%Paper: 
%From: "Manfred B. Lindner" <Y29@vm.hd-net.uni-heidelberg.de>
%Date: Thu, 17 Jun 93 16:04:29 CET

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\def\slimits{\mbox{\tiny
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\ \vskip -.8cm
\ \hskip 12.1cm HD--THEP--93--19

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\vskip 2.3cm
\begin{center}
      {\Large\sc\bf The Phenomenological Viability of} \\
      \ \\
      {\Large\sc\bf Top Condensation}\\
\vskip 1.7cm
      {\sc Manfred Lindner\footnote{Heisenberg Fellow}}\\

\vskip .8cm
      {\sl  Institut f\"ur Theoretische Physik\\
      der Universit\"at Heidelberg\\
      Philosophenweg 16, D--W--6900 Heidelberg}\\
\end{center}

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{\def\thefootnote{}
 \footnote{Email: Y29 at VM.URZ.UNI--HEIDELBERG.DE}
}

\vskip 2.6cm
   \begin{center}{\Large\bf Abstract}\end{center}
\par \vskip .05in
We discuss how the full dynamics of top condensation models can
modify the relations between the physical top mass, the amount
of custodial $SU(2)$ violation and the weak gauge boson masses.
It is emphasized that it is possible to get phenomenologically
acceptable relations between $\Delta\rho$, $m_t$ and $M_W$ and
that in addition the scale of new physics can be chosen to be
${\cal O}(TeV)$ such that a fine--tuning problem is avoided.
} % end all local definitions

\vfill
\noindent
{\sl To appear in the Proceedings of the 4th Hellenic School
     on Elementary Particle Physics}

\newpage
%------- body of paper ---------------------------------------------

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The formation of a $\bar t t$ condensate by some ``pairing force'' could be
responsible for a dynamical breaking of the \EW symmetry \cite{BHL,Hab}.
A theory creating such a condensate would naturally explain a heavy
top mass, would be very helpful to avoid Flavour Changing Neutral Current
(FCNC) problems  and would be very attractive due to its economy.
Simple initial realizations of top condensation were based on effective
Nambu--Jona-Lasinio (NJL) models with non--renormalizable
four--fermion interactions. This led however to discussions about higher
dimensional operators \cite{Hase} which depend crucially on how
``effective'' or ``fundamental'' the four--fermion interactions are.
Subsequently fundamental four--fermion theories where proposed
\cite{L4f} while other authors justify an effective NJL description
of strongly coupled broken gauge theories \cite{TOPCOLOR,U1U1,King,Bon,Mar}.

Independently of such questions we study here the phenomenological
viability of top condensation ideas by assuming essentially only
that we know the solution $\Sigma_t(p^2)$ of some relevant Schwinger--Dyson
(``gap'') equation for the dynamically generated top mass. Thus we
pretend to know the \EW symmetry breaking top propagator to be
\beq
S_t({\rm p}^2) = \frac{i}{\slash{{\rm p}} -\Sigma_t({\rm p}^2)}~,
\label{SF}
\eeq
with the (pole) top mass $m_t=\Sigma_t(m_t^2)$. All other quarks and
leptons are assumed to be massless. Without specifying the gap equation
we assume furthermore that for the theory under consideration\footnote{This
is e.g. justified for asymptotically free theories where chiral
symmetry breaking disappears as $p^2\rightarrow\infty$.}
$\Sigma_t(p^2)\stackrel{p^2\rightarrow\infty}{\bf\longrightarrow}0$
and that there is only one unique solution for $m_t$.

The breaking of the \EW symmetry (i.e. $\Sigma_t\neq 0$) is assumed
to be the result of unspecified new strong forces acting only on the
known quarks and leptons and especially on the $t-b$ doublet.
The emergence of a top condensate breaks global symmetries and the
resulting \GBs are ``eaten'' in a dynamical Higgs mechanism such that
$W$ and $Z$ become massive. Presumably such a theory does not change
significantly if the weak $U(1)_Y$ coupling $g_1$ is sent to
zero\footnote{Indirectly (via vacuum alignment) a small $U(1)_Y$ coupling
could be very important such that $g_1=0$ should be understood as the result
of the limiting procedure $g_1\rightarrow 0$.}. In the limit $g_1=0$ the
corrections which give mass to the $W_3$ and $W_\pm$ propagators must
be induced by the fermions which are representations under both
$SU(2)_L$ and the new strong force. We should therefore study the
contributions of $\Sigma_t$ to the vacuum polarizations of the $W$
and $Z$ propagators. In an expansion in powers of $g_2^2$ the leading
contribution is given by diagrams which connect the $W_\pm$ or $W_3$
line to a fermion pair from both sides. There are two ways how these
four internal fermion lines can be connected: By inserting twice the full
fermionic propagators or by inserting once the full four--fermion
Kernel of the new, strong interaction. Note that in leading order
$g_2^2$ the fermion propagators and the Kernel do not contain any
\EW gauge boson propagation themself since this would cost at least an
extra power of $g_2^2$. Insertions of fermionic vacuum polarizations
into higher order \EW loop diagrams, for example, are suppressed
by corresponding powers of $g_2^2$. Thus in leading order $g_2^2$, but
exact in the new strong coupling, the $W$ propagator is graphically
represented by Fig.~\ref{F1}. The first contribution is a generalization
of the leading \SM diagrams with hard fermion masses replaced by $\Sigma$'s,
i.e. the sum of all one particle irreducible diagrams which contribute
to the dynamically generated fermion masses. The second contribution
contains the exact Kernel $K$ of the strong forces
responsible for condensation and it is useless to expand this Kernel
perturbatively in powers of the coupling constant of the new strong force.
The Goldstone theorem tells us however that the Kernel must contain poles
of massless \GBs due to the breaking of global symmetries by the fermionic
condensates. This is symbolically expressed by the second line of
Fig.~\ref{F1}, where $\tilde K$ does not contain any further poles of
massless particles. But $\tilde K$ may (and typically will) contain
all sort of massive bound states which could e.g. be vectors, Higgs--like
scalars etc. in all possible channels.

