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\title{MASS ENHANCEMENT AND CRITICAL BEHAVIOR IN TECHNICOLOR THEORIES}
\author{Thomas Appelquist, John Terning\\
Department of Physics, Yale University, New Haven, CT 06520 \\ and\\L.C.R.
Wijewardhana \\ Department of Physics, University of Cincinnati, Cincinnati,
OH 45221}
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\begin{abstract}
      Quark and lepton masses in technicolor theories can be enhanced if the
high energy, extended technicolor (ETC) interactions play an important  
role in electroweak symmetry breaking. This happens when the ETC coupling 
and the technicolor gauge coupling at high energies lie 
close to a certain 
critical line. The enhancement has been associated with the existence of 
composite scalars made mainly of technifermions, with
masses small compared to the ETC scale. The initial study of these states was 
carried out with the technicolor gauge coupling 
neglected. In this paper we investigate the properties of
such scalars including the technicolor gauge
interactions. We find that for realistic values of the gauge coupling, the 
scalars will not be narrow resonances. Mass and
width estimates are made and  some comments about the phenomenology of these
states are included. 
 
\end{abstract}
 
 
 
The recent revival of interest in technicolor theories of electroweak symmetry
breaking has been stimulated partly by the observation that these theories
should not necessarily be viewed as scaled-up versions of QCD. Momentum
components well
above the confinement scale $\Lambda_{tc}$ can play a more important role than
they do in QCD --  with important consequences such as the generation of very
large fermion masses. Walking technicolor \cite{walk} is one example of this
phenomenon. Another possibility is that the higher energy extended technicolor
(ETC) interactions,
which must be present to generate the masses of ordinary fermions, can play an
important direct role, along with the technicolor interactions, in the
electroweak breaking, leading to even larger fermion masses
\cite{AETW,Takeuchi}. This can
take place only if the combination of the ETC coupling and the technicolor
coupling at the ETC scale is sufficiently close to a certain critical
curve \cite{ASTW}.
 
It has recently been suggested that this ETC-driven enhancement is
associated with the appearance of composite scalars that are
light compared to the ETC scale \cite{CCL}. The enhanced
fermion mass then arises from an effective Yukawa coupling of the
fermion to the scalar which develops a vacuum value from the technicolor
interactions. Here we summarize a study of the light composite
scalars generated by near-critical high energy interactions. We conclude that
unless the technicolor coupling at the ETC scale is unrealistically weak and 
the ETC coupling is very close to the critical curve, these light
states will have large widths.
 
We consider a single doublet of technifermions $\Psi = (U,D)$ subject
to a confining technicolor force and an additional, higher energy, attractive
ETC interaction. The latter is approximated by an effective, $SU(2)_L \times
U(1)$ invariant, four-fermion coupling \cite{AS}
\begin{equation}
{\cal L}_{4f} = {{8\pi^2 \lambda}\over{N_{tc}\Lambda^2}} (\bar{\Psi}_L^i U_R)
(\bar{U}_R \Psi_L^i) ~,
\label{eq:L4f}
\end{equation}
where $i$ is a summed $SU(2)_L$ index, $N_{tc}$ is the number of technicolors,
$\Lambda$ is the ETC mass scale, and $\lambda$ is the interaction strength
of the ETC interactions. Implicit technicolor indices are also summed in each
fermion bilinear.
 
Only the $U_R$ and not the $D_R$ has been included in the ETC
interaction, assuming that terms involving the latter will be weaker. Thus,
while the technicolor interactions will generate a dynamical mass for both the
U and D, the above ETC interaction will contribute only to the U mass. This
anticipates that when ETC couplings to ordinary
fermions are included, the mechanism being explored here will be especially
important for the generation of the t-quark mass.
 
We begin by recalling some features of dynamical chiral symmetry breaking
driven by the combination of a gauge interaction and a high energy, ETC,
four-fermion interaction. Suppose that the physics of interest
takes place at energies well above the confinement scale $\Lambda_{tc}$. It
should be possible to
describe this physics in terms of $\lambda$ and $\alpha(\Lambda)$ (the
technicolor coupling at the ETC scale), with technicolor running neglected to
first approximation. The running of the technicolor coupling can then be
included perturbatively. We will adopt this procedure and return later to a 
discussion of the conditions under which the approach is reliable.
 
