\alpha_c$ in a momentum
range up to some cutoff $\Lambda_* < \Lambda$. The well-known solution to
this simplified model (often referred to in the literature as quenched
QED) is a non-vanishing
dynamical mass $\Sigma(p)$ falling monotonically as a
function of $p$ from some value $\Sigma(0)$ \cite{classicrefs,SD}. For
$\alpha_{*} \rightarrow \alpha_c$ from above ($N_f \rightarrow N_f^c$
from below),
$\Sigma(0)$ exhibits the behavior
\beq
\Sigma(0) \approx \Lambda_*
\exp\left({{- \pi}\over{\sqrt{{{\alpha_*}\over{\alpha_c}} -1}}}\right) ~.
\label{critical}
\eeq
Thus the order parameter $\Sigma(0)$ is predicted to vanish
non-analytically as $\alpha_{*} \rightarrow \alpha_c$.
We expect a similar critical behavior in the full theory. After all, the
intrinsic scale
$\Lambda$ introduced in Eq. (\ref{Lambda}), where $\alpha(\Lambda) \approx
0.78 \,\alpha_*$, plays the role of
an ultraviolet cutoff. Asymptotic freedom sets in beyond this scale and the
dynamical mass function falls rapidly ($\sim 1/p^2$). Indeed we find that with
a
running coupling the
critical behavior is exponential as above, but that the coefficient
in the exponential depends on the details of physics at scales on
the order of $\Lambda$. It is not universally $-\pi$.
This can be understood
analytically in the following manner. Following Ref. \cite{TABrazil},
the gap equation can be converted to differential form with appropriate
boundary conditions, and the solution to the linearized equation
can be written as
\beq
\Sigma(p) = {c \Sigma(0)^2\over p } \sin{ \int^{p}_{a\Sigma(0)} { dk\over
k}\sqrt{\alpha(k)/\alpha_c-1}}
\eeq
for momenta $p$ below the scale $\Lambda_c$ at which
$\alpha(\Lambda_c)= \alpha_c$, where $c$ is chosen so that
$\Sigma(\Sigma(0)) = \Sigma(0)$.
We have dropped terms explicitly proportional to derivatives of $\alpha(k)$
since the coupling is near the fixed point in this range and we have taken the
lower limit of the integral to be of order $\Sigma(0)$ ($a = {\cal O}(1) $).
For $
k > \Lambda_c$,
the solution takes a different form, expressible in terms of a hyperbolic
sine function
when the running is slow. The two solutions must match at $ p = \Lambda_c$
and the upper solution must satisfy the ultraviolet boundary condition.
Note that $\Lambda_c / \Lambda$
vanishes like $(r - 1)^{1/b\alpha_*}$ as $r \rightarrow 1$,
where $r \equiv \alpha_* / \alpha_c$.
The matching condition at $\Lambda_c$ says simply that
\beq
\int^{\Lambda_c}_{a\Sigma(0)} { dk\over
k}\sqrt{\alpha(k)/\alpha_c-1}
\eeq
takes on some value depending on the details of the upper solution. It can
be seen to be finite in the limit $r \rightarrow 1$ and it must be less
than $\pi$
if the dynamical mass is to remain positive for all momenta. (Solutions
with nodes also exist,
but a computation of the vacuum energy \cite{CJT,Bardeen} indicates that the
nodeless solution represents the stable ground state.) Because $\alpha(k)
\approx \alpha_*$ for small momenta, it can then be seen that $1/ \log (
\Lambda_c / \Sigma(0))$ vanishes like $ \sqrt{r-1}$ as $r \rightarrow 1$.
Since $\Lambda_c / \Lambda$ behaves like $(r - 1)^{1/b\alpha_*}$, it
follows that $1/ \log (\Lambda / \Sigma(0))$ also vanishes like $
\sqrt{r-1}$ as $r \rightarrow 1$.
