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\begin{center}{{\tenbf COMMENTS ON TECHNICOLOUR MODEL BUILDING}
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{\tenrm Thomas Appelquist and John Terning\footnote{Talk presented by 
John Terning at Beyond the Standard Model III, June 21-24, 1992,
Carleton University, Ottawa, Canada.}\\}

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{\tenit Department of Physics,Yale University\\}
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{\tenit New Haven, CT 06511, U.S.A.\\}
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{\tenrm ABSTRACT}}
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{\rightskip=3pc
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 \noindent
We discuss constraints on extended technicolour model building, mechanisms
for producing small neutrino
masses, and develop a potentially realistic model. 
\vglue 0.6cm}
{\elevenbf\noindent 1. Introduction}
\vglue 0.4cm
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\elevenrm
Recent work\cite{walking,strongETC} shows that technicolour (TC) theories 
can be made compatible with
%demonstrates that there are no serious 
%technical obstacles to describing 
the observed particle mass spectrum (with $\rho$ close to 1 and without
excessive flavor changing neutral 
currents (FCNC's)).
% in terms of a TC model.  
Though such an exercise is interesting as a sort of existence proof, one worries
that since there are more parameters (gauge groups, gauge couplings, 
four-fermion couplings, etc.) 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 construct models with fewer parameters than 
the standard model.
Such models should make testable predictions for physics below 100 GeV, 
and have the potential to be ruled out by experiment.
In principle an extended technicolour (ETC) model would fit this bill.
In reality it is unlikely that we would be able to accurately 
analyze the non-perturbative
dynamics involved, so we might be forced to parameterize 
some of our ignorance.  Even so, we could still hope to have fewer
parameters than observables, and hence a testable model.
\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 requires that the mass of the gauge 
boson that connects the 
$s$ quark to technifermions be at least 200 TeV, and that the mass of the gauge 
boson that connects the $s$ quark to the $d$ quark be at least 1000 TeV.  On 
the other hand, to obtain a
$t$ quark mass above 100 GeV we expect
an ETC scale of order 5 TeV.  Such arguments suggest 
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 the simplest
possibility is
to require that the right-handed $t$ and right-handed $b$ be in different
reps of the ETC group ($SU(2)_L$ gauge invariance requires the
left-handed t and b to be in the same representation).  

The $S$ parameter can also be worrisome, since experiments
seem to be finding $S$ to be around $-1 \pm 1$, whereas QCD-like
TC theories have large positive contributions to S (which grow with the 
number of technicolours (N$_{TC}$)).  Of course, $S$ could be quite different
in non-QCD-like theories\cite{S}.
However, the constraint on the $S$ parameter seems to suggest that N$_{TC}$ 
should be kept as small as possible, i.e. N$_{TC} = 2$. 
Recently Sundrum \cite{Sundrum} has shown that vacuum alignment need not be
a problem for $SU(2)_{TC}$.

Eichten and Lane\cite{EL}, 
have shown that the absence of a visible axion implies a limit on
the number of spontaneously broken global U(1) symmetries.
This led them to require quark-lepton unification in ETC models. 
In general, a realistic ETC theory should not have any exact, spontaneously 
broken, global symmetries, since this will lead to massless 
Goldstone bosons.  Thus we cannot have repeated 
reps of the ETC gauge group.

Finally an ETC theory must explain why neutrinos are so light.
\vglue 0.5cm
{\elevenbf \noindent 3. A Word About MAC}
\vglue 0.4cm
The established folklore states that fermion
condensates occur in the most attractive channel (MAC)
\cite{RDSC},  which is determined by one gauge boson exchange. 
Peskin \cite{RDSC} has shown that when the gauge symmetry is
unbroken, the MAC analysis is equivalent to minimizing
an effective potential truncated at two loops.
However, when the gauge bosons do acquire masses from the
fermion condensate there are pure gauge boson contributions to the
effective potential which have not yet been considered.  