With running neglected, dynamical mass generation can be studied in linearized 
ladder approximation \cite{ASTW,ALM}. In
Landau gauge \cite{AMNW}, the gap equation for the dynamical mass
$\Sigma(p)$ of the U takes the following form after angular integration:
 
\begin{equation}
\Sigma(p)  =  {{3\alpha C_2(R)}\over{4\pi}} \int^{\Lambda^2} {{dk^2}\over
{M^2}} \Sigma(k)   +  \lambda \int^{\Lambda^2}
{{dk^2}\over {\Lambda^2}} \Sigma(k)  ~~+~~...~,
\label{eq:gap}
\end{equation}
 
\noindent
where $\alpha \equiv \alpha(\Lambda)$, $C_2(R)$ is the quadratic Casimir
of the technifermion representation $R$, $M$ is the maximum of $p$ or $k$, and
the ellipses represent terms of order $\alpha^2$, $\alpha \lambda$, and higher.
 
Analysis of this equation in the $\alpha, \lambda$ plane \cite{ASTW} reveals
that a
critical curve separates the broken phase ($\Sigma \neq 0$) from the symmetric
phase ($\Sigma = 0$). For $\lambda \leq 1/4$, the broken phase exists only
for $\alpha > \alpha_c \equiv \pi/3C_2(R)$. For $\lambda \geq 1/4$, the critical
curve separating the two phases is defined by
\begin{equation}
\lambda_{\alpha} =\left[{1\over2} + {1\over2}\eta\right]^2 ~,
\label{eq:cc}
\end{equation}
where $\eta \equiv \sqrt{1-\alpha/\alpha_c}$. The broken phase then exists only
for $\lambda > \lambda_{\alpha}$.
 
When the running of the technicolor gauge coupling
is re-introduced, this distinction between broken and symmetric phases is
blurred. The growth of the coupling at momenta near $\Lambda_{tc}$ will in fact
always break the chiral symmetry at these lower scales. The critical curve in
$\lambda$ and $\alpha$ is therefore, loosely speaking, the dividing
line between the regime where the high energy interactions (four-fermion and
technicolor combined) are able to break the symmetry, and the regime
where they are too weak. With $\lambda$ and $\alpha$ in the weak,
"symmetric"
regime the low energy breaking of the symmetry by the growth of the
technicolor coupling near $\Lambda_{tc}$ will typically lead to
$\Sigma(0) \sim \Lambda_{tc} \ll \Lambda$.
 
It is this regime, where the spontaneous breaking is dominated by the "low
energy" technicolor interaction, that is of principal interest in this paper. 
To set the stage, however, we first suppose that $\alpha$ and 
$\lambda$ lie in the "broken" phase. The behavior of $\Sigma(p)$, as the
critical curve is approached can be derived from the gap equation.
Since in a realistic theory $\alpha < \alpha_c$, we summarize the critical
behavior only for this case. As $\lambda \rightarrow \lambda_{\alpha}$ with 
$\alpha$ fixed,
one finds that~\cite{ASTW}
\begin{equation}
{{\Sigma(0)}\over{\Lambda}} \sim
 (\lambda / \lambda_{\alpha} - 1)^{1\over{2\eta}} ~.
\label{eq:sig0}
\end{equation}
 
\noindent Thus the critical behavior involves a (gauge independent)~\cite{AMNW}
anomalous dimension that depends on the strength of the long range interaction.
With technicolor running reinstated, this behavior will persist as long as
$\lambda/\lambda_{\alpha} - 1$ is such that $\Sigma(0) \gg
\Lambda_{tc}$.
 
 
We now return to the regime $\lambda < \lambda_{\alpha}$, where
$\Sigma(0)
\sim \Lambda_{tc}$. There it has been shown that for a range of $\lambda$
near $\lambda_{\alpha}$,
 the high energy mass of the technifermion $\Sigma(\Lambda)$
takes the form \cite{Takeuchi}
\begin{equation}
\Sigma(\Lambda) \sim {{g^2<\bar\psi \psi>_{\lambda=0}}\over
{\Lambda^2(1 - \lambda/\lambda_{\alpha})}}  ~.
\label{eq:sigLambda}
\end{equation}
 
\noindent Here, $<\bar\psi \psi>_{\lambda=0}$ is the technifermion condensate
in a pure technicolor theory normalized at $\Lambda$, and
$g^2 = 4\pi^2\lambda$. If an ordinary fermion (quark or
lepton) is coupled to the technifermion by an ETC interaction with strength
of order
$g^2/\Lambda^2$, then its mass is also given by Eq.~\ref{eq:sigLambda}.
This expression exhibits
the ETC-driven mass enhancement as $\lambda \rightarrow \lambda_{\alpha}$. It
will break down once
$ \Lambda (1-\lambda/\lambda_{\alpha})^{1/2} \sim \Lambda_{tc}$.
 