The \GB contributions\footnote{Which are essential for a gauge
invariant dynamical Higgs mechanism.} shown in the second line of
Fig.~\ref{F1} were used by Pagels and Stokar \cite{PaSto} to obtain
a relation between the $\Sigma$'s and the \GB decay constants.
Their derivation uses the fact that only the
\GBs contribute a term proportional $p_\mu p_\nu/p^2$ to the $W$
polarization at vanishing external momentum, but this method ignores possible
contributions from $\tilde K$ which enter indirectly via the use of Ward
identities. The $p_\mu p_\nu/p^2$ contributions to $\Pi_{\mu\nu}$ are
balanced (up to small corrections from $\tilde K$) by $g_{\mu\nu}$ terms
created by the first diagram on the {\em rhs} of Fig.~\ref{F1}.
Following ref.~\cite{BluLi} we derive a relation between the $\Sigma$'s
and the \GB decay constants from these $g_{\mu\nu}$ terms. The result
can be compared with the Pagels--Stokar relation and we will see that
contributions from $\tilde K$ are significantly suppressed. Let us
therefore work with rescaled fields such that gauge couplings appear
in the kinetic terms of the gauge boson Lagrangian like
$\left(-1/4g^2\right)\left(W_{\mu\nu}\right)^2$.
Since we do not include any propagating $W$ bosons we need not gauge fix
at this stage. The inverse $W$ propagator can be written as
\beq
\frac{1}{g_2^2}D_{W,\mu\nu}^{-1}(p^2) = \frac{1}{g_2^2}
  \left(-g_{\mu\nu}+\frac{p_\mu p_\nu}{p^2}\right)~p^2
- \Pi_{\mu\nu}(p^2)~,
\label{invprop}
\eeq
with the polarization tensor
$\Pi_{\mu\nu}(p^2)=\left(-g_{\mu\nu}p^2+p_\mu p_\nu\right)\Pi(p^2)$.
At vanishing external momentum the first fermion loop on the {\em rhs}
of Fig.~\ref{F1} contributes to $\Pi_{\mu\nu}$
\beq
\Pi_{\mu\nu} = -iZ^2N_c~\int\frac{d^4k}{(2\pi)^4}~
\frac{
{\sl Tr}\left[\Gamma_\mu(\slash k+\Sigma_1(k))\Gamma_\nu
(\slash k+\Sigma_2(k))\right]
}
{(k^2-\Sigma_1(k)^2)(k^2-\Sigma_2(k)^2)}~,
\label{Pimunu}
\eeq
where $N_c$ is the number of colors, $Z^{-1}=\sqrt{2}, 2$
in the charged and neutral channel, respectively,
$\Gamma_\alpha=\left(\frac{1-\gamma_5}{2}\right)\gamma_\alpha$, and
${\bf +i\epsilon}$ is generally implied in the denominator.
In the neutral channel we get corrections from $\bar t t$
(i.e. $\Sigma_1=\Sigma_2=\Sigma_t$), $\bar b b$
(i.e. $\Sigma_1=\Sigma_2=\Sigma_b\equiv 0$) and in the charged channel
contributes only $\bar t b$ or $\bar b t$
(i.e. $\Sigma_1=\Sigma_t$, $\Sigma_2=\Sigma_b\equiv 0$).
By naive power counting eq.~(\ref{Pimunu}) has quadratic and logarithmic
divergences. Since we assumed
$\Sigma_i(p^2)\stackrel{p^2\rightarrow\infty}{\bf\longrightarrow}0$
for the top quark and all other fermions
we find that the divergences of $\Pi_{\mu\nu}(p^2)$ are identical to those
calculated for $\Sigma_i\equiv0$. It makes therefore sense to split
$\Pi_{\mu\nu}(p^2)=\Pi^0_{\mu\nu}(p^2)+\Delta\Pi_{\mu\nu}(p^2)$ where
$\Pi^0_{\mu\nu}$ is defined as $\Pi_{\mu\nu}$ for $\Sigma_i\equiv0$.
$\Pi^0_{\mu\nu}$ is then an uninteresting $\Sigma_i$ independent constant
which contains all divergences and needs renormalization. Contrary the
interesting $\Sigma_i$ dependent piece
$\Delta\Pi_{\mu\nu}=\Pi_{\mu\nu}-\Pi^0_{\mu\nu}$
is finite, even when the external momentum is sent to zero. Thus
\bea
\Delta\Pi_{\mu\nu} &=&
  -iZ^2N_c~\int\frac{d^4k}{(2\pi )^4}~\left\{
    \frac{
    {\sl Tr}
    \left[\Gamma_\mu(\slash{k}+\Sigma_1)\Gamma_\nu(\slash{k}+\Sigma_2)\right]
           }{(k^2-\Sigma_1^2)(k^2-\Sigma_2^2)}
  -
    \frac{
    {\sl Tr}\left[\Gamma_\mu\slash{k}\Gamma_\nu\slash{k}\right]
           }{k^4}
                                 \right\} \\
                  &=&
  - g_{\mu\nu}~\frac{Z^2N_c}{(4\pi)^2} \int\limits_0^\infty dk^2~
    \frac{k^2(\Sigma_1^2+\Sigma_2^2) - \Sigma_1^2\Sigma_2^2}
    {(k^2-\Sigma_1^2)(k^2-\Sigma_2^2)}~,                 \label{finalDPi}
\eea
where angular integration was performed in Euclidean space and subsequently
continued back to Minkowski space. Under the integral one has as usual
${\sl Tr}\left[\Gamma_\mu\slash{k}\Gamma_\nu\slash{k}\right]=-g_{\mu\nu}k^2$
and ${\sl Tr}\left[\Gamma_\mu\Gamma_\nu\right] = 0$. Note that this
separation procedure for $\Delta\Pi_{\mu\nu}$ does not spoil gauge
invariance.

