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\begin{document}
\pagestyle{empty}
\begin{flushright}
NMCPP/98-17\\
\\
\end{flushright}
\vspace*{1cm}
\begin{center}
{\bf\Large Chiral Symmetry and Quark Confinement}\\
\vspace*{1.0cm}
{\bf Kevin Cahill\footnote{kevin@kevin.phys.unm.edu
\quad http://kevin.phys.unm.edu/$\tilde{\ }$kevin/}}\\
\vspace{.2cm}
\vspace{.2cm}
New Mexico Center for Particle Physics\\
Department of Physics and Astronomy\\
University of New Mexico\\
Albuquerque, NM 87131-1156\\
\vspace{.50cm}
\vspace{.50cm}
\vspace*{1.0cm}
{\bf Abstract}
\end{center}
\begin{quote}
In the physical vacuum of $QCD$,
the energy density of light-quark fields 
strongly coupled to slowly varying gluon fields
can be negative,
and so a condensate of
pairs of quarks and antiquarks
of nearly opposite momenta forms
which breaks chiral symmetry, confines quarks,
and makes gluons massive.
\end{quote}
\vfill
\begin{flushleft}
\today\\
\end{flushleft}
\eject

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\section{The $QCD$ Vacuum}
The thesis of this note is
that chiral-symmetry breaking, quark confinement,
and the short range of the strong force
all arise from the same feature of $QCD$,
namely that the energy density
of strongly coupled light~\cite{Gribov} quarks can be negative.
The hamiltonian $H_q$ of the $u$, $d$, and $s$ quarks
\beq
H_q = %\int \! d^3x \, \cH_u(x) = \int \! d^3x \, 
\sum_{f=u,d,s} \int \! d^3x \, 
\bar \psi_f \left( \vec \c \cdot \vec \nabla 
- i g \c^0 A_{0a} \frac{\l_a}{2} 
- i g \vec \c \cdot \vec A_a \frac{\l_a}{2} + m_f \right) \psi_f
\label {H}
\eeq
can assume large negative mean values 
due to the term $ - g \int \! d^3x \vec J_a \cdot \vec A_a $
when the gauge field $ \vec A_a $
varies slowly with a modulus
$ |\vec A_a| $ 
that exceeds $ m_u/g $ by a sufficient margin~\cite{CahillHerling}.
For nearly constant gauge fields $ \vec A_a $,
the states that drive the energy lowest
are condensates~\cite{Wilczek} of pairs of light quarks and 
antiquarks of opposite momenta;
in such pairs the color charges cancel, but 
the color currents add.
When $ g |\vec A_a| >> m_d $,
the $u$ and $d$ quarks play very similar roles,
but pairs of $s$ quarks and antiquarks are important
only when $ g |\vec A_a| \gg m_s $\@.
\par
If the gauge fields are not only slowly varying
but also essentially abelian,
in the sense that $ g f_{abc} A_\mu^b A_\nu^c $
is small 
(\emph{e.g.}, because $ A^a_\mu(x) \simeq C^a(x) V_\mu(x) $),
then the energy of the gauge fields is also small.
If an essentially abelian gauge field,
\emph{e.g.,} $|\vec A_8|$, is relatively constant 
over a sphere of radius $R$ beyond which it either
remains constant or slowly drops to zero,
then its energy density near the sphere
can be of the order of
$|\vec A_8|^2/R^2$ or less 
while that of the light-quark
condensate can be as negative as $- |g \vec A_8|^4$. 
The physical vacuum
of $QCD$ is therefore a linear combination 
of states, each of which is approximately
a coherent~\cite{Glauber} state $ | \vec A \rangle $ of 
a slowly varying, essentially abelian gauge field $ \vec A_a(x) $
and an associated condensate of pairs of
$u$ and $d$ quarks and $ \bar u $ and $ \bar d $ 
antiquarks of nearly opposite momenta:
\beq
| \Omega \rangle \simeq \int \!\!\! D\vec A_a \, f(\vec A_a)
\!\!\prod_{S(A,u)}
\!\!a^\dagger( \vec p_i, \s, u_i ) a^{c\dagger}( \vec q_j, \tau, u_j )
%u^\dagger_{c}( \vec p, \s ) \bar u^\dagger_{c'}( \vec p', \s' )
\!\!\prod_{S(A,d)}
\!\!a^\dagger( \vec p_i, \s, d_i ) a^{c\dagger}( \vec q_j, \tau, d_j )
%d^\dagger_c( \vec p, \s ) \bar d^\dagger_{c'}( \vec p', \s' )
| \vec A \rangle.
