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\title{{\bf MASS ENHANCEMENT AND CRITICAL BEHAVIOR IN TECHNICOLOR THEORIES}}
\author{THOMAS APPELQUIST, JOHN TERNING\thanks{Presented by John Terning}\\
{\em Department of Physics, Yale University, New Haven, CT 06520}\\
\vspace{0.3cm}
 and\\
\vspace*{0.3cm}
L.C.R. WIJEWARDHANA 
\\ {\em Department of Physics, University of Cincinnati, Cincinnati,
OH 45221}}
\begin{document}
\maketitle

\begin{center}
\parbox{13.0cm}
{\begin{center} ABSTRACT \end{center}
{\small \hspace*{0.3cm} 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.}}
\end{center}
  
The recent revival of interest in technicolor theories of electroweak symmetry
breaking has been stimulated partly by the observation that momentum
components well
above the confinement scale $\Lambda_{tc}$ can play a more important role than
they do in QCD.  In particular 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}.  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 attractive
ETC interaction. The latter is approximated by an effective, $SU(2)_L \times
U(1)$ invariant, four-fermion coupling 
\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.
 
We begin by recalling some features of dynamical chiral symmetry breaking
driven by the combination of gauge and
four-fermion interactions. 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 $\alpha \equiv 
\alpha(\Lambda)$ (the
technicolor coupling at the ETC scale).
With the running of $\alpha$ neglected, dynamical mass 
generation can be studied in linearized 
ladder approximation \cite{ASTW,ALM}.
Analysis of  the gap equation for the dynamical mass
$\Sigma(p)$ of the U fermion
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 $\alpha > \alpha_c \equiv \pi/3C_2(R)$,
where $C_2(R)$ is the Casimir of the fermion representation.
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, the 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. 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, where typically
$\Sigma(0) \sim \Lambda$, and the low energy breaking regime,
where typically
$\Sigma(0) \sim \Lambda_{tc} \ll \Lambda$. 
It is this regime, where the spontaneous breaking is dominated by the ``low
energy" technicolor interaction, that is of principal interest in this paper. 
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 {{4\pi^2\lambda <\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 $\Lambda$.
 If an ordinary fermion (quark or
lepton) is coupled to the technifermion by an ETC interaction with strength
of order
$\lambda/\Lambda^2$, 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. \citenum{CCL}, it was suggested that this enhancement can
be attributed to the
existence of a light scalar particle of mass $M \sim \Lambda
(1 - \lambda/\lambda_{\alpha})^{1/2}$. It couples to the
technifermion
and then develops a vacuum value due to the technicolor interactions, producing
a ``tadpole" diagram, thus leading to Eq.~\ref{eq:sigLambda}.
 The discussion of scalar formation in Ref.
\citenum{CCL} was
restricted to a pure four-fermion theory, 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.

The effective four-fermion interaction in Eq.~\ref{eq:L4f} 
can be eliminated in favor of four auxiliary scalar fields \cite{ATW},
 each with mass $\Lambda$.
The question of whether light scalars exist as
narrow or even broad resonances can be addressed by constructing the 
inverse scalar propagator  $\Delta^{-1}(p)$, for the auxillary 
field $\sigma$, which
couples to $U\bar{U}$, for example. 
To compute $\Delta^{-1}(p)$, we will evaluate 
Feynman graphs with two scalars
coupled to one fermion loop with any number of 
ladder gauge boson exchanges \cite{ATW}, 
and with
external momentum $p$ on the scalar legs. This
requires knowledge of $\Gamma(p,k)$, the 1PI
$\sigma U\bar{U}$ vertex with momentum $p$ flowing along the scalar line.
In the full ladder approximation, and neglecting derivatives of $\Gamma(p,k)$ 
(details are given in ref.~\citenum{ATW}) $\Delta^{-1}(p)$ 
can be expressed in terms of
$\Gamma(0,k)$. Using the known form \cite{AMNW} of $\Gamma(0,k)$:
\begin{equation}
\Gamma(0,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 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}
In the regime $\lambda < \lambda_{\alpha}$ of special interest here,
the zero-momentum limit of the scalar inverse
propagator gives 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 $\lambda_{\alpha}$. Thus
 we recover the fermion mass enhancement
formula (Eq.~\ref{eq:sigLambda}).

Before proceeding further, we examine some simple limiting cases. 
In the NJL limit ($\alpha \rightarrow 0$) it can be shown that
Eq.~\ref{eq:Delta-1} reproduces the usual logarithmically supressed width to 
mass ratio.
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
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.

We next consider under what conditions the scalar resonance will be narrow.
$\Delta(p)$ will
describe a narrow resonance if $\theta$ is small. (In this case,
the width to mass ratio is approximately equal to $\theta$.)
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) 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 $\Delta(p)$
(Fig.~1).  The two figures correspond to different values 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 $M(0)$.  As $\alpha/\alpha_c$ is increased the curve shifts down
(relative to $M(0)$) and broadens (relative to the position of the peak). 

\vspace*{9cm}
\begin{center}
{\small Fig. 1.  Graphs of Im $\Delta(p)$ for different values of 
$\lambda/\lambda_{\alpha}$
and $\alpha/\alpha_c$.  The curves in each graph
are normalized so that the peak value of the $\alpha/\alpha_c = 0.01$ curve
equals 1.}
\end{center}

As a specific example, consider the case in which $\lambda$ is tuned to within
$1\%$ of $\lambda_{\alpha}$ (Fig.~1a), giving $M(0)/\Lambda \approx 1/10$. 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 a full width at half maximum of roughly the same
order.  Even with a great deal of fine tuning (Fig.~1b), 
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, so it is not surprising that they can be narrow.  
The effect of the gauge interaction is to shift the 
sensitivity of $Z$ towards the infrared; $Z^{1/2}$ is then proportional
to $(\Lambda^2/p^2)^{1/2 - {\eta}/2}$. This large denominator factor, however, 
will be cancelled 
by $\Gamma(0,k)$ (Eq.~\ref{eq:Gamma}),
which also enters into the computation of the widths.
 There is therefore no reason for the resonances to be narrow. 

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 
will not be 
narrow resonances for realistic values of the technicolor gauge coupling.
 
\noindent \medskip{\bf Acknowledgements}
\\ 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.

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\end{document}