The \GB decay constants $F_i^2$ are the poles of $\Pi(p^2)$ at vanishing
external momentum. For our definition of $\Pi_{\mu\nu}$ we find that
$F_i^2$ is identical to eq.~(\ref{finalDPi}) without the factor
$-g_{\mu\nu}$. Using $Z$ for the charged and neutral channel one finds
\beq
F_\pm^2 = \frac{N_c}{32\pi^2}\int\limits_0^\infty dk^2~
          \frac{\Sigma_t^2}{k^2-\Sigma_t^2}~,\quad
F_3^2   = \frac{N_c}{32\pi^2}\int\limits_0^\infty dk^2~
          \frac{k^2\Sigma_t^2-\frac{1}{2}\Sigma_t^4}{(k^2-\Sigma_t^2)^2}~,
\label{FF}
\eeq
such that
\beq
F_3^2-F_\pm^2 = \frac{N_c}{64\pi^2}\int\limits_0^\infty dk^2~
                \frac{\Sigma_t^4}{(k^2-\Sigma_t^2)^2}~.
\label{finaldF2}
\eeq
Eq.~(\ref{FF}) for $F_\pm^2$ is equivalent to the result obtained by Pagels
and Stokar \cite{PaSto} from the $q_\mu q_\nu/q^2$ contributions of \GBs
to $\Pi_{\mu\nu}$. The result for the neutral channel in eq.~(\ref{FF})
looks however somewhat different. But by using the integral identity
\beq
\int\limits_0^\infty dx ~\frac{x^2f(x)^\prime - f(x)^2}
{\left( x-f(x) \right)^2} = f(\infty )~,
\label{intid}
\eeq
for $x=k^2$ and $f=\Sigma_t^2$ we can rewrite eq.~(\ref{FF}) into
\beq
F_3^2 = \frac{N_c}{32\pi^2}\int\limits_0^\infty dk^2~k^2~
        \frac{\Sigma_t^2- k^2\Sigma_t\Sigma_t^\prime}{(k^2-\Sigma_t^2)^2}~,
\label{PASTOres}
\eeq
where $\Sigma_t^\prime = d\Sigma_t /dk^2$. Even though this looks now
formally similar to the Pagels--Stokar result it differs by a factor 2
in front of the derivative term in the nominator of eq.~(\ref{PASTOres}).
This difference may appear less important, but we will see
that in the limit of a hard top mass our method produces the correct
$\rho$--parameter, while the Pagels--Stokar result produces 3/2 times the
correct answer. Additionally our expression leads also to a better numerical
estimate of $f_\pi$ if we follow the methods of ref.~\cite{PaSto}.

The $\rho$--parameter \cite{rhoinvent} is defined as $\rho:=F_\pm^2/F_3^2$
which can now be written as
\beq
\rho = 1+\Delta\rho =
\frac{F_\pm^2}{F_3^2} = \left(1+\frac{(F_3^2-F_\pm^2)}{F_\pm^2}\right)^{-1}
\simeq 1 - 2~\frac{(F_3^2-F_\pm^2)}{v^2}~,
\label{rhoaprox}
\eeq
and from eq.~(\ref{finaldF2}) we find the contribution of the $t-b$
doublet
\beq
\Delta\rho = \frac{-N_c}{32\pi^2v^2}~\int\limits_0^\infty dk^2~
             \frac{\Sigma_t^4}{(k^2-\Sigma_t^2)^2}~,
\label{finalrho}
\eeq
where we used $F_\pm^2=v^2/2$ with $v\simeq 175~GeV$ in the denominator.
Model independent parametrizations of radiative corrections parametrize
the information contained in $\Delta\rho$ essentially in the variables
$T$ \cite{PesTa} or $\epsilon_1$ \cite{Alta}.

With the expressions for $\Delta\rho$ in eq.~(\ref{finalrho}) and
$F_i^2$ in eq.~(\ref{FF}) we can calculate for given
$\Sigma_t(p^2)\stackrel{p^2\rightarrow \infty}{\longrightarrow}0$ three
independent observable quantities which are one of the weak gauge boson
masses (either $M_W^2 = g_2^2F_\pm^2$ or $M_Z^2 = (g_1^2+g_2^2)F_3^2~$),
$\Delta\rho$ and furthermore the physical top mass $m_t$. These three
quantities are dominated by different momenta and therefore
$\Sigma\neq constant$ leads to a different relation than a constant, i.e.
hard mass. It is instructive to look at the degree of convergence of the
involved integrals. The \GB decay constants $F_i^2$ are formally log.
divergent, but are finite with our assumption on $\Sigma_t(p^2)$. In that
case renormalization is not needed, but due to the formal log. divergence
$\Sigma_t$ contributes with equal weight at all momentum scales. In other
words, the magnitude of $F_i^2$ depends crucially on the high energy tail
of $\Sigma_t$. The difference $F_\pm^2 - F_3^2$ has better convergence
properties and is always finite, even for $\Sigma_t(p^2)=constant$. This
implies that $\Delta\rho$ is finite, as it should be, and it is most
sensitive to infrared scales somewhat above $m_t$. Finally $m_t$ is of
course only sensitive to one point, namely $m_t=\Sigma_t(m_t^2)$.