In Ref. \cite{CCL}, it was suggested that this enhancement can
be attributed to the
existence of a light scalar particle of mass $M \sim \Lambda
(1 - \lambda/\lambda_{\alpha})^{1/2}$ ~\cite{carp}. It couples to the
technifermion
and then develops a vacuum value due to the technicolor interactions, producing
a "tadpole" diagram and leading to the result (\ref{eq:sigLambda}).
This estimate is reliable as long as
$\lambda_{\alpha} - \lambda$ is large enough that $M \gg \Lambda_{tc}$.
 The discussion of scalar formation in Ref.
\cite{CCL} was
restricted to a pure four-fermion theory \cite{carp}, i.e. a Nambu-Jona-Lasinio
(NJL) model \cite{NJL}.
Here we include the technicolor gauge interactions and specifically
address the issue of the existence of a light physical scalar.
 
 
To get an idea of what to expect, recall that in the regime outside the
critical curve, the high energy interactions are strong enough to break the
chiral symmetry and produce three massless scalar Goldstone bosons. They will
also produce a massive scalar whose mass vanishes as the critical curve is
approached.
Since the phase transition in the non-running theory is second order
\cite{ASTW},
corresponding scalar resonances produced by the high energy
interactions are also expected
on the weak side of the critical curve, whose masses
approach zero as the critical curve is approached.  In this phase, still 
neglecting running, chiral symmetry ensures that the scalar
resonances are degenerate.
To explore the region of this phase near the critical curve,
auxillary fields are introduced for these light scalar
degrees of freedom.
 
Four auxiliary fields are introduced by replacing the
effective four-fermion interaction in Eq.~\ref{eq:L4f} by
\begin{equation}
{\cal{L}}_{aux} =- {{g}\over{\sqrt{N_{tc}}}}{\bar{\Psi}_L M ({U_R \atop 0})
~+~h.c.}
- {{\Lambda^2}\over {4}}Tr~M^{\dag}M,
\label{eq:aux}
\end{equation}
where $M = \sigma + i\vec{\tau}\cdot \vec{\pi}$. The $SU(2)_L \times U(1)$
symmetry is manifest in this Lagrangian.  These fields do not, in general,
have definite parity, since the original interaction (Eq.~\ref{eq:L4f}) is 
parity violating.  We will refer to all of these fields as scalars.  
The study of critical behavior in terms of the auxiliary fields begins with the
quadratic term in the effective potential. The classical piece can be read off
from ${\cal{L}}_{aux}$ and the quantum piece can be computed in ladder
approximation by evaluating the
graphs depicted in Fig.~1 at zero external momentum \cite{BLM,AMNW}. The sum of
graphs can be written as
\begin{equation}
V^{(2)}_{ladder} = {{(\sigma^2+{\vec{\pi}}^2)}\over{2}} g^2 \int {{d^4k}\over{(
2\pi)^4}} {{\Gamma(k)}\over{k^2}} ~ ,
\label{eq:Vladder}
\end{equation}
 
\noindent where $\Gamma(k)$ is the full ladder-approximation to the 1PI
$\sigma U\bar{U}$ vertex with zero momentum flowing along the scalar line.
$\Gamma(k)$ can be evaluated by solving the Dyson-Schwinger equation
which gives \cite{AMNW}
\begin{equation}
\Gamma(k) = {{{1}\over {{{1}\over {2}} + {{1}\over {2}} \eta }}}
 \left({{k^2}\over {\Lambda^2}}\right)^{- {{1}\over{2}} + {{1}\over {2}}\eta}~.
\label{eq:Gamma}
\end{equation}
 
\noindent Substituting this expression into Eq.~\ref{eq:Vladder} and
combining the result with the classical term then gives the full quadratic
effective potential
\begin{equation}
V^{(2)} = {\Lambda^2\over {2}} \left[1 - {{\lambda}\over {({{1}\over {2}} +
{{1}\over {2}}}\eta)^2}\right] (\sigma^2 + {\vec{\pi}}^2)~.
\label{eq:V2}
\end{equation}
 
\noindent The stability of this potential at the origin is determined by
the sign of the coefficient of ${{1}\over {2}}(\sigma^2 +{\vec{\pi}}^2)$.
The critical curve (Eq.~\ref{eq:cc}) is determined by setting this
coefficient to zero.
 