\label {qcd vacuum}
\eeq
Here the sets $ S(A,u) $ and $ S(A,d) $ specify
the momenta $\vec p, \vec q$, spins $\s, \tau$,
and colors $i, j$ of the quarks and antiquarks, 
the operator $ a^{c\dagger}( \vec q, \tau, d_j ) $
creates a $d$ antiquark of momentum $ \vec q $,
spin $\tau$, and color $j$,
the function $ f(\vec A_a) $ is a weight function, 
and the $s$ quarks have been suppressed.
\par
In what follows I shall compute the energy 
density of a such a light-quark condensate
for the case of a constant gauge field 
$ \vec A_8 $.  It will turn out that if
$ g |\vec A_8| $ is of the order of a GeV,
then the mean value 
$ \langle \thalf ( \bar u u + \bar d d ) \rangle $
of the light-quark condensate
is about $ (260\, $MeV$)^3$ as required
by soft-pion physics.
\par
Because a color-electric field moves 
quarks in one direction and antiquarks
in the opposite direction,
the quark condensate of 
the $QCD$ vacuum in this model
is not stable in the presence 
of color-electric fields.
Thus volumes of space that are traversed
by color-electric fields have less quark-antiquark condensate 
and hence a higher energy density than that of the physical vacuum.
Consequently the surface of a hadron
is exposed to a pressure that is equal
to the difference between the energy density 
$ \rho_\Omega $\@  
of the physical vacuum outside the hadron
and the energy density \( \rho_h \) inside the hadron
which, due to the color-electric fields,
is somewhat higher than $ \rho_\Omega $\@.
This pressure \( p \)
\beq
p \simeq \rho_h - \rho_\Omega 
\eeq
confines quarks because
it squeezes their color-electric fields.
Thus quarks are confined not because
of the energy of their color-electric fields
but because their color-electric fields
are excluded by the physical vacuum.
In the example which follows,
a very small decrease in the quark-antiquark condensate
results in a pressure \( p \)
of the order of $ ( 1\, $GeV$)^4$\@.

\section{A Particular Condensate}
Let us consider the case of a constant
gauge field $ \vec A_8 $ that points in the
direction 8 of color space;  
the energy density and quark condensate
associated with a slowly varying gauge 
field that points in an arbitrary direction
in color space should be similar.
If we call the quark colors
red, green, and blue, then the condensate
will be made of red and green $u$, $d$, and $s$
quarks of momentum $ \vec p $ and both spin indices $ \s $;
red and green $u$, $d$, and $s$ antiquarks of momentum $ - \vec p $
and both spin indices $ \s $;
blue $u$, $d$, and $s$
quarks of momentum $ - \vec p $ and both spin indices $ \s $;
and blue $u$, $d$, and $s$
antiquarks of momentum $ \vec p $  
and both spin indices $ \s $.
The domains of integration $S(A,u)$ and $S(A,d)$
for the $ u $ and $d$ quarks will be very
similar when the gauge field $ \vec A_8 $ is intense,
but the domain for the $s$ quarks will be smaller.
The component $ |\Omega_A \rangle $
of the $QCD$ vacuum associated with the gauge field 
$ \vec A_8 $ will then be 
\beq
| \Omega_A \rangle = 
\prod_{S(A,u)}
a^\dagger( \vec p_i, \s, u_i ) a^{c\dagger}( - \vec p_i, \s, u_i )
%u^\dagger_c( \vec p, \s ) \bar u^\dagger_{c}( - \vec p, \s )
\prod_{S(A,d)}
a^\dagger( \vec p_i, \s, d_i ) a^{c\dagger}( - \vec p_i, \s, d_i )
%d^\dagger_c( \vec p, \s ) \bar d^\dagger_{c}( - \vec p, \s )
\, | \vec A_8 \rangle 
\label {cond}
\eeq
apart from the $s$ quarks.
These products over momentum, spin, and color
are defined by box quantization in a volume \(V\),
and so the mean value of the hamiltonian \( H_q \)
in the state \( | \Omega_A \rangle \) is really
an energy density.