Since we have no reliable theoretical guide for which fermion
condensates actually form, we will take a more 
phenomenological approach, i.e. we will allow
condensates that are consistent with producing the standard model as
a low energy effective theory.
\vglue 0.5cm
{\elevenbf \noindent 4. Why Are Neutrinos Light? \hfil}
\vglue 0.4cm
A simple explanation for the
fact that neutrinos ($\nu$'s) are extremely light
was originally pointed out
by Sikivie et. al.\cite{Sikivie} 
Consider the following left-handed and charge-conjugated 
right-handed fermions labeled with their $SU(3)_{ETC}\otimes
SU(2)_L\otimes U(1)_Y$ charges: 
\begin{eqnarray}
E_L, \tau_L; N_L, \nu_{\tau L} & \sim &({\bf 3},{\bf 2})_{-1} \nonumber \\
E_R^c,\tau_R^c & \sim & ({\bf \overline{3}},{\bf 1})_{2}\nonumber \\
N_R^c,\nu_R^c & \sim &({\bf \overline{3}},{\bf 1})_{0}\nonumber \\
N_{1 R}^c,\nu_{1 R}^c & \sim &({\bf \overline{3}},{\bf 1})_{0}\nonumber \\
N_{2 R}^c,\nu_{2 R}^c & \sim & ({\bf 3},{\bf 1})_{0} ~,\nonumber 
\end{eqnarray}
where a {\bf 3} of $SU(3)_{ETC}$ corresponds to two technicolours and
one family. 
Note that there are two extra sets of $N_R^c$'s and $\nu_R^c$'s in conjugate
reps.  If
($N_R^c$,$\nu_R^c$) condenses with ($N_{1 R}^c$,$\nu_{1 R}^c$) 
then $SU(3)_{ETC}$
breaks down to $SU(2)_{TC}$, and it is ($N_{2 R}^c$,$\nu_{2 R}^c$) 
which survive to
be the partners of ($N_L$, $\nu_{\tau L}$). Now the usual one-ETC gauge 
boson exchange graph that feeds
down masses to the $\tau$ will not give a mass to the $\nu_\tau$, 
since
this graph is identically zero.  To feed down a mass to the $\nu_\tau$ 
there must
be some mixing between different ETC gauge bosons. If the required mixing
can only be produced by a loop containing $\nu$'s and $N$'s, and we now 
consider the low-energy effective theory where the heavy ETC gauge bosons and
$N$'s have been integrated out, then we see that the diagram for the neutrino 
mass
consists of a four-$\nu$ interaction vertex with two lines closed off by a 
mass insertion, a form familiar from the Nambu--Jona-Lasinio gap equation.  
Thus if the effective four-$\nu$ coupling constant is 
sub-critical, the $\nu_\tau$ stays massless.    However, if there are other 
fermions that can produce the mixing, then the mass of the $\nu_{\tau}$
will be suppressed relative to the mass of the $\tau$ by a factor of the
mixing mass squared over the ETC gauge boson mass squared.

\vglue 0.5cm
{\elevenbf \noindent 5. A Recipe for an ETC Model \hfil}
\vglue 0.4cm
We will now proceed to construct an ETC model.  We will assume that 
the TC group is $SU(2)_{TC}$, and that there is one family of technifermions.
We will require that:
1) there are no exact non-Abelian global symmetries,
2) quarks and leptons are unified,
3) fermions appear only in antisymmetric irreps,  (then the
model only
contains ${\bf 3}$'s and ${\bf \overline{3}}$'s of colour.\cite{Georgi})
4) all gauge anomalies vanish.
With these requirements we can proceed rather straightforwardly.  We will 
gauge as many symmetries as possible without inducing proton decay; this
can be done by putting all 20 $SU(2)_L$ doublets in a 
single representation.
A search for the smallest irrep of the smallest gauge group
produces the ${\bf 36}$ of $SU(9)$.   The simplest way
to include the charge-conjugated right-handed fermions is to have two 
${\bf \overline{36}}$'s, but this fermion content by itself would lead to
no isospin breaking, and no mixing angles in the CKM matrix.  This 
disaster can be avoided if we also add in extra fermions that can
mix with some components of the ${\bf \overline{36}}$'s.  The remaining
antisymmetric irreps of $SU(9)$ are: the ${\bf 9}$, the ${\bf 84}$, and
the ${\bf 126}$.  The ${\bf 9}$ is too small to be interesting, so 
we can add an ${\bf 84}$ and an ${\bf \overline{84}}$
or a ${\bf 126}$ and a ${\bf \overline{126}}$.

We can now explicitly write down an ETC model.  The gauge group
is $SU(9)\otimes SU(2)_L\otimes U(1)_R$, and we will take the fermion content 
to be:
\begin{eqnarray}
({\bf 36},{\bf 2})_0 & & ({\bf 126}, {\bf 1})_0 \nonumber \\
({\bf \overline{36}},{\bf 1})_{-1} ({\bf \overline{36}},{\bf 1})_1
& & ({\bf \overline{126}}, 
{\bf 1})_0 ~.\nonumber 
\end{eqnarray}

The remaining content of the model lies in specifying the pattern of symmetry
breaking.  As we have mentioned before we are forced to proceed 
phenomenologicaly, so without further adue we assume that a condensate
in the channel $({\bf \overline{36}},1)_1\times({\bf \overline{36}},1)_{-1}
\rightarrow ({\bf \overline{126}},1,)_0$ forms around $10^4$ TeV.   
Below $10^4$TeV, the gauge symmetry is then broken down to $SU(5)_{ETC}\otimes
SU(4)_{PS}\otimes SU(2)_L \otimes U(1)_R$, where PS denotes the usual 
Pati-Salam group.