This can also be seen in a direct, numerical solution of the integral gap
equation. In Landau gauge and after Wick rotation to Euclidean space, this
equation can be written in the form
\begin{eqnarray}
\ \Sigma(p) &=&
{1\over4}\int {{dk^2}\over{M^2}} {k^2\Sigma(k)\over k^2+\Sigma(k)^2 }{
\alpha(M^2)\over{\alpha_c}}
\end{eqnarray}
where $M = \max(p,k)$ and the approximation $\alpha((p-k)^2) \approx
\alpha(M^2)$
has been made before doing the angular integration. We solve this equation
with a numerical ultraviolet cutoff
much larger than $\Lambda$ and plot $\log (\Sigma(0)/\Lambda_c)$ versus
$ 1 / \sqrt{r-1}$ in Figure 1. The
result is insensitive to the numerical cutoff and
exhibits straight line behavior as $r \rightarrow1$. The slope of the line
is $0.82 \pi$.
If the theory is modified in some way at scales on the order of $\Lambda$,
straight line behavior is still exhibited, but with a slope depending
on the details of the modification.
Thus the only feature of the critical behavior determined purely by the
infrared,
fixed point behavior is that $1/ \log (\Lambda / \Sigma(0))$ vanishes
like $ \sqrt{r-1}$ as $ r \rightarrow 1$.
Below the scale of the dynamical mass $\Sigma(p)$, the quarks
decouple, leaving a pure gauge theory behind. One might worry that this would
invalidate the above analysis since it relies on the fixed point
which only exists when the quarks contribute to the $\beta$ function. This
is not a problem, however, since when
$\Sigma(0) \ll \Lambda$, the dominant momentum
range in the gap equation, leading to the above critical behavior
(\ref{critical}), is $\Sigma(0) < p < \Lambda$. In this range, the quarks are
effectively massless and the coupling does appear to be approaching an infrared
fixed point. Below the scale $\Sigma(0)$ confinement sets in. The
confinement scale can be estimated by noting that
at the decoupling scale $\Sigma(0)$, the effective coupling constant is of
order $\alpha_c$. A simple estimate using the above
expressions then shows that the confinement scale is roughly the same order
as the chiral symmetry breaking scale, $\Sigma(0)$.
If $N_f $ is
reduced sufficiently below $N_{f}^c$ so that $\alpha_*$ is not close to
$\alpha_c$, both $\Sigma(0)$ and the confinement scale
become of order $\Lambda$. The linear approximation to the gap equation
is then no longer valid and it is no longer the
case that higher order contributions to the effective potential can be
argued to be small. The methods of this paper are then no longer useful.
From the behavior of $\Sigma(0)$ near the transition, the corresponding
behavior of the Goldstone boson decay constant,
the quark condensate, and other physical scales can be estimated. We
return to this question after considering further
the nature of the chiral phase transition we have just described.
The smooth vanishing of the order parameter $\Sigma(0)$, Eq.
(\ref{critical}), suggests that the chiral symmetry phase
transition at $N_f = N_{f}^c$ ($\alpha_{*} = \alpha_{c}$) might be second
order. In a second order transition, however, an infinite
correlation length is associated with a set of scalar and pseudoscalar
degrees of freedom, with vanishing masses, described by an
effective Landau-Ginzburg Lagrangian. In the broken phase, the Goldstone
bosons are massless and the other scalar masses vanish at
the transition. There are no other light degrees of freedom. In the
symmetric phase, the scalars and pseudoscalars form a
degenerate multiplet. The situation here is quite different. We first
demonstrate this by showing that in the symmetric phase, there are no light
scalar and pseudoscalar degrees of freedom. We then
comment more generally on the physics of the transition.
\subsection{The Symmetric Phase}
To search for light, scalar and pseudoscalar degrees of freedom in the
symmetric phase, we examine
the color-singlet quark-antiquark scattering amplitude in the same
(RG-improved ladder) approximation leading to the above
critical behavior. If the transition is second order, then poles
should appear which move to zero momentum as we approach the
transition. We take the incoming (Euclidean) momentum of the initial
quark and
antiquark to be $q/2$, but
keep a non-zero momentum transfer by assigning outgoing momenta
$q/2 \pm p$ for the final quark
and antiquark. Any light scalar resonances should make their presence
known by producing pole in the scattering amplitude (in the complex
$q^2$ plane).