We would like to study now corrections in the relation between $m_t$,
$M_W$ and $\Delta\rho$ when $\Sigma_t$ is the solution of a hypothetical
Schwinger--Dyson equation which deviates from $\Sigma_t=m_t= constant$.
First we would like to see if the correct \SM result emerges for a $t-b$
doublet when $\Sigma_t\rightarrow m_t$. Therefore we set
\beq
\Sigma_t(p^2) = m_t\,\Theta(\Lambda^2-p^2)~,
\label{topan1}
\eeq
and ignore again the $b$ quark mass. From eq.~(\ref{finalrho}) we obtain
for our ansatz
\beq
\Delta\rho
= \frac{N_cm_t^2}{32\pi^2v^2}\left(\frac{1}{1-m_t^2/\Lambda^2}\right)
  ~\stackrel{\Lambda\rightarrow\infty}{\longrightarrow}~\Delta\rho^{SM}
= \frac{N_c\alpha_{em}}{16\pi\sin^2\theta_W\cos^2\theta_W}
  ~\frac{m_t^2}{M_Z^2}~.
\label{rhotheta}
\eeq
Note that in the limit $\Lambda\rightarrow\infty$ (i.e. a hard, constant
top mass) we obtain correctly the leading \SM value while the Pagels--Stokar
relation would produce 3/2 times the \SM result.
For finite $\Lambda$ eq.~(\ref{rhotheta}) describes furthermore the
modification of the \SM result due to a physical high energy momentum
cutoff. Such a cutoff makes $\Delta\rho$ a little bit more positive than
in the \SM which implies for a fixed experimental value of $\Delta\rho$
a lower top mass prediction. From eq.~(\ref{FF}) it is in addition
possible to determine $M_W$ for the ansatz eq.~(\ref{topan1})
\beq
M_W^2
 = g_2^2F_\pm^2
 = \frac{g_2^2N_c}{32\pi^2}
         ~\int\limits_0^{\Lambda^2}dk^2~\frac{\Sigma_t^2}{k^2-\Sigma_t^2}
 =\frac{g_2^2N_c}{32\pi^2}~m_t^2~
        \ln{\left(\frac{\Lambda^2-m_t^2}{m_t^2}\right)}~.
\label{MWmttheta}
\eeq
Taking as experimental input $M_W=80.14\pm 0.27~GeV$,
$\Delta\rho = 0.005\pm 0.008$, $\alpha_{em}^{-1}(M_Z^2) = 127.8\pm 0.1$
and $\sin^2\theta_W^{eff}(M_Z^2) = 0.2318\pm 0.0007$ we plot in
Fig.~\ref{F2} the two central top mass values resulting from
eqs.~(\ref{rhotheta}) and (\ref{MWmttheta}) as a function of $\Lambda$
(dashed lines).

The ansatz eq.~(\ref{topan1}) can be viewed as the result of a
Nambu--Jona-Lasinio (NJL) gap equation of top condensation as for
example in the model of Bardeen, Hill and Lindner (BHL) \cite{BHL}.
In fact a NJL gap equation is the simplest conceivable Schwinger--Dyson
equation where $\Sigma_t$ is forced to be a constant. Fig.~\ref{F2} shows
clearly that ultra high values of $\Lambda$ and the experimental errors are
required to get the two top mass values in agreement. For such high
$\Lambda$ the effective Lagrangian is valid for many orders of magnitude
which led in the BHL analysis to the so--called ``renormalization group
improvement''. This means in the current language
that $\Sigma_t=constant$ is replaced by
$\Sigma_t=g_t(p^2)v$, where $v=175~GeV$ and $g_t(p^2)$ is the solution
of the one--loop renormalization group equation. In BHL the predicted
top mass is then the ``effective fixedpoint'' of the renormalization
group flow. The same result could be seen in eq.~(\ref{MWmttheta}) since
the effective fixedpoint dictates the shape of $\Sigma(p^2)$ for many
orders of magnitude. The BHL scenario has however phenomenological problems.
First the very high value of $\Lambda$ is nothing else then the old
hierarchy problem which appears now as a fine--tuning of the four--fermion
coupling $G$. Furthermore the infrared fixedpoint prediction is higher
than the dashed curve resulting from eq.~(\ref{MWmttheta}) which is
shown in Fig.~\ref{F2} and has (within newest experimental errors) no
intersection with the line resulting from eq.~(\ref{rhotheta}). Thus this
simplest scenario seems unacceptable even for very high values of $\Lambda$.

Remembering that $\Delta\rho$ and $M_W$ are sensitive to details
of $\Sigma_t$ in a different way we should ask ourselfs if the above
problems can be solved by modifications of the solution $\Sigma_t(p^2)$.
The answer is of course yes, and we illustrate now the two most
important type of changes: The addition of a slowly falling tail
and/or the addition of a ``bump'' somewhat above $m_t$.

First we consider a very rough ansatz for a ``bump'' between
$\Lambda_1$ and $\Lambda$ with $m_t<\Lambda_1<\Lambda$
by modifying eq.~(\ref{topan1})
\beq
\Sigma_t(p^2) = \left\{
\begin{array}{ll}
0                 & {\rm for~} p^2 > \Lambda^2~;\\
\sqrt{r}\cdot m_t & {\rm for~} \Lambda_1^2 \leq p^2 \leq \Lambda^2~;\\
m_t               & {\rm for~} p^2<\Lambda_1^2~,
\end{array}\right.
\label{topan2}
\eeq
where $\Sigma$ is changed between $\Lambda_1$ and $\Lambda$. For $r>1$
there is an extra ``bump'' between $\Lambda_1$ and $\Lambda$ which affects
$\Delta\rho$. For $\Lambda^2,\Lambda_1^2 \gg m_t^2,rm_t^2$ we get
\beq
\Delta\rho \simeq
\frac{N_cm_t^2}{32\pi^2v^2}\left(1+\frac{m_t^2}{\Lambda^2}
        -\left[\frac{m_t^2(\Lambda^2-\Lambda_1^2)}
         {\Lambda^2\Lambda_1^2}~(r^2-1)\right]
\right)~,
\label{rhobump}
\eeq
where extra contributions due to $r\neq 1$ and $\Lambda_1\neq\Lambda$
are isolated in square brackets. We can see that the bump counteracts
the effect of the cutoff and makes $\Delta\rho$ less positive. In
principle the bump can even be chosen to make $\Delta\rho$ vanish.
The relation eq.~(\ref{MWmttheta}) between $m_t$ and $M_W$ becomes also
modified. For $\Lambda^2,\Lambda_1^2 \gg m_t^2,rm_t^2$ we get approximately
\beq
M_W^2 \simeq \frac{g_2^2N_c}{32\pi^2}~m_t^2~
\left(\ln{\left(\frac{\Lambda^2-m_t^2}{m_t^2}\right)}
        +\left[(r-1)\ln{\left(\frac{\Lambda^2}{\Lambda_1^2}\right)}\right]
\right)~,
\label{MWmtbump}
\eeq
where extra contributions due to the bump are again isolated in square
brackets.