In the regime $\lambda < \lambda_{\alpha}$ of special interest here,
the quadratic effective potential
gives the zero-momentum limit of the scalar inverse
propagator, that is the "zero-momentum mass"
\begin{equation}
M(0) = \Lambda(1 - \lambda/\lambda_{\alpha})^{1/2} ~.
\label{eq:M0}
\end{equation}
$M(0)$ will be small compared to $\Lambda$ if nature
provides us with a $\lambda$ close to $\lambda_{\alpha}$. Recalling that the 
scalar $\sigma$
will develop a small vacuum value, we recover the fermion mass enhancement
formula (Eq.~\ref{eq:sigLambda}) \cite{Takeuchi}, together with its tadpole
interpretation \cite{CCL}.
 
To explore the new physics at scales well below $\Lambda$ generated by
near-critical high energy interactions, it is necessary to move beyond zero
momentum. This amounts to the construction of the effective action of the low 
energy
theory. The question of whether the light scalars exist as
narrow or even broad resonances can be addressed by constructing the quadratic
term in the effective action of the scalar fields. For the $\sigma$ field,
for example, it can be written in the Euclidean-space form 
\begin{equation}
W^{(2)}[\sigma] = - \int {{d^{4}p}\over{(2\pi)^4}}
{1\over2} \Delta^{-1}(p)\sigma(p) \sigma(-p) ~,
\label{eq:W}
\end{equation}
where $\sigma(p)$ is the momentum-space field and $\Delta^{-1}(p)$ is the
inverse scalar propagator at momentum $p$.
 
To compute $\Delta^{-1}(p)$, one must evaluate the graphs shown
in Fig.~1 with some non-zero external momentum $p$ in the scalar legs
\cite{effaction}. This
requires knowledge of $\Gamma(p,k)$, the full ladder-approximation to the 1PI
$\sigma U\bar{U}$ vertex with momentum $p$ flowing along the scalar line.
A complete computation of $\Gamma(p,k)$ is difficult but an approximation will
be adequate for our purposes. We begin by considering a
Taylor series expansion about $p=0$ of $\Delta^{-1}(p)$.
In the ladder approximation one can show, by use of the
Dyson-Schwinger equation for $\Gamma(p,k)$, that two terms
contribute to the second derivative (with respect to $p^\mu$) of the graphs in
question (See Fig 2).  One term has two derivatives on a fermion propagator and
a $\Gamma(p,k)$ at each scalar vertex; the other term has one derivative on a
propagator, a $\Gamma(p,k)$ at one vertex,
and a derivative of $\Gamma(p,k)$ at the other vertex.  The derivative of
$\Gamma(p,k)$ is higher order in $\alpha$ than $\Gamma(p,k)$. By solving the 
Dyson-Schwinger
equation for the partial derivative of $\Gamma(p,k)$ with respect to $p$,
evaluated at $p = 0$, one can show that the second term is also numerically
smaller than the first (evaluated at $p=0$ \cite{IR}) for
$\alpha \leq \alpha_c$.  

Higher
derivatives of $\Delta^{-1}(p)$ will also contain terms with and without
derivatives of $\Gamma(p,k)$.  In what follows we will drop all terms involving
derivatives of $\Gamma(p,k)$, assuming their sum is numerically smaller than 
the sum of terms with no derivatives of $\Gamma(p,k)$. These
terms in the Taylor expansion contain only graphs with derivatives on
the fermion propagator.  They can easily be summed.
 
This resummation gives a graph with
a $\Gamma(k)$ at each scalar vertex, and the external momentum $p$ flowing
through one fermion line.  Performing the trace and the angular integrations we
obtain
\begin{equation}
\Delta^{-1}(p) - \Delta^{-1}(0) =
{{g^2}\over{8\pi^2}} \int_{0}^{p^2}
dk^2 \Gamma^{2}(k) \left({{k^2}\over{p^2}} - 2 \right)
-p^2{{g^2}\over{8\pi^2}} \int_{p^2}^{\Lambda^2}
dk^2 {{\Gamma^{2}(k)}\over{k^2}} ~.
\label{eq:dDelta-1}
\end{equation}
 
\noindent  Combining the
result for the effective potential (Eq.~\ref{eq:V2}) with the above kinetic
term (and Wick rotating so that $p^2$ is positive for a time-like Minkowski
momentum)
we find
\begin{equation}
\Delta^{-1}(p) = - \Lambda^2 \left(a{{p^2}\over{\Lambda^2}}+ b
\left({{p^2}\over{\Lambda^2}}\right)^\eta
\left[\cos(\eta\pi)-i\sin(\eta\pi)\right]+ 1-{\lambda\over{\lambda_{\alpha}}}
\right) ~,
\label{eq:Delta-1}
\end{equation}
 