If the Dirac indices of the initial quark and antiquark are $\lambda$ and
$\rho$, and the those of the final state quark and antiquark are $\sigma$
and $\tau$, then the scattering
amplitude can be written for sufficiently small $q$ as:
\begin{equation}
T_{\lambda \rho \sigma \tau}(p,q) = \delta_{\lambda \rho} \delta_{\sigma
\tau}\,
{{1}\over{p^2}} \,T(p,q) + . . .~,
\label{fullscatt}
\end{equation}
where the dots indicate pseudoscalar, vector, axial-vector, and tensor
components, and we have factored out $1/p^2$ to make $T(p,q)$
dimensionless. We contract
Dirac indices so that we obtain the Schwinger-Dyson (SD) equation for the
scalar
s-channel scattering amplitude, $T(p,q)$, containing only t-channel
gluon exchanges. If $p^2 \gg q^2$, then $q^2$ will simply act as an
infrared cutoff in the loop integrations.
The SD equation in the scalar channel
is:
\begin{eqnarray}
T(p,q) &=& {{\alpha_*}\over{\alpha_c}} \pi^2+
4\pi^2 \lambda \,{{ p^2}\over{\Lambda_*^2}}+
{{\alpha_*} \over{4\alpha_c}}\left(
\int_{q^2}^{p^2} {{dk^2}\over{k^2}} \,T(k,q) +
\int_{p^2}^{\Lambda_*^2} {{dk^2}\over{k^2}} \,T(k,q) \,{{ p^2}\over{k^2}}
\right)
\nonumber \\
& &+
\lambda \int_{q^2}^{\Lambda_*^2} {{dk^2}\over{k^2}} \,T(k,q) \,{{
p^2}\over{\Lambda_*^2}} ~.
\label{T}
\end{eqnarray}
For the purpose of this discussion
we neglect the running of the gauge coupling
$\alpha$ up to the scale $\Lambda_*$.
This is a good approximation at the low momenta of interest here,
where the coupling is near the infrared fixed point $\alpha_*$. For
convenience, we use Landau gauge ($\xi=1$) where the quark
wavefunction renormalization vanishes. The issue of gauge invariance is
addressed
in the Appendix.
The first term in Eq. (\ref{T}) is simply one gluon
exchange, while the second term arises from a chirally symmetric,
four-quark interaction, i.e. a Nambu---Jona-Lasinio (NJL) \cite{NJL}
interaction, which we have introduced here for purposes of this analysis. It
allows us to make contact with the familiar
study of light degrees of freedom in the NJL theory when it
is near-critical.
For momenta $p^2>q^2$, Eq.~(\ref{T}) can be converted to a
differential equation:
\begin{equation}
p^4{d^2 \over (dp^2)^2}\,\, T =
-{{\alpha_*}\over {4\alpha_c}}\,\,T~,
\label{diff}
\end{equation}
with appropriate boundary conditions determined from Eq.~(\ref{T}).
The solutions of Eq.~(\ref{diff}) have the form.
\begin{equation}
T(p,q) = A
\left({p^2 \over \Lambda_*^2}\right)^{{1 \over 2} + {1 \over 2} \eta}
+B
\left({p^2 \over \Lambda_*^2}\right)^{{1 \over 2} - {1 \over 2} \eta} ~,
\label{sol}
\end{equation}
where the coefficients $A$ and $B$ are functions of $q^2 / \Lambda_*^2$,
and for $\alpha_*<\alpha_c$,
\begin{equation}
\eta= \sqrt{1-\alpha_*/\alpha_c}~.
\label{eta}
\end{equation}
The coefficients $A$ and $B$ can be
determined by substituting the solution back into Eq.~(\ref{T}). This
gives:
\begin{equation}
A ={{-2 \pi^2}\over{\left(1+\eta\right)^2}}
{{\left(1-\eta\right) \left(1- {{\lambda}\over{\lambda_*}}\right)
\left( {{q^2}\over {\Lambda_*^2}}\right)^{-{1\over 2}+{1\over 2}\eta} }
\over { 1- {{\lambda}\over{\lambda_\alpha}}+
\left({{\lambda}\over{\lambda_\alpha}}-
\left({{1-\eta}\over{1+\eta}}\right)^2\right)
\left({{q^2}\over {\Lambda_*^2}}
\right)^\eta } }
{}~,
\label{A}
\end{equation}
and
\begin{equation}
B =
{{2 \pi^2\left(1-\eta\right)
\left(1- {{\lambda}\over{\lambda_\alpha}}\right)
\left({{q^2}\over {\Lambda_*^2}}\right)^{-{1\over 2}+{1\over 2}\eta} }
\over { 1- {{\lambda}\over{\lambda_\alpha}}+
\left({{\lambda}\over{\lambda_\alpha}}-
\left({{1-\eta}\over{1+\eta}}\right)^2\right)
\left({{q^2}\over {\Lambda_*^2}}
\right)^\eta } }~,
\label{B}
\end{equation}
where
\begin{equation}
\lambda_\alpha \equiv \left[{1 \over 2} + {1 \over 2} \eta
\right]^2 ~,
\label{lambda_alpha}
\end{equation}
and
\begin{equation}
\lambda_* \equiv \left[{1 \over 2} - {1 \over 2} \eta
\right]^2 ~.