Now we add a slowly falling high energy tail to the last ansatz
eq.~(\ref{topan2})
\beq
\Sigma_t(p^2) = \left\{
\begin{array}{ll}
{\rm equation~(\ref{topan2})}                   &{\rm for~}p^2<\Lambda^2~;\\
\sqrt{r}
m_t~\left(\frac{p^2}{\Lambda^2}\right)^{-\alpha}&{\rm for~}p^2>\Lambda^2~,
\end{array}\right.
\label{topan3}
\eeq
where $\alpha>0$ is assumed. This high energy tail which is parametrized
by $\alpha$ leads to
\beq
\Delta\rho \simeq
\frac{N_cm_t^2}{32\pi^2v^2}\left(1+\frac{m_t^2}{\Lambda^2}
        -\left[\frac{m_t^2(\Lambda^2-\Lambda_1^2)}
         {\Lambda^2\Lambda_1^2}~(r^2-1)\right]
        -\left\{\frac{r^2}{4\alpha+1}~\frac{m_t^2}{\Lambda^2}\right\}
\right)~,
\label{rhotail}
\eeq
and
\beq
M_W^2 =\frac{g_2^2N_c}{32\pi^2}~m_t^2~
\left(\ln{\left(\frac{\Lambda^2-m_t^2}{m_t^2}\right)}
        +\left[(r-1)\ln{\left(\frac{\Lambda^2}{\Lambda_1^2}\right)}\right]
        +\left\{\frac{r}{2\alpha}\right\}\right)~,
\label{MWmttail}
\eeq
where the extra corrections due to the tail are isolated in curly
brackets.

Note that we are looking for a scenario which simultaneously avoids
the fine--tuning problem and which is phenomenologically acceptable.
Consequently $\Lambda$ and $\Lambda_1$ should be $TeV$--ish and
the top mass values required from the $\Delta\rho$-- and $M_W$--data
should agree. This requires consequently some gap equation with a
generic condensation scale ${\cal O}(TeV)$ capable of producing a bump,
and a tail -- maybe of the type discussed in ref.~\cite{BluLi}.
The asymptotic high energy behaviour
of $\Sigma_t$ might be described by a renormalization group equation
if the spectrum of the theory does not contain further mass thresholds.
This would imply a logarithmic tail and the parameter $\alpha$ should be
very small. We could for example fix $\alpha$ in the minimal scenario by
expanding the Higgs less one--loop renormalization group equation
for $g_t$ in the \SM. This would lead to $\alpha \simeq 0.04$.
%%% this is the medium value of (8g_3^2+9g_2^2/4+17g_1^2/12)/16\pi^2 %%%
For such small values of $\alpha$ the tail leads to mild effects
in the $\rho$--parameter and drastic changes in the $M_W$--$m_t$ relation.

We can illustrate the effects of the combined bump and tail
by plotting  eqs.~(\ref{rhotail}) and (\ref{MWmttail}) in Fig.~\ref{F2}
as solid lines for the parameters $r=2$, $\Lambda=2\Lambda_1$,
$\Lambda_1=2m_t$ and $\alpha=0.04$.
The small value of $\alpha$ (corresponding to a logarithmic high energy
tail of $\Sigma_t$) influences mostly the $M_W$--$m_t$ relation while
the bump affects essentially only the $\Delta\rho$--$m_t$ relation.
Taking into account experimental and theoretical errors the two top mass
values agree for low values of $\Lambda$ consistent with the above
assumptions and avoiding fine--tuning. We have thus illustrated that
structured solutions of $\Sigma_t$ can solve the fine--tuning problem,
i.e. allow for $\Lambda$--values within a few $TeV$. Furthermore the
predicted $m_t$--$M_W$--$\Delta\rho$ relations are modified to be
consistent with the data on $M_W$ and $\Delta\rho$. The predicted
top mass differs however typically somewhat from its \SM
value -- something that will only be tested by a direct search for the
top quark. A bump and a tail as discussed could for example be
relevant in models of top condensation where heavy gauge bosons trigger
condensation \cite{U1U1} or in bootstrap scenarios where the
$t$--channel effects of a composite Higgs are non--negligible \cite{Nambu}.

There are other \EW observables which are sensitive to the top mass value
like for example the $Z{\bar b}b$ vertex. If $m_t$ is replaced by
$\Sigma_t$ in the relevant diagrams then one finds however that the
top mass dependence is replaced by sensitivity to $\Sigma_t$ at
low momenta. Thus in a first approximation these quantities
depend essentially on the pole mass. There are however corrections
which should become observable if high enough precision can be reached.

In summary we find that top condensation models are both phenomenological
viable and natural if $\Sigma_t$ has suitable structure. The calculation
of $\Sigma_t$ from first principles is however in general very difficult
for a given model due to the non--perturbative nature of the relevant
Schwinger--Dyson equation. But a reliable probe of the discussed effects
will emerge when the Fermilab Collider starts to push the direct top
mass limits into the \SM window. If the above ideas are relevant then
the top quark should not be found precisely in the often cited \SM window
but somewhat higher.

%------------------------------------------------------------------------
%
% References

\newpage
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\bibitem{Hab}Further developments are reviewed in M. Lindner,
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\bibitem{Hase}A. Hasenfratz, P. Hasenfratz, K. Jansen, J. Kuti and
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\bibitem{L4f}J. Zinn--Justin, Nucl. Phys. B367 (1991) 105;\\
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\bibitem{TOPCOLOR}C.T. Hill, Phys. Lett. B266 (1991) 419.
\bibitem{U1U1}M. Lindner and D. Ross, Nucl. Phys. B370 (1992) 30.
\bibitem{King}S.F. King, Phys. Rev. D45 (1992) 990.
\bibitem{Bon}R. B\"onisch, Phys. Lett. B268 (1991) 394.
\bibitem{Mar}S.P. Martin, Phys. Rev. D44 (1991) 2892 and D46 (1992) 2197.
\bibitem{BluLi}A. Blumhofer and M. Lindner, Heidelberg preprint
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\bibitem{PaSto}H.Pagels and S. Stokar, Phys. Rev. D20 (1979) 2947;\\
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\end{thebibliography}
%------------------------------------------------------------------------
% Figures