\noindent where
\begin{equation}
a = {\lambda\over{2\lambda_{\alpha}(1-\eta)}} ~,
\label{eq:a}
\end{equation}
\begin{equation}
b = {\lambda\over{\lambda_{\alpha}\eta(1-\eta^2)}} ~.
\label{eq:b}
\end{equation}
Recall that we are restricting our attention to the case $\lambda > 1/4$ 
($\alpha
< \alpha_c$). Before proceeding further with the question of the possible
existence of narrow
resonances for realistic values of the coupling constants, we examine
Eq.~\ref{eq:Delta-1} in some simple limiting cases.
 

In the NJL limit ($\alpha \rightarrow 0$) we find
\begin{equation}
\Delta_{NJL}^{-1}(p) =
p^2{\lambda\over2}\left[\ln\left({{\Lambda^2}\over{p^2}}\right)+ i\pi\right]
 - \Lambda^2(1-\lambda) ~.
\label{eq:DeltaNJL-1}
\end{equation}
\noindent
Poles of $\Delta_{NJL}(p)$ occur for complex $p^2$.  We parameterize the 
location of the pole nearest the physical region by a mass and a width, i.e., 
the pole occurs at $p^2 \equiv (M_{NJL} - {i\over2}\Gamma_{NJL})^2$.  For
$\lambda$ close to 1 this
yields (keeping only leading terms in $\Gamma_{NJL}/M_{NJL}$) the physical mass
\begin{equation}
M_{NJL}^2 \approx
{{2(1-\lambda)}\over{\lambda\ln\left({\lambda\over{2(1-\lambda)}}\right)}}
\Lambda^2 ~,
\label{eq:MNJL}
\end{equation}
 
\noindent and narrow width
\begin{equation}
{{\Gamma_{NJL}}\over{M_{NJL}}} \approx
{\pi\over{\ln\left({\lambda\over{2(1-\lambda)}}\right)}} ~.
\label{eq:GammaNJL}
\end{equation}
Thus the familiar NJL results are recovered in this limit.
To obtain Eq.~\ref{eq:DeltaNJL-1} from
Eq.~\ref{eq:Delta-1}, an expansion in
${{\alpha}\over{2\alpha_c}} \ln({{\Lambda^2}\over{p^2}})$
and ${{\alpha}\over{2\alpha_c}}$ has been made, keeping
only the zeroth order term.  Note that this means dropping a term of order
$\lambda{{\alpha}\over{2\alpha_c}}$ relative to the $1-\lambda$ term. To obtain
Eqs.~\ref{eq:MNJL} and \ref{eq:GammaNJL} we have assumed that
$\ln\left({\lambda\over{2(1-\lambda)}}\right) \gg 1$.  Therefore these two
equations should give good approximations for the mass and width only when
\begin{equation}
{{\alpha}\over{2\alpha_c}} \ll {{1-\lambda}\over{\lambda}} \ll
{{1}\over{\ln\left({\lambda\over{2(1-\lambda)}}\right)}} \ll 1 ~.
\label{eq:validNJL}
\end{equation}
 
For the more realistic case of finite $\alpha$, we examine the location of the
poles of the propagator as $\lambda$ approaches $\lambda_{\alpha}$. The poles
again occur for
complex $p^2$ so we set $p^2 = p_0^2 \exp(-i\theta)$. For
$\eta < 1$, and $\lambda$ very close to $\lambda_{\alpha}$, we can neglect
${{p^2}\over{\Lambda^2}}$ relative to
$\left({{p^2}\over{\Lambda^2}}\right)^\eta$ in the real part of
$\Delta^{-1}(p)$. We then find zeros of $\Delta^{-1}(p)$ at
\begin{equation}
p_0 \approx
\Lambda \left({{1-{\lambda\over{\lambda_{\alpha}}}}\over{b}}\right)^
{1\over{2\eta}} ~, \label{eq:p0}
\end{equation}
\begin{equation}
\theta \approx
\left({{m-\eta}\over\eta}\right)\pi+{{a\sin({{m-\eta}\over\eta}\pi)}
\over{b\eta}}\left({{1-{{\lambda}\over{\lambda_{\alpha}}}}\over{b}}\right)
^{{{1-\eta}\over\eta}} ~,
\label{eq:theta}
\end{equation}
\noindent where $m$ is an odd integer.  We expect the physical pole to
correspond to $m = 1$, since it is the closest pole to the physical region.
Eq.~\ref{eq:p0} describes how the mass scale of the pole position vanishes as
$\lambda$ is tuned toward the critical curve. It is interesting to note that
this scaling law is different from that of the zero-momentum mass of the
scalar (Eq.~\ref{eq:M0}).  It is, however, the same scaling law as
exhibited by $\Sigma(0)$ in the broken phase (Eq.~\ref{eq:sig0}).  We expect
the same scaling behavior for the mass of the scalar in the broken phase.
Eqs.~\ref{eq:p0} and \ref{eq:theta} give a reliable
description of the pole position of $\Delta(p)$ only when
\begin{equation}
a\left({{1-{\lambda\over{\lambda_{\alpha}}}}\over{b}}\right)^{1\over{\eta}}
\ll 1 - {\lambda\over{\lambda_{\alpha}}} ~.
\label{eq:self-con}
\end{equation}
It can be seen from this condition that the scaling behavior sets in more
rapidly as $\alpha$ is increased toward $\alpha_c$.
 