\label{lambdatilde}
\end{equation}
If we denote the location of the poles of the functions $A$ and $B$ in the
complex $q^2$ plane by $q_0^2$, we
then have
\begin{equation}
|q_0^2| = \Lambda_*^2
\left({| \lambda_\alpha- \lambda |}\over
{|\lambda- \lambda_*|}\right)^{1 \over \eta}~.
\label{q0}
\end{equation}
We see immediately that as $\lambda \rightarrow \lambda_\alpha$ (the
critical NJL coupling)
for $\alpha_* <\alpha_c$ the pole approaches the origin $q_0^2 = 0$,
indicating the existence of light degrees of freedom.
This is to be expected for a second order phase transition. As
$\alpha_*$ is increased the corresponding particles become broad
resonances \cite{ATW1}.
Of course in this region our analysis is not complete, precisely because
of the existence of the light scalar and pseudoscalar degrees of freedom.
These light degrees of freedom must be incorporated into the analysis,
for example they will have an effect on the two loop $\beta$ function.
Furthermore as discussed by Chivukula et. al. \cite{criticalhier} one
generally expects that, with more than two flavors of quarks, as
$\lambda$ is tuned towards $\lambda_\alpha$ the theory undergoes a
Coleman-Weinberg transition \cite{colemanweinberg} to the chirally broken
phase before $\lambda$ reaches $\lambda_\alpha$.
Now consider the limit $\eta \rightarrow 0$ ($\alpha_* \rightarrow \alpha_c$),
with
$\lambda < 1/4$, we have
\begin{eqnarray}
|q_0^2| &\rightarrow &
\Lambda_*^2 \left(1+{{\eta}\over{1/4-\lambda}}\right)^{1\over\eta}
\nonumber \\
&\rightarrow & \Lambda_*^2 \exp\left({{4}\over{1-4\lambda}}\right) ~.
\label{q0limit}
\end{eqnarray}
Thus we see that at $\alpha_* \rightarrow \alpha_c$, with $\lambda<1/4$,
there are no
poles in the complex $q^2$-plane with $q_0^2 \ll\Lambda_*$. There are
therefore no
light scalar and pseudoscalar degrees of freedom to constitute an effective
Landau-Ginzburg theory, so the chiral phase
transition is not second order along the line
$\alpha_*=\alpha_c$. This is in agreement with the analysis of Ref.
\cite{Miransky}.
Now imagine starting out with $\alpha_* < \alpha_c$ and $\lambda \approx
\lambda_\alpha$, so that
we have a light scalar resonance,
and then dialing the parameters so that $\alpha_*$ increases and $\lambda$
decreases in such a way that we approach the critical line $\alpha_*=\alpha_c$.
We then see from Eqs. (\ref{q0}) and (\ref{lambdatilde}) that we must
first
cross the line $\lambda = \lambda_*$, and that as we approach this line,
the mass of the scalar
grows and actually diverges. Thus the scalar resonance disappears from the
physical spectrum before we reach $\alpha_* = \alpha_c$.
Even before we reach this point, the width of the scalars becomes as large as
their mass, and they can no longer be considered resonances.
There is nothing special about the scalar and pseudoscalar channels in the
above
analysis. A similar analysis of the other channels, such as vector and
axial-vector,
would also reveal that there are no light excitations in the symmetric
phase near the critical coupling $\alpha_c$.