\newpage

\ \vskip -1.5cm

\centerline{\huge\bf Figures}

%    FIGURE 1

\begin{figure}[ht]
\special{psfile=corfig1.psf hscale=70 vscale=70 hoffset=-100 voffset=180
                angle=-90}
\vspace*{0.8cm}
\unitlength1cm
    \begin{picture}(15,3.6)
    \put(7.95,2.75){$K$}
    \put(12.45,0.55){$\tilde K$}
    \put(7.95,1.2){$\frac{i}{q^2}$}
    \end{picture}
\caption{\label{F1}
{\sl The $W$ propagator in leading order $g_2^2$ and exact
in the new non--perturbative interactions. Fermionic self--energies
are represented as fat dots and the four--fermion Kernel $K$
is represented by a fat circle. In the second line the Kernel is
split into \GB contributions (which arise due to the broken global
symmetries with some non--trivial vertex function) and $\tilde K$
(which has no further massless poles).}}
\end{figure}

%    FIGURE 2

\begin{figure}[hb]
\special{psfile=corfig2.psf hscale=60 vscale=60 hoffset=-25 voffset=10
                angle=-90}
\vspace*{6.5cm}
\unitlength1cm
    \begin{picture}(15,4)
    \end{picture}
\caption{\label{F2}
{\sl The predicted (pole) top mass $m_t$ versus $\Lambda$
using $\Delta\rho$ and $M_W$ as experimental input. The upper dashed
line follows from eq.~(\ref{MWmttheta}) and the lower dashed line
from eq.~(\ref{rhotheta}). The solid lines follow from the combined
bump and tail ansatz for $\Sigma_t$ showing that low values of
$\Lambda$ are then possible.}}
\end{figure}