The fermion content of the theory below $10^4$ TeV is:
\begin{eqnarray}
({\bf 5},{\bf 4},{\bf 2})_0 & ({\bf 10},{\bf 1},{\bf 2})_0 & ({\bf 1}, {\bf 6},
{\bf 2})_0 \nonumber \\
({\bf \overline{5}},{\bf \overline{4}},{\bf 1})_{-1} 
({\bf \overline{5}},{\bf \overline{4}},{\bf 1})_{1} & 
({\bf \overline{10}},{\bf 1},{\bf 1})_{-1} 
({\bf \overline{10}},{\bf 1},{\bf 1})_{1} &  \nonumber \\
({\bf 5},{\bf \overline{4}},{\bf 1})_0 &
({\bf 10},{\bf 6},{\bf 1})_0 ({\bf \overline{10}},{\bf 4},{\bf 1})_0 & 
({\bf \overline{5}},{\bf 1},{\bf 1})_0
({\bf 1},{\bf 1},{\bf 1})_0 \nonumber \\
({\bf \overline{5}},{\bf 4},{\bf 1})_0 &
({\bf \overline{10}},{\bf 6},{\bf 1})_0 ({\bf 10},{\bf \overline{4}},
{\bf 1})_0 
& ({\bf 5},{\bf 1},{\bf 1})_0
({\bf 1},{\bf 1},{\bf 1})_0 ~,\nonumber 
\end{eqnarray}
where the first two lines correspond to the ${\bf 36}$ and 
${\bf \overline{36}}$'s respectively.
Next we imagine that a condensate forms in the channel $({\bf 10},{\bf 6},
{\bf 1})_0 \times ({\bf \overline{5}},{\bf \overline{4}},{\bf 1})_1
\rightarrow ({\bf 5},{\bf 4},{\bf 1})_1$ around 1000 TeV.  This breaks
the gauge symmetry down to $SU(4)_{ETC}\otimes SU(3)_C\otimes SU(2)_L
\otimes U(1)_Y$. We can now see that the
$({\bf 5},{\bf 4},{\bf 2})_0$, the
$({\bf \overline{5}},{\bf \overline{4}},{\bf 1})_{-1}$, and the
$({\bf \overline{5}},{\bf \overline{4}},{\bf 1})_{-1}$ contain particles with
quantum numbers corresponding to three families of ordinary fermions (plus
$\nu_R$'s) and one family of technifermions. Note that
the first family splits off at this scale. We can also see
why $\nu_R^c$'s and right-handed down-type quarks ($d_R^c$'s) are special in 
this model.
Singlets under $SU(4)_{PS} \otimes SU(2)_L \otimes U(1)_R$ will have the 
standard model quantum numbers of $\nu_R$'s, while particles that transform as
$({\bf 6},{\bf 1})_0$ under $SU(4)_{PS} \otimes SU(2)_L \otimes U(1)_R$
will split into particles with standard model quantum
numbers $({\bf \overline{3}},{\bf 1})_{2/3}$
and $({\bf 3},{\bf 1})_{-2/3}$ which correspond respectively 
to $d_R^c$'s
 and an exotic quark which can obtain  a mass with a $d_R^c$ that is gauge
invariant under $SU(3)_C\otimes SU(2)_L \otimes U(1)_Y$.  In fact the
condensate which forms at 1000 TeV generates just such a mass connecting
a $({\bf \overline{4}},{\bf \overline{3}},{\bf 1})_{2/3}$ with a
a$({\bf 4}, {\bf 3}, {\bf 1})_{-2/3}$.  Thus there will 
be extra mixing available for $\nu_R^c$'s and $d_R^c$'s. 