That this should be the case is not surprising. With the transition
governed by a long-range gauge force with an infrared fixed point,
approximate conformal invariance should be exhibited at momentum scales small
compared to $\Lambda$ in the symmetric phase. (For further discussions on this
point see Ref. \cite{miryam}.) Thus no light
scales will be present, in contrast to phase transitions governed by short
range forces as in the NJL or the finite temperature theories.
\subsection{The Broken Phase}
In the broken phase near the transition, one light scale, $\Sigma(0)$,
appears. It is therefore natural (in the assumed absence of instanton
effects) to expect
that the entire physical spectrum of the theory will be set by $\Sigma(0)$
and scale to zero with it as $N_f \rightarrow N_f^c$
from below. This point has been stressed recently by Chivukula
\cite{chivukula}. Thus there will clearly be no effective
Landau-Ginzburg Lagrangian. No finite set of light degrees of freedom can
be isolated in the broken phase in the limit $N_f \rightarrow N_f^c$,
and no light degrees of freedom (other than quarks and gluons)
exist in the symmetric phase!
Within this general picture, it is important to describe the spectrum of
resonances in more detail. If, for example,
a near-critical theory is the basis for a technicolor theory of
electroweak symmetry breaking \cite{postmodern}, then the the
light scale $\Sigma(0)$ will correspond to the electroweak scale and the
spectrum of resonances at this scale will have a direct
impact on precision electroweak measurements. In particular, the $S$
parameter \cite{S} will depend sensitively on this
spectrum. An especially interesting question in this regard is whether
parity doubling or even inversion of parity partners appears in this light
spectrum as
$N_f^c$ is approached.
The Goldstone boson decay constant $F_\pi$ is also proportional to
$\Sigma(0)$. A simple dimensional estimate
suggests that $F_\pi^2 \approx N \Sigma^{2}(0)/16 \pi^{2}$. Because of the
dominance of the fixed point at scales below $\Lambda$, this is
clearly a ``walking" theory. If the coupling stays close to $\alpha_c$ then
the
dynamical mass $\Sigma(p)$ falls roughly like $1/p$ in
this range. As a consequence, the condensate
$\langle \bar{q}^i_L q^j_R \rangle$ is
enhanced well above the value it would have in a QCD-like
theory. A simple estimate gives $< \bar{q}^i_L q^j_R > \approx N
\Sigma(0)^2 \Lambda /16 \pi^2 $.
Finally, it is important to note that with the entire spectrum of physical
states collapsing to zero with $\Sigma(0)$ at the transition, the analysis
of the transition using only the quark and gluon degrees of freedom is open
to question. It seems reasonable, however, to conjecture that these states
will
not be important at the momentum scales $ \Sigma(0) < k < \Lambda$
dominating the transition.
Some evidence for this is provided by estimates of higher order effects to
which we now turn.
\section{Higher Order Estimates}
We have so far analyzed the chiral symmetry breaking phase transition
using the ladder gap equation, i.e. the SD equation with the lowest order
kernel,
and the running gauge coupling determined by the two-loop $\beta$
function. In order to consider higher order effects we first
develop
a gauge-invariant technique to estimate the critical coupling without
relying on the intricacies of the SD equation.
In Ref. \cite{ALM}, it was noted that to lowest order the SD criticality
condition
can be written in the form
\beq
\gamma(2- \gamma) = 1 ~,
\label{crit}
\eeq
where $\gamma$ is the anomalous dimension of the quark mass operator.
To all orders in perturbation theory, this condition is gauge
invariant (since $\gamma$ is gauge invariant) and is equivalent to the
condition
\cite{rainbow}
$\gamma =1$ mentioned previously in the text. However if these
conditions are truncated at a finite order in perturbation theory they
lead to different results. We will take Eq.~(\ref{crit}) to define the critical
coupling order by order, since it allows us to reproduce the
known leading order result.
Through three loops $\gamma$ is given
in the $\overline{\rm MS}$ scheme by \cite{nikhef}
\beq
\gamma = \gamma_0 \alpha + \gamma_1 \alpha^2 +
\gamma_2 \alpha^3 + ...