\end{document}
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 1448  356 d 1448  360 d 1452  370 d 1452  376 d 1450  380 d 1448  382 d
 1448  360 r 1454  370 d 1454  378 d 1452  380 d 1448  382 d 1444  382 d
 1412  406 r 1414  400 d 1420  396 d 1430  394 d 1436  394 d 1446  396 d
 1452  400 d 1454  406 d 1454  410 d 1452  416 d 1446  420 d 1436  422 d
 1430  422 d 1420  420 d 1414  416 d 1412  410 d 1412  406 d 1414  402 d
 1416  400 d 1420  398 d 1430  396 d 1436  396 d 1446  398 d 1450  400 d
 1452  402 d 1454  406 d 1454  410 r 1452  414 d 1450  416 d 1446  418 d
 1436  420 d 1430  420 d 1420  418 d 1416  416 d 1414  414 d 1412  410 d
 1412  446 r 1414  440 d 1420  436 d 1430  434 d 1436  434 d 1446  436 d
 1452  440 d 1454  446 d 1454  450 d 1452  456 d 1446  460 d 1436  462 d
 1430  462 d 1420  460 d 1414  456 d 1412  450 d 1412  446 d 1414  442 d
 1416  440 d 1420  438 d 1430  436 d 1436  436 d 1446  438 d 1450  440 d
 1452  442 d 1454  446 d 1454  450 r 1452  454 d 1450  456 d 1446  458 d
 1436  460 d 1430  460 d 1420  458 d 1416  456 d 1414  454 d 1412  450 d
  901  372 r  939  372 d  897  374 r  939  374 d  897  374 r  927  352 d
  927  384 d  939  366 r  939  380 d  897  406 r  899  400 d  905  396 d
  915  394 d  921  394 d  931  396 d  937  400 d  939  406 d  939  410 d
  937  416 d  931  420 d  921  422 d  915  422 d  905  420 d  899  416 d
  897  410 d  897  406 d  899  402 d  901  400 d  905  398 d  915  396 d
  921  396 d  931  398 d  935  400 d  937  402 d  939  406 d  939  410 r
  937  414 d  935  416 d  931  418 d  921  420 d  915  420 d  905  418 d
  901  416 d  899  414 d  897  410 d  897  446 r  899  440 d  905  436 d
  915  434 d  921  434 d  931  436 d  937  440 d  939  446 d  939  450 d
  937  456 d  931  460 d  921  462 d  915  462 d  905  460 d  899  456 d
  897  450 d  897  446 d  899  442 d  901  440 d  905  438 d  915  436 d
  921  436 d  931  438 d  935  440 d  937  442 d  939  446 d  939  450 r
  937  454 d  935  456 d  931  458 d  921  460 d  915  460 d  905  458 d
  901  456 d  899  454 d  897  450 d  389  378 r  391  376 d  393  378 d
  391  380 d  389  380 d  385  378 d  383  374 d  383  368 d  385  362 d
  389  358 d  393  356 d  401  354 d  413  354 d  419  356 d  423  360 d
  425  366 d  425  370 d  423  376 d  419  380 d  413  382 d  411  382 d
  405  380 d  401  376 d  399  370 d  399  368 d  401  362 d  405  358 d
  411  356 d  383  368 r  385  364 d  389  360 d  393  358 d  401  356 d
  413  356 d  419  358 d  423  362 d  425  366 d  425  370 r  423  374 d
  419  378 d  413  380 d  411  380 d  405  378 d  401  374 d  399  370 d
  383  406 r  385  400 d  391  396 d  401  394 d  407  394 d  417  396 d
  423  400 d  425  406 d  425  410 d  423  416 d  417  420 d  407  422 d
  401  422 d  391  420 d  385  416 d  383  410 d  383  406 d  385  402 d
  387  400 d  391  398 d  401  396 d  407  396 d  417  398 d  421  400 d
  423  402 d  425  406 d  425  410 r  423  414 d  421  416 d  417  418 d
  407  420 d  401  420 d  391  418 d  387  416 d  385  414 d  383  410 d
  383  446 r  385  440 d  391  436 d  401  434 d  407  434 d  417  436 d
  423  440 d  425  446 d  425  450 d  423  456 d  417  460 d  407  462 d
  401  462 d  391  460 d  385  456 d  383  450 d  383  446 d  385  442 d
  387  440 d  391  438 d  401  436 d  407  436 d  417  438 d  421  440 d
  423  442 d  425  446 d  425  450 r  423  454 d  421  456 d  417  458 d
  407  460 d  401  460 d  391  458 d  387  456 d  385  454 d  383  450 d
 1950 3000 r  150 3000 d 1950 3000 r 1950 2936 d 1821 3000 r 1821 2968 d
 1693 3000 r 1693 2968 d 1564 3000 r 1564 2968 d 1436 3000 r 1436 2936 d
 1307 3000 r 1307 2968 d 1179 3000 r 1179 2968 d 1050 3000 r 1050 2968 d
  921 3000 r  921 2936 d  793 3000 r  793 2968 d  664 3000 r  664 2968 d
  536 3000 r  536 2968 d  407 3000 r  407 2936 d  279 3000 r  279 2968 d
  150 3000 r  150 2968 d 2067 1530 r 2109 1530 d 2067 1532 r 2109 1532 d
 2067 1524 r 2067 1538 d 2109 1524 r 2109 1554 d 2099 1554 d 2109 1552 d
 2081 1576 r 2083 1570 d 2087 1566 d 2093 1564 d 2097 1564 d 2103 1566 d
 2107 1570 d 2109 1576 d 2109 1580 d 2107 1586 d 2103 1590 d 2097 1592 d
 2093 1592 d 2087 1590 d 2083 1586 d 2081 1580 d 2081 1576 d 2083 1572 d
 2087 1568 d 2093 1566 d 2097 1566 d 2103 1568 d 2107 1572 d 2109 1576 d
 2109 1580 r 2107 1584 d 2103 1588 d 2097 1590 d 2093 1590 d 2087 1588 d
 2083 1584 d 2081 1580 d 2081 1616 r 2083 1612 d 2085 1610 d 2089 1608 d
 2093 1608 d 2097 1610 d 2099 1612 d 2101 1616 d 2101 1620 d 2099 1624 d
 2097 1626 d 2093 1628 d 2089 1628 d 2085 1626 d 2083 1624 d 2081 1620 d
 2081 1616 d 2083 1612 r 2087 1610 d 2095 1610 d 2099 1612 d 2099 1624 r
 2095 1626 d 2087 1626 d 2083 1624 d 2085 1626 r 2083 1628 d 2081 1632 d
 2083 1632 d 2083 1628 d 2097 1610 r 2099 1608 d 2103 1606 d 2105 1606 d
 2109 1608 d 2111 1614 d 2111 1624 d 2113 1630 d 2115 1632 d 2105 1606 r
 2107 1608 d 2109 1614 d 2109 1624 d 2111 1630 d 2115 1632 d 2117 1632 d
 2121 1630 d 2123 1624 d 2123 1612 d 2121 1606 d 2117 1604 d 2115 1604 d
 2111 1606 d 2109 1612 d 2059 1660 r 2063 1656 d 2069 1652 d 2077 1648 d
 2087 1646 d 2095 1646 d 2105 1648 d 2113 1652 d 2119 1656 d 2123 1660 d
 2063 1656 r 2071 1652 d 2077 1650 d 2087 1648 d 2095 1648 d 2105 1650 d
 2111 1652 d 2119 1656 d 2067 1686 r 2109 1672 d 2067 1686 r 2109 1700 d
 2073 1686 r 2109 1698 d 2109 1666 r 2109 1678 d 2109 1692 r 2109 1706 d
