_\mu
\end{array} \right) _R ~~
\left( \begin{array}{lccr}
0 & \nu_e & n & n \\
- \nu_e & 0 & n & n \\
- n & - n & 0 & \nu_\sigma \\
- n & - n & - \nu_\sigma & 0  
\end{array} \right) _R  ~~.
\end{equation}
The fermions in the {\bf 6} obtain Majorana mixing masses of order the breaking
scale which we take to be 1000 TeV.  Ignoring the heavy {\bf 6}, the $SU(4)$ 
gauge interactions are vector-like.  We next imagine that $SU(4)$ is broken
down to $SU(3)$ by some as yet unspecified sector of the model (eg. quarks
and techniquarks), which splits off the second generation at an assumed scale
of 200 TeV.  Finally, we assume that the attractive channel (but not the most
attractive) for ${\nu_\tau}_R$ condensation (${\bf3}\otimes{\bf3}\rightarrow
{\bf {\overline 3}}$) gives a Majorana mass of order
25 TeV (the assumed breaking scale) to ${\nu_\tau}_R$.  This condensation
breaks $SU(3)$ down to $SU(2)_{TC}$.  

We can now discuss the neutrino mass spectrum of our model.  
The ${\nu_\sigma}_L$ can only
receive a mass which is suppressed by scales higher than 1000 TeV so
it is approximately massless.  The ${\nu_e}_L$  mass is of order
$3 \times 10^{-4}$ eV as discussed before, so the ${\nu_e}_L$ and 
${\nu_\sigma}_L$ are both in the appropriate mass range to allow the MSW 
mechanism to operate.  Also
as before, the ${\nu_\tau}_L$ mass is of order
40 keV.  The case of the $\nu_\mu$ is more interesting.
We have assumed that there is no Majorana mass produced directly, but one
may feed down from the ${\nu_\tau}_R$ through ETC interactions.  The model 
contains ETC gauge bosons with masses of 200 TeV that connect ${\nu_\mu}_R$
with the ${\nu_\tau}_R$.  The dynamical Majorana mass connects the 
${\nu_\tau}_R$ with the conjugate ${\nu_\tau}_R$, and the same gauge boson
takes the ${\nu_\tau}_R$ back to a ${\nu_\mu}_R$.  $SU(3)$ symmetry prevents
us from connecting the gauge boson lines, but when $SU(3)$ breaks to 
$SU(2)_{TC}$
there will in general be some mixing of the order of the breaking scale squared.
Thus we can estimate the ${\nu_\mu}_R$ Majorana mass by one
factor of the condensate (which we approximate by $4 \pi (25 $TeV$)^3$)
a factor of the gauge boson mixing and four inverse powers
of the gauge boson mass:
\begin{equation}
M \approx 4 \pi {{(25 {\rm TeV})}^5 \over {(200 {\rm TeV})^4}} \approx 
80 {\rm GeV} ~~.
\end{equation}   
So we find an approximate mass for the left-handed $\nu_\mu$ given by:
\begin{equation}
m_{\nu_\mu} \approx {{(0.1 {\rm GeV})^2} \over {80 {\rm GeV}}} \approx 
130 {\rm keV} ~~.
\end{equation}   
Which is close to the region required for the absence of neutrinoless double
beta decay (with an assumed 1\% $\nu_\tau-\nu_e$ mixing).  This model does
not contain a dark matter candidate, but
the possibility of the $\nu_\mu$ being heavier than
the $\nu_\tau$, which at first sight seems implausible, actually arises
naturally in the model.
\vglue 0.5cm
{\elevenbf \noindent 6. Conclusions \hfil}
\vglue 0.4cm
We have discussed how some classes of simple ETC models can be ruled out by 
simply counting quarks.  We have made some general speculations on neutrino
masses and also presented a
toy model of leptons that gives interesting neutrino masses.  We have not 
achieved the goal we set in the introduction, but
perhaps these observations will help in the task of constructing
a realistic ETC model.
\vglue 0.5cm
{\elevenbf \noindent 7. Acknowledgements \hfil}
\vglue 0.4cm
J.T. would like to thank S. Chivukula, E. Eichten, A. Nelson, M. Soldate, and 
M. White for helpful discussions.
\vglue 0.5cm
{\elevenbf\noindent 8. References \hfil}
\vglue 0.4cm
 