\eeq
where
\begin{eqnarray}
\gamma_0 &=&
{{3\,C_2(R)}\over {2\,\pi }} \\
\gamma_1 &=&
{{1 }\over
{16\,{{\pi }^2}}}
[ 3\,{{C_2(R)}^2} - {{10\,C_2(R)\,N_f}\over 3} + {{97\,C_2(R)\,N}\over 3}
] \\
\gamma_2 &=&
{{1 }\over {64\,{{\pi }^3}}}[
129\,{{C_2(R)}^3} - {{70\,C_2(R)\,{N_f^2}}\over {27}} -
{{129\,{{C_2(R)}^2}\,N}\over 2} +
{{11413\,C_2(R)\,{{N}^2}}\over {54}} \nonumber \\
&& +
C_2(R)\,N_f\,N\,\left( -{{556}\over {27}} - 48\,\zeta(3) \right) +
{{C_2(R)}^2}\,N_f\,\left( -46 + 48\,\zeta(3) \right)]
\end{eqnarray}
Inserting this result in Eq. (\ref{crit}) and truncating to one-loop we find
\beq
2 \gamma_0 \alpha = 1.
\eeq
Solving for $\alpha$ we find a one-loop estimate of the critical coupling
that agrees with standard result:
\beq
\alpha_c^{(1)}= {{ \pi }\over{3 \, C_2(R)}} =
{{2 \pi \,N}\over{3\left(N^2-1\right)}}~.
\eeq
At two-loops the critical condition is
\beq
2 \gamma_0 \alpha + 2 \gamma_1 \alpha^2 - \gamma_0^2 \alpha^2 =1.
\label{twoL}
\eeq
Solving for $\alpha$ we find a two-loop estimate of the critical coupling:
\beq
\alpha_c^{(2)}= {{36\,\pi }\over {45\,C_2(R) - 97\,N + 10\,N_f}}
\pm {{{\sqrt{24}}\,\pi \,{\sqrt{9\,C_2(R) + 97\,N - 10\,N_f}} }\over
{{\sqrt{C_2(R)}}\,\left( -45\,C_2(R) + 97\,N - 10\,N_f \right) }}
~.
\eeq
The $+$ sign gives the positive root. We compare this with the one-loop
estimate
by taking N large and using the value $N_f \approx 4 N$ corresponding to
criticality:
\beq
\alpha_c^{(2)} \approx {{( \sqrt{11808} -72 )\, \pi }\over{69 N}} \approx
{{1.67}\over{ N}}.
\eeq
Numerically it can be seen that the ${\cal O}(\alpha^2)$ terms in the
criticality condition, Eq. (\ref{twoL}),
evaluated
at $\alpha = \alpha_c^{(2)}$ are typically about 25\% to 30\% of the leading
term for
$N_f \approx 4 N$. It can also be seen numerically that for for $N_f
\approx 4 N$
the four-loop term \cite{nikhef} in $\gamma$
is larger than the three-loop term, so it is not appropriate to go beyond two
loops in
this expansion for these values of $N_f$, and we should only use the
three-loop
term as an estimate of the error in our calculation.
Through three-loops, the $\beta$ function is given by
\begin{displaymath}
\beta(\alpha)=-b\alpha^2-c\alpha^3-d\alpha^4
\end{displaymath}
where $b$ and $c$ are given by Eqs. (\ref{b}) and (\ref{c}), and in the MS
scheme,
\begin{eqnarray}
d &=& {1\over 32\pi^2} \left( {2857 N^3-1415 N^2 N_f + 79 N(N_f)^2\over
54} - {205N\over 18}C_2(R) N_f \right. \\
\nonumber
&& \,\,\,\,\,\, \,\,\,\,\,\, \,\,\,\,\,\, \,\,\,\,\,
+ \left. {{11}\over{9}}C_2(R) (N_f)^2+ C_2(R)^2 N_f \right)
\end{eqnarray}
Since the three-loop term is scheme dependent we cannot obtain a scheme
independent answer without going to the same order in $\beta$ and $\gamma$, so
we will only use the three-loop term for error estimates.
In Table 1 we list some numerical results. We have computed the value of
$N_f^c$
for $SU(N)$ gauge theories for values of $N$
ranging form 2 to 10, showing the results at different orders in perturbation
theory. In section 4 (using the leading order estimate of the critical
coupling)
it was shown that
$N_f^c$ goes like $4N$ for large $N$. We see that going to two loops in the
criticality condition produces a small shift in this relation. We also list the
estimated value of the critical coupling at one and two loops.