We next consider under what conditions the scalar resonance will be narrow.
With a pole located at $p^2 = p_0^2 \exp(-i\theta)$, $\Delta(p)$ will
describe a narrow resonance if $\theta$ is small. In that case, it is
reasonable to parameterize the location of the pole by a mass and a width.  That
is, we set $p_0^2 \exp(-i\theta) = (M_\sigma - {i\over2}\Gamma_\sigma)^2$, which
yields
\begin{equation}
M_\sigma =
p_0 \left({{1+\cos(\theta)}\over{2}}\right)^{1\over2} \approx p_0  ~,
\label{eq:Msigma}
\end{equation}
\begin{equation}
{{\Gamma_\sigma}\over{M_\sigma}} =
{{2\sin(\theta)}\over{1+\cos(\theta)}} \approx \theta    ~.
\label{eq:Gammasigma}
\end{equation}
When is $\theta$ in fact small? We first observe 
that for finite $\alpha$ ($\eta < 1$), $\theta$ (Eq.~\ref{eq:theta}) 
does not approach zero as $\lambda \rightarrow
\lambda_{\alpha}$. Therefore, the width to mass ratio is not suppressed
(as in the NJL case, $\alpha = 0$) as the critical curve is approached.
For small but nonzero $\alpha$, this expression gives $\theta \rightarrow
{\alpha\over{2\alpha_c}} \pi$ as $\lambda \rightarrow \lambda_{\alpha}$. Thus,
as the
mass scale of the scalar state is made small by approaching the critical curve,
it is not described by a narrow Breit-Wigner resonance unless $\alpha$ is quite
small.
 
Having considered these special limiting cases, we now consider more generic
values of the coupling constants. A description of the
resonance structure of the theory is provided by a plot of Im $\Delta(p)$. We
do this by evaluating the general expression for $\Delta(p)$
Eq.~\ref{eq:Delta-1}, and plotting Im $\Delta(p)$ versus
$p/M(0)$ (in Fig.~3) for various values of $\lambda/\lambda_{\alpha}$ and
$\alpha/\alpha_c$.  Each figure corresponds to a different (small) value of
$M(0)/\Lambda =(1-\lambda/\lambda_{\alpha})^{1/2}$. In each case, a resonant
curve exists for the smallest value of $\alpha/\alpha_c$, peaked at a momentum
 smaller than $M(0)$. The curve then shifts down
(relative to $M(0)$) and broadens (relative to the position of the peak) as 
$\alpha/\alpha_c$ is increased. Both the
shifting down of the peak and the broadening are consistent with the limiting
cases discussed above.

As a specific example, consider the case in which $\lambda$ is tuned to within
$1\%$ of $\lambda_{\alpha}$ (Fig.~3b), giving $M(0)/\Lambda \approx 1/10$. The 
likely value of $\alpha$ ($\equiv
\alpha(\Lambda)$) depends on the details of the technicolor theory. If
$\Lambda$ is in a range between, say, $30$ TeV and $1000$ TeV, and if the
technicolor coupling either runs normally or walks at a rate attainable in a
realistic theory, $\alpha/\alpha_c$ will be somewhere between roughly $0.2$
and $0.5$. This is a range within which a broad Breit-Wigner curve exists,
peaked
roughly around $0.3M(0)$. With $\Lambda = 100$ TeV, for example, the peak would
be around $3$ TeV, with a full width at half maximum of roughly the same
order.  With less fine tuning (Fig.~3a), the resonance is broad even in the
NJL limit.  With a great deal of fine tuning (Fig.~3d), the resonance is narrow
for very small $\alpha$ but becomes broad once 
$\alpha/
\alpha_c$ is as large as say 0.2.
 