\par
The only quark operators that have non-zero mean values
in the state $| \Omega_A \rangle $ are those
that destroy and create the same kind of quark
or antiquark.  Thus if we normally order the
quark hamiltonian (\ref{H}), then the part 
of the magnetic term 
$ H_{qm} = - g \int \!\! d^3x \, \vec J_a \cdot \vec A_a $
that involves the field $ \psi_d(x) $ of the $d$ quark 
\beq
\psi_{\ell d}(x) = \sum_\sigma \! \int \! \!\! \frac{d^3p}{(2\pi)^{3/2}}
\left[ u_\ell(\vec p, \s, d_i ) e^{ip\cdot x} a(\vec p, \s, d_i)
+v_\ell(\vec p, \s, d_i ) e^{-ip\cdot x} a^{c \dagger} (\vec p, \s, d_i)
\right]
\label {psi}
\eeq
has a mean value
\bea
E_{dm} & = & \langle \Omega_A | H_{dm} | \Omega_A \rangle = 
\langle \Omega_A | \left( - g \int \!\! d^3x 
\, \vec J_a^d \cdot \vec A_a \right) | \Omega_A \rangle \nn\\
& = & \mbox{}
\langle \Omega_A | \left( -ig \int \!\! d^3x
\, \bar \psi_d \vec \c \cdot \vec A_a \frac{\l_a}{2} \psi_d \right)
| \Omega_A \rangle
\eea
given by
\bea
E_{dm} & = & g \vec A_8 \cdot \sum_{\s,i}
\frac{\l^8_{ii}}{2}
\int_{S(A,d)} \!\!\!\!\!\!\! d^3p \, \left[
u^\dagger(\vec p_i, \s, d_i ) 
\c^0 \vec \c 
\,u(\vec p_i, \s, d_i ) \right. \nn\\
& & \mbox{} 
\left. \qquad \qquad - v^\dagger(- \vec p_i, \s, d_i )
\c^0 \vec \c 
\,v(- \vec p_i, \s, d_i ) \right].
\eea
The spin sums~\cite{Weinberg}
\beq
\sum_{\s} u_\ell(\vec p, \s, d_i ) u^\ast_{\ell'} (\vec p, \s, d_i )
= \frac{1}{2p^0} \left[ \left( p^\mu \c_\mu + i m_d\right) \c^0 
\right]_{\ell \ell'}
\label {sumu}
\eeq
and
\beq
\sum_{\s} v_\ell(\vec p, \s, d_i ) v^\ast_{\ell'} (\vec p, \s, d_i )
= \frac{1}{2p^0} \left[ \left( p^\mu \c_\mu - i m_d\right) \c^0 
\right]_{\ell \ell'}
%\sum_{\s} v(\vec p, \s, d_i ) v^\dagger (\vec p, \s, d_i )
%= \frac{1}{2p^0} \left( p^\mu \c_\mu - i m_d\right) \c^0 
\label {sumv}
\eeq 
imply the trace relations
\beq
u^\dagger(\vec p_i, \s, d_i )
\c^0 \vec \c
\, u(\vec p_i, \s, d_i ) = 
v^\dagger( \vec p_i, \s, d_i )
\c^0 \vec \c
\, v( \vec p_i, \s, d_i ) = - \frac{ 2 \vec p}{p^0}
\eeq
and
\beq
u^\dagger(\vec p_i, \s, d_i ) \, u(\vec p_i, \s, d_i )
= v^\dagger( \vec p_i, \s, d_i ) \, v( \vec p_i, \s, d_i )
= 2. 
\eeq
Thus the magnetic energy density of the $d$ quarks
and antiquarks of color $i$
in the constant gauge field $ \vec A_8 $ is
\beq
E_{dm} = - 2 g \l^8_{ii} \int_{S(A,d)} \!\! 
\frac{ d^3p }{  p^0} \,
\vec A_8 \cdot \vec p .
\eeq
The same spin sums imply that 
in the state $ | \Omega_A \rangle $,  
the mean value of the color charge density 
$ J_a^{0d} = \psi_d^\dagger \thalf \l_a \psi_d $
and that of the color current 
$ \vec J_a^d = - \psi_d^\dagger \c^0 \vec \c \thalf \l_a \psi_d $ 
for $ a \ne 8$ both vanish.
Thus the mean value of the
second term of the hamiltonian (\ref{H}) is zero.
\par
The mean value of the hamiltonian \( H_q \)
for $d$ quarks and antiquarks of color $i$
in the state \( | \Omega_A \rangle \)
is therefore
\beq
E_{{di}} = \int_{S(A,d,i)} \!\!  d^3p  
\left(
4 p^0 - 2 g \l^8_{ii} \frac{\vec A_8 \cdot \vec p}{ p^0} 
\right)
\eeq   
where the domain of integration $S(\vec A_8,d,i) $
is the set of momenta $ \vec p $ for which the integrand is negative
\beq
g \l^8_{ii} \vec A_8 \cdot \vec p > 2 ( \vec p^2 +m_d^2 ) .