The next stage of breaking is taken to occur around 100 TeV, where a
condensate forms in the channel $({\bf 4},{\bf 1},{\bf 2})_0\times
({\bf 6},{\bf 1},{\bf 2})_0 \rightarrow ({\bf \overline{4}},{\bf 1},{\bf 1})_0$,
which breaks the gauge symmetry down to $SU(3)_{ETC}\otimes SU(3)_C \otimes
SU(2)_L \otimes U(1)_Y$. The second family splits off at this scale.
Once all the group decompositions are performed we see that we have the 
correct fermion content to perform the trick discussed in section 4, 
not only for $\nu_R^c$'s, but also for the $d_R^c$'s.
Thus at the lowest ETC scale, around 10 TeV, we have condensates forming in
the following channels: 
$({\bf \overline{3}},{\bf \overline{3}},{\bf 1})_{2/3} \times
({\bf \overline{3}},{\bf 3},1)_{-2/3} \rightarrow
({\bf 3},{\bf 1},{\bf 1})_0$ (i.e. ``$d_R^c$"'s),
$({\bf \overline{3}},{\bf 1},{\bf 1})_0 \times 
({\bf \overline{3}},{\bf 1},{\bf 1})_0 \rightarrow
({\bf 3},{\bf 1},{\bf 1})_0$ (i.e. ``$\nu_R^c$"'s), and
$({\bf 3},{\bf 1},{\bf 2})_0 \times ({\bf 3},{\bf 1}, {\bf 2})_0 \rightarrow
({\bf \overline{3}},{\bf 1},{\bf 1})_0$.  This breaks the gauge symmetry down
to $SU(2)_{TC}\otimes SU(3)_C \otimes
SU(2)_L \otimes U(1)_Y$, and we have a one family TC theory.

It should be noted that the mechanisms for generating the masses of the $t$
and the $b$ are totally different.  The $t$ gets its mass through the standard
one ETC gauge boson exchange, while the $b$ mass is suppressed by the trick
of Sikivie et. al., also the $b_R^c$ mixes with exotic quarks.  Thus
this model has no problem accomodating a large $t$-$b$ mass splitting, but
it still remains to explain the $t$-$\tau$ mass splitting. As with the first
two families, we can expect about a factor of 10 enhancement from QCD 
and walking effects \cite{colour}.  The remaining factor of 10 
ehancement could arise 
from a 10\% ``fine" tuning of four-fermion couplings in the effective theory 
below 10 TeV \cite{strongETC}.

We also note that our choice of fermion content 
allows us to implement the Nelson-Barr solution to the strong-CP problem
\cite{CP}.  The fundamental theory is assumed to be CP conserving. If
CP is spontaneously broken by complex phases in the masses of particles 
coming from the ${\bf 126}$,  then CP violating phases will appear in 
the CKM matrix, but the determinant of the
mass matrix can be real, so the effective strong CP violating parameter 
$\overline{\theta}$ is identically zero.
\vglue 0.5cm
{\elevenbf \noindent 6. Conclusions \hfil}
\vglue 0.4cm
We have constructed a potentially realistic (i.e. not obviously wrong)
ETC model, which incorporates all the right ingredients: $m_t \gg m_b$,
$m_\nu \approx 0$, a family hierarchy, no bad FCNC's, no massless techniaxion,
and no strong CP problem.  It remains to be seen whether this model can 
survive a more detailed scrutiny, in particular whether it can produce
the observed masses and mixing angles. 
\vglue 0.5cm
{\elevenbf \noindent 7. Acknowledgements \hfil}
\vglue 0.4cm
J.T. would like to thank S. Barr, M. Einhorn, B. Holdom, L. Krauss,  
and R. Sundrum for helpful discussions.
\vglue 0.5cm
{\elevenbf\noindent 8. References \hfil}
\vglue 0.4cm
 
\begin{thebibliography}{9}
 
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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;
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(1987) 568.

\bibitem{strongETC}
T. Appelquist et.al., {\em Phys. Lett.} {\bf 220B}, 223 (1989);
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{\elevenbf B221} (1989) 177;
V.A. Miransky, M. Tanabashi, and K. Yamawaki, {\elevenit Mod. Phys. Lett.} 
{\elevenbf A4} (1989) 1043.

\bibitem{S}E. Gates and J. Terning {\elevenit Phys. Rev. Lett.} {\elevenbf 67}
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\bibitem{Sundrum}R. Sundrum, Lawrence Berkeley Lab. preprint {\elevenbf 32107}.

\bibitem{EL} E. Eichten and K. Lane, {\elevenit Phys. Lett.} {\elevenbf B90}
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\bibitem{RDSC}J.M. Cornwall, {\elevenit Phys. Rev} {\elevenbf D10} (1974) 500;
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\bibitem{colour}B. Holdom, {\elevenit Phys. Rev. Lett.} {\elevenbf 60} (1988) 
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\bibitem{CP}A.E. Nelson, {\elevenit Phys. Lett.} {\elevenbf B136} (1984) 387;
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