The curves of Fig.~3 show clearly that the tightly bound light scalar objects
produced by
near-critical high energy interactions will not in general be narrow
resonances.
Even with $\lambda$ tuned very close to $\lambda_\alpha$, 
the width to mass ratio
will only be small if $\alpha$ is unrealistically small for a technicolor
theory.
 
To understand the origin of these results, it is convenient to frame the 
discussion in terms of the wavefunction renormalization factor $Z$ of the
scalar. In
the NJL limit, $Z$ is sensitive to high momentum components and is 
proportional to $\ln{\Lambda^2/p^2}$.  The scalar 
couplings to fermions
are inversely proportional to $Z^{1/2}$. Therefore, these states are weakly 
coupled in the limit $\Lambda^2/p^2 \gg 1$, which arises when $\lambda 
- 1 \ll 1$.   
It is not surprising then that the NJL resonances can be narrow 
(Eq.~\ref{eq:GammaNJL}).  
The effect of the long range gauge interaction is to shift the 
sensitivity of $Z$ towards the infrared. $Z^{1/2}$ will be large (proportional
to $(\Lambda^2/p^2)^{1/2 - {\eta}/2}$) due to the contribution of small 
momentum components. This large denominator factor, however, will be cancelled 
by a corresponding $\Lambda$ dependence in  $\Gamma(k)$ (Eq.~\ref{eq:Gamma}),
which also enters into the computation of the decay widths. The partial and
full decay widths depend only on momentum components at the scale of the
resonance peak and are independent of $\Lambda$. There is therefore no reason 
for the resonances to be narrow. 
 
In a more realistic theory, there will also be strong ETC interactions coupling
the t quark to the U, as well as to itself. The t is therefore expected to play 
a role equal to the U in the determination of critical behavior and the
enhancement of fermion mass. Even though its coupling to the scalar 
channels will be strong at the ETC scale, it will be much
more weakly coupled than the U at the scale associated with the resonances. This
can be seen directly from the above discussion. The Z factor of the scalar 
 is large (as described above) in the
region of the resonance.  We are assuming here that only one set of parameters 
is tuned close to critical and that there is therefore only one set of light 
scalars.
Since the t has no technicolor interactions, however, 
there is no vertex correction to compensate this factor. 

For any of our conclusions to be useful in a technicolor theory,
it is important to discuss the effect on the light physics of reinstating the 
running of the technicolor coupling. We have already pointed out that the 
running turns what we have been calling
the "symmetric" phase into a broken phase with the breaking taking place on the
order of the confinement scale $\Lambda_{tc}$.  This produces the fermion
mass enhancement formula of Eq.~\ref{eq:sigLambda} with its tadpole
interpretation.
 
What effect does the running have on the resonance spectrum just discussed and
other features of the low energy physics?  First of all, the technifermions and
the techni-gauge bosons will be confined at scales on the order of
$1/\Lambda_{tc}$.  If the light scales discussed above (those
appearing in the graphs of Fig.~3) are large compared to $\Lambda_{tc}$, the
confinement can be expected to have little effect on the mass and total width.
The ladder computations for the total width reported above will remain
reliable in analogy to the way QCD can be used to describe R
in $e^+e^-$ annihilation.
The dominant decay products of the scalar resonances will be 
the technicolor-singlet
technihadrons, including the longitudinally polarized $W$'s and $Z$'s.  The
various partial decay rates will depend in detail on the confinement dynamics
as they do in QCD.  

If these resonances are not too far above $\Lambda_{tc}$, they may be
accessible at the the SSC or at a very high energy $e^{+}e^{-}$ collider. Since
their strongest couplings are likely to be to the U-type
techniquark, the dominant production mechanism will probably involve
$U\bar{U}$ production followed by emission of the resonant state from one of
these heavy fermions. Whether the
resonances can be detected will depend on the partial and full widths, as well
as detector capabilities, backgrounds, etc.
 