We see that even though the percentage shift of the value
of $N_f^c$
is small, the higher order terms of the beta function make a
significant contribution at the critical point. For $N_c$ between 3 and 10 we
estimate
that the error in $N_f^c$ at two-loops is about 12\% from the truncation of the
$\beta$ function and about 10\% from the truncation of $\gamma$, while for
$N_c=2$
the errors are somewhat larger, around 14\% from each. It is important to
emphasize
that these are simply numerical estimates of the next to leading
contributions. Even at large N, there is no obvious small parameter here
leading to a controlled expansion. Thus the smallness of still higher order
terms is not guaranteed.
\begin{table}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|}
\hline
$N_c$ & $N_f^c$ (1,2) & $N_f^c$ (2,2) & $N_f^c$ (2,3) & $\alpha_c^{(1)}$
& $\alpha_c^{(2)}$ \\
\hline\hline
2 & 7.86 & 8.27 & 7.12 & 1.4 & 1.11 \\ \hline
3 & 11.9 & 12.4 & 10.9 & 0.785 & 0.595 \\ \hline
4 & 15.9 & 16.6 & 14.6 & 0.559 & 0.412 \\ \hline
5 & 20.0 & 20.8 & 18.3 & 0.436 & 0.317 \\ \hline
6 & 24.0 & 24.9 & 22. & 0.359 & 0.258 \\ \hline
7 & 28.0 & 29.1 & 25.7 & 0.305 & 0.218 \\ \hline
8 & 32.0 & 33.3 & 29.4 & 0.266 & 0.189 \\ \hline
9 & 36.0 & 37.4 & 33.1 & 0.236 & 0.166 \\ \hline
10 & 40.0 & 41.6 & 36.8 & 0.212 & 0.149 \\ \hline
\end{tabular}
\end{center}
\caption{Estimates of $N_f^c$. The two numbers in parentheses
give the order used in the critical condition on $\gamma$ and the
$\beta$ function. The comparison of the (2,2) and (2,3) give an
estimate of the error in truncating the $\beta$ function at two-loops.}
\end{table}
\section{Summary and Conclusions}
In this paper, we have explored features of the chiral phase transition
in $SU(N)$ gauge theories. We have argued that the transition takes place
at a relatively large value of $N_f$ ($N_f^c \approx 4N$) where the infrared
coupling is determined by a fixed point accessible in the loop
expansion of the $\beta$ function, and that the transition can be studied
using a ladder gap equation. Our higher order estimates suggest that
the estimate of $N_f^c$ is good to about 20\%.
To phrase things in physical terms, the effect of the light quarks
is to screen the long range force, eventually disordering the
system and taking it to the symmetric phase. That the transition takes
place at a relatively large value of $N_f$ means that the quarks are
relatively ineffective at long range screening.
With an infrared fixed point governing the transition, the order parameter
vanishes in a characteristic exponential fashion and all physical scales
vanish in the same way. There is
no finite set of light degrees of freedom that can be identified to form an
effective, Landau-Ginzburg theory. In the symmetric phase ($N_f > N_f^c$) ,
no light degrees of freedom are formed as $N_f \rightarrow N_f^c$. Thus the
transition is continuous but not conventionally second order. The validity
of the approach is considered by estimating higher order terms in both the
$\beta$ function and the anomalous dimension of the mass operator.
In Ref. \cite{AppSel}, it was noted that single instanton effects in a theory
with an infrared fixed point seem capable of triggering a chiral phase
transition at similarly large
values of $N_f/N$. A detailed computation was carried only out for an $SU(2)$
gauge theory but the analysis indicated that this could be the case at larger
values of $N$ as well.
It is interesting to compare our results with the phase structure of
supersymmetric $SU(N)$ theories where exact results are available
\cite{Seiberg}. In such theories there is also a large range of $N_f$ where
the theory is asymptotically free and an infrared fixed point occurs. A
transition
to a strongly coupled phase occurs at $N_{\rm f,SUSY}^c = 3 N/2$. Thus it
seems
plausible that
infrared fixed points are fairly generic in asymptotically free gauge theories
with a large number of flavors. One prominent difference between the
supersymmetric
and non-supersymmetric cases is that the strongly coupled phase
$N+1