\begin{thebibliography}{9}
 
\bibitem{walking}
B. Holdom, {\elevenit Phys. Lett.} {\elevenbf B150} (1985) 301;
T. Appelquist, D. Karabali, and L.C.R. Wijewarhanda, {\elevenit Phys. Rev.
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K. Yamawaki, M. Bando, and K. Matumoto,{\elevenit Phys. Rev.Lett.} 
{\elevenbf 56} (1986) 1335 
T. Appelquist and L.C.R. Wijewarhanda, {\elevenit Phys. Rev} {\elevenbf D35}
(1987) 774;
T. Appelquist and L.C.R. Wijewarhanda, {\elevenit Phys. Rev} {\elevenbf D36}
(1987) 568.

\bibitem{strongETC}
T. Appelquist, M. Einhorn, T. Takeuchi, and L.C.R.
Wijewardhana, {\em Phys. Lett.} {\bf 220B}, 223 (1989);
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K. Matumoto {\elevenit Prog. Theor. Phys. Lett.} {\elevenbf 81} (1989) 277;
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{\elevenbf A4} (1989) 1043.

\bibitem{KMEN}S. King and S. Mannan, U. of Southampton preprint {\elevenbf
SHEP 90/91-22}; M. Einhorn and D. Nash, to be published in 
{\elevenit Nucl. Phys.} {\elevenbf B}.

\bibitem{S}E. Gates and J. Terning {\elevenit Phys. Rev. Lett.} {\elevenbf 67}
(1991) 1840;
R. Sundrum and S. Hsu, Lawrence Berkeley Lab. preprint {\elevenbf
LBL-31066};
see also the talks by T. Appelquist and T. Takeuchi in these proceedings.

\bibitem{EL} E. Eichten and K. Lane, {\elevenit Phys. Lett.} {\elevenbf B90}
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\bibitem{group}R. Slansky, {\elevenit Phys. Rep.} {\elevenbf 79} (1981) 1. 

\bibitem{Georgi} H. Georgi, {\elevenit Nucl. Phys.} {\elevenbf B156} (1979)
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\bibitem{GMRS} M. Gell-Mann, P. Ramond, and R. Slansky, in {\elevenit
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\bibitem{FS} E. Farhi and L. Susskind, {\elevenit Phys. Rev.} {\elevenbf D20}
(1979) 3404.
\bibitem{Bag} J. Bagger and S. Dimopoulos, {\elevenit Nucl. Phys.} {\elevenbf
B244} (1984) 247; and references therein.
\bibitem{Raby}G. Guidice and S. Raby, Ohio State University preprint
{\elevenbf OSU-DOE-ER-01545-550}.
\bibitem{models}
B. Holdom, {\elevenit Phys. Rev.} {\elevenbf D23} (1981) 1637;
{\elevenit Phys. Lett.} {\elevenbf B246} (1990) 169;
U. of Nagoya preprint {\elevenbf DPNU-91-27} 
to be published in the 1991 Nagoya Spring School Lectures.
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1233;
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327. 
\bibitem{Simpson}J.J. Simpson, {\elevenit Phys. Rev. Lett.} {\elevenbf 54}
(1985) 1891;
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\bibitem{Nelson}
M. Dugan et. al. {\elevenit Phys. Rev. Lett.} {\elevenbf 54} (1985) 2302;
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L.M. Krauss, {\elevenit Phys. Lett.} {\elevenbf B263} (1991) 441.


\end{thebibliography}
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