\label {domain}
\eeq
The set $S(\vec A_8,d,i) $ is empty unless 
\( g \l^8_{ii} |\vec A_8| > 4 m_d \), 
which requires
the effective magnitude of the gauge field 
to be large compared to the mass $ m_d $ of the $d$ quark.
\par
Since the quark energy density \( E_{fi} \) 
depends upon the flavor $f$ and the color $i$ only through
the dimensionless ratio
\beq
r = \frac{4 m_f }{g \l^8_{ii} |\vec A_8|},
\eeq
we may write it as the integral 
\beq
E_{fi} = - 32 \pi (g \l^8_{ii} |\vec A_8|)^4
\int_{1-\sqrt{1-r^2}}^{1+\sqrt{1-r^2}}
dx \, \frac{x ( 2x - x^2 - r^2 )^2 }{ \sqrt{ x^2 + r^2 }}
\eeq
which has the value
\bea
E_{fi} & = & \mbox{} - 32 \pi (g \l^8_{ii} |\vec A_8|)^4 \nn\\ 
& & \mbox{} \left[
\left( \frac{1}{5} (x^2+r^2)^2
- x^3 - \frac{1}{2} r^2 x
+ \frac{\textstyle 4}{\textstyle 3} (x^2 - 2 r^2 )
\right)
\sqrt{x^2 + r^2} \right. \nn\\
& & \left. \qquad \qquad 
\mbox{} + \frac{1}{2} r^4 {\rm arcsinh}{\left(\frac{x}{r}\right)}
\right]_{1-\sqrt{1-r^2}}^{1+\sqrt{1-r^2}} .
\eea
For small $r$ the energy density $E_{fi}$ is approximately
\beq
E_{fi} \simeq - 32 \pi (g \l^8_{ii} |\vec A_8|)^4
\left( \frac{16}{15} 
- 4 \left(\frac{4 m_f}{g \l^8_{ii} |\vec A_8|} \right)^2 \right).
\eeq
Summing over the three colors, we get
\beq
E_{f} \simeq - 64 \pi (g |\vec A_8|)^4
\left( \frac{16}{15}
- 4 \left(\frac{4 m_f}{g |\vec A_8|} \right)^2 \right)
\eeq 
which displays isospin symmetry 
when \( 4 m_f \ll g | \vec A_8 | \).
If the gauge field is moderately strong
\( 4 m_s > g \l^8_{ii} | \vec A^8 | \gg 4 m_d \),
then the energy density of the $u$--$d$ condensate is
\beq
E_{ud} \simeq - 128 \pi (g |\vec A_8|)^4
\left( \frac{16}{15}
- 4 \left(\frac{4 m_{ud}}{g |\vec A_8|} \right)^2 \right)
\eeq 
where \( m_{ud}^2 = ( m_u^2 + m_d^2 )/2 \).
For stronger gauge fields,
\( g \l^8_{ii} | \vec A^8 | >> 4 m_s \),
the energy density of the light-quark condensate is
\beq
E_{uds} \simeq - 192 \pi (g |\vec A_8|)^4
\left( \frac{16}{15}
- 4 \left(\frac{4 m_\ell}{g |\vec A_8|} \right)^2 \right)
\eeq 
where \( m_\ell^2 = ( m_u^2 + m_d^2 + m_s^2 )/3 \).

\section{The Breakdown of Chiral Symmetry}
The quark condensate occasioned by the constant
gauge field $ \vec A_8 $ gives rise to a mean value
of the space average of $ \thalf ( \bar u u + \bar d d ) $,
which is an order parameter that traces
the breakdown of chiral symmetry.
By using the spin sums (\ref{sumu}) and (\ref{sumv})
and the expansion (\ref{psi}) of the Dirac field,
we find for this order parameter 
\bea
\langle {\thalf} ( \bar u u + \bar d d ) \rangle 
& = & \langle \Omega_A | \int \!\! d^3x {\thalf} ( \bar u u + \bar d d )
| \Omega_A \rangle 
\nn \\ & = & 
{\thalf} \sum_{f=u}^d \sum_{i=1}^3
\sum_\s \int_{S(A,f,i)} \!\! d^3p \left[
u^\dagger(\vec p_i, \s, f_i ) i \c^0  
u(\vec p_i, \s, f_i ) \right