Whether the resonances lie enough above $\Lambda_{tc}$ so that our
estimates neglecting the running are reliable, depends on $\alpha/\alpha_c$ and
the smallness of $\lambda - \lambda_{\alpha}$. Suppose first that
$\alpha/\alpha_c \equiv \alpha(\Lambda)/\alpha_c $ is small, say, 
$\approx 0.1$. If $\Lambda$ is in the range 
from $100$ TeV to $1000$ TeV, this corresponds to a normally running theory.
If $\lambda - \lambda_{\alpha}$ is not too small, the resonance mass
will then sit well above $\Lambda_{tc}$, where the running coupling
remains quite close to $\alpha(\Lambda)$. In this case, it can also be seen 
that the small $\alpha$ expansion
of $\Delta^{-1}(p)$ should be reasonably convergent. The use of the
zeroth-order term (Eq.~\ref{eq:DeltaNJL-1}) should give a good first
approximation. Higher order corrections involving both ladder exchange and
running coupling corrections could then be computed simultaneously. 

If $\alpha/\alpha_c$ is larger, then the anomalous dimension in $\Gamma(k)$ is
large and the full form of $\Delta^{-1}(p)$ must be retained. Still, the 
neglect of running can be a good first approximation.
Suppose that $\alpha/\alpha_c \approx 0.5$. With $\Lambda$ in the range between
$100$ TeV and $1000$ TeV, this corresponds to a rather slowly running theory.
With $\lambda - \lambda_{\alpha}$ small but not too small, the resonance mass
will be small compared to $\Lambda$ but well above $\Lambda_{tc}$. In this
case, it is walking that justifies the use of a constant technicolor coupling
as a first approximation. 
Computations showing the the effect of running in different cases will be 
reported in a future publication.
              
If the parameters are such that the resonance curves are centered at
energies of order $\Lambda_{tc}$, then the computation of $\Delta(p)$ described
above will not be quantitatively reliable.  The zero-momentum mass $M(0)$ may
still be above $\Lambda_{tc}$ and the estimate of Eq.~\ref{eq:M0} still 
reliable.
Confinement effects, however, could become important in the description of the
resonances which then could mix with the technicolor states.  Disentangling the
experimental signals in this case may be difficult.
 
An important question is whether the light scalar
resonances are able to mediate flavor-changing neutral currents. If we restrict
our attention to CP conserving interactions, then possible off-diagonal 
couplings of ordinary fermions to these resonances will not  produce 
unacceptable flavor-changing neutral currents
if the zero-momentum boson masses, $M(0)$, are above
$1.5$ TeV \cite{Ellis3}. The contribution of the scalars to flavour-changing
neutral current processes involving the t quark may be much larger.  This
could be of immediate interest if the t is discovered in the next few years.
 
Finally, it is worth pointing out that the results described here will not be
qualitatively changed if the effective four-fermion interaction is replaced by
a realistic interaction such as the exchange of a heavy spin-one boson 
\cite{AS}. The low
energy physics doesn't depend on the details of the high energy physics, but
only on the symmetries and whether the couplings are tuned relatively close to
criticality.

To conclude, we have studied the properties of light composite scalars which 
are present in technicolor theories with  near-critical ETC interactions. 
We have shown that these scalars (which can enhance quark and lepton masses) 
will not be 
narrow resonances for realistic values of the technicolor gauge coupling.
In addition we have pointed out some phenomenological consequences of
these resonances. 
 
\noindent \medskip\centerline{\bf Acknowledgements}
\vskip 0.15 truein
We thank S. Chivukula, A. Cohen, D. Kosower, K. Lane, S. Macdowell, and V.
Miransky for helpful
discussions.  This work was supported in part by the Natural Sciences and
Engineering Research Council of Canada and by the U.S. Department of Energy
under contracts DE-AC-02-76ERO3075 and DE-FG-02-84ER40153. T.W.A and L.C.R.W.
acknowledge the Aspen Center for Physics for its hospitality during the 
summer of 1990.
\vskip 0.15 truein 

\vfill\eject
\noindent \medskip\centerline{\bf Figure Captions}
\vskip 0.15 truein
Fig. 1.  The quantum corrections to the quadratic effective action.  The wavy 
lines are
techni-gauge bosons, the solid lines are technifermions, and the dashed lines
represent the scalar field.

Fig. 2.  The second derivative of $\Delta^{-1}(p)$.  The external momentum $p$
flows through the upper lines, and the slashes indicate a derivative with 
respect to $p_\mu$.

Fig. 3.  Graphs of Im $\Delta(p)$ for different values of 
$\lambda/\lambda_{\alpha}$,
and different values of $\alpha/\alpha_c$.  The curves in each graph
are normalized so that the peak value of the $\alpha/\alpha_c = 0.01$ curve
equals 1.


\vfill\eject
 
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\end{document}