 2059 1746 r 2123 1710 d 2075 1762 r 2073 1766 d 2067 1772 d 2109 1772 d
 2069 1770 r 2109 1770 d 2109 1762 r 2109 1780 d 2073 1860 r 2077 1862 d
 2067 1862 d 2073 1860 d 2069 1856 d 2067 1850 d 2067 1846 d 2069 1840 d
 2073 1836 d 2077 1834 d 2083 1832 d 2093 1832 d 2099 1834 d 2103 1836 d
 2107 1840 d 2109 1846 d 2109 1850 d 2107 1856 d 2103 1860 d 2067 1846 r
 2069 1842 d 2073 1838 d 2077 1836 d 2083 1834 d 2093 1834 d 2099 1836 d
 2103 1838 d 2107 1842 d 2109 1846 d 2093 1860 r 2109 1860 d 2093 1862 r
 2109 1862 d 2093 1854 r 2093 1866 d 2093 1878 r 2093 1902 d 2089 1902 d
 2085 1900 d 2083 1898 d 2081 1894 d 2081 1888 d 2083 1882 d 2087 1878 d
 2093 1876 d 2097 1876 d 2103 1878 d 2107 1882 d 2109 1888 d 2109 1892 d
 2107 1898 d 2103 1902 d 2093 1900 r 2087 1900 d 2083 1898 d 2081 1888 r
 2083 1884 d 2087 1880 d 2093 1878 d 2097 1878 d 2103 1880 d 2107 1884 d
 2109 1888 d 2067 1916 r 2109 1930 d 2067 1918 r 2103 1930 d 2067 1944 r
 2109 1930 d 2067 1910 r 2067 1924 d 2067 1938 r 2067 1950 d 2059 1958 r
 2063 1962 d 2069 1966 d 2077 1970 d 2087 1972 d 2095 1972 d 2105 1970 d
 2113 1966 d 2119 1962 d 2123 1958 d 2063 1962 r 2071 1966 d 2077 1968 d
 2087 1970 d 2095 1970 d 2105 1968 d 2111 1966 d 2119 1962 d 1082  250 r
 1082  278 d 1080  250 r 1080  278 d 1080  256 r 1076  252 d 1070  250 d
 1066  250 d 1060  252 d 1058  256 d 1058  278 d 1066  250 r 1062  252 d
 1060  256 d 1060  278 d 1058  256 r 1054  252 d 1048  250 d 1044  250 d
 1038  252 d 1036  256 d 1036  278 d 1044  250 r 1040  252 d 1038  256 d
 1038  278 d 1088  250 r 1080  250 d 1088  278 r 1074  278 d 1066  278 r
 1052  278 d 1044  278 r 1030  278 d 1020  265 r 1020  285 d 1019  289 d
 1016  290 d 1014  290 d 1012  289 d 1010  286 d 1019  265 r 1019  285 d
 1018  289 d 1016  290 d 1024  273 r 1014  273 d   6 lw lt2  150  972 r
  415  993 d  615 1026 d  746 1059 d  842 1092 d  917 1125 d  978 1158 d
 1029 1191 d 1072 1224 d 1109 1257 d 1142 1289 d 1171 1322 d 1197 1355 d
 1220 1388 d 1242 1421 d 1261 1454 d 1279 1487 d 1296 1520 d 1311 1553 d
 1325 1586 d 1339 1618 d 1351 1651 d 1363 1684 d 1374 1717 d 1385 1750 d
 1394 1783 d 1404 1816 d 1413 1849 d 1421 1882 d 1429 1914 d 1437 1947 d
 1445 1980 d 1452 2013 d 1458 2046 d 1465 2079 d 1471 2112 d 1477 2145 d
 1483 2178 d 1489 2211 d 1494 2243 d 1499 2276 d 1504 2309 d 1509 2342 d
 1514 2375 d 1518 2408 d 1523 2441 d 1527 2474 d 1531 2507 d 1535 2539 d
 1539 2572 d 1543 2605 d 1546 2638 d 1550 2671 d 1554 2704 d 1557 2737 d
 1560 2770 d 1564 2803 d 1567 2836 d 1570 2868 d 1573 2901 d 1576 2934 d
 1579 2967 d 1581 3000 d 1632  862 r 1626  895 d 1624  928 d 1624  961 d
 1624  993 d 1624 1026 d 1624 1059 d 1624 1092 d 1624 1125 d 1624 1158 d
 1624 1191 d 1624 1224 d 1624 1257 d 1624 1289 d 1624 1322 d 1624 1355 d
 1624 1388 d 1624 1421 d 1624 1454 d 1624 1487 d 1624 1520 d 1624 1553 d
 1624 1586 d 1624 1618 d 1624 1651 d 1624 1684 d 1624 1717 d 1624 1750 d
 1624 1783 d 1624 1816 d 1624 1849 d 1624 1882 d 1624 1914 d 1624 1947 d
 1624 1980 d 1624 2013 d 1624 2046 d 1624 2079 d 1624 2112 d 1624 2145 d
 1624 2178 d 1624 2211 d 1624 2243 d 1624 2276 d 1624 2309 d 1624 2342 d
 1624 2375 d 1624 2408 d 1624 2441 d 1624 2474 d 1624 2507 d 1624 2539 d
 1624 2572 d 1624 2605 d 1624 2638 d 1624 2671 d 1624 2704 d 1624 2737 d
 1624 2770 d 1624 2803 d 1624 2836 d 1624 2868 d 1624 2901 d 1624 2934 d
 1624 2967 d 1624 3000 d lt0 1335  862 r 1350  895 d 1362  928 d 1373  961 d
 1384  993 d 1394 1026 d 1403 1059 d 1412 1092 d 1421 1125 d 1429 1158 d
 1437 1191 d 1444 1224 d 1451 1257 d 1458 1289 d 1464 1322 d 1471 1355 d
 1477 1388 d 1482 1421 d 1488 1454 d 1493 1487 d 1499 1520 d 1504 1553 d
 1509 1586 d 1513 1618 d 1518 1651 d 1522 1684 d 1527 1717 d 1531 1750 d
 1535 1783 d 1539 1816 d 1542 1849 d 1546 1882 d 1550 1914 d 1553 1947 d
 1557 1980 d 1560 2013 d 1563 2046 d 1566 2079 d 1570 2112 d 1573 2145 d
 1575 2178 d 1578 2211 d 1581 2243 d 1584 2276 d 1587 2309 d 1589 2342 d
 1592 2375 d 1594 2408 d 1597 2441 d 1599 2474 d 1602 2507 d 1604 2539 d
 1606 2572 d 1608 2605 d 1611 2638 d 1613 2671 d 1615 2704 d 1617 2737 d
 1619 2770 d 1621 2803 d 1623 2836 d 1625 2868 d 1627 2901 d 1628 2934 d
 1630 2967 d 1632 3000 d 1113  862 r 1396  895 d 1439  928 d 1451  961 d
 1455  993 d 1456 1026 d 1456 1059 d 1456 1092 d 1456 1125 d 1456 1158 d
 1456 1191 d 1456 1224 d 1456 1257 d 1456 1289 d 1456 1322 d 1456 1355 d
 1456 1388 d 1456 1421 d 1456 1454 d 1456 1487 d 1456 1520 d 1456 1553 d
 1456 1586 d 1456 1618 d 1456 1651 d 1456 1684 d 1456 1717 d 1456 1750 d
 1456 1783 d 1456 1816 d 1456 1849 d 1456 1882 d 1456 1914 d 1456 1947 d
 1456 1980 d 1456 2013 d 1456 2046 d 1456 2079 d 1456 2112 d 1456 2145 d
 1456 2178 d 1456 2211 d 1456 2243 d 1456 2276 d 1456 2309 d 1456 2342 d
 1456 2375 d 1456 2408 d 1456 2441 d 1456 2474 d 1456 2507 d 1456 2539 d
 1456 2572 d 1456 2605 d 1456 2638 d 1456 2671 d 1456 2704 d 1456 2737 d
 1456 2770 d 1456 2803 d 1456 2836 d 1456 2868 d 1456 2901 d 1456 2934 d
 1456 2967 d 1456 3000 d
e
%%Trailer
EndPSPlot
%--- END OF FIGURE 2 ---------------------------

____________________________________________________________________
M. Lindner, Institut f. Theoretische Physik, Universitaet Heidelberg,
Philosophenweg 16, D-W-6900 Heidelberg, Germany; Tel. D-6221-569-414,
FAX : D-6221-569-331, Email Y29@VM.URZ.UNI-HEIDELBERG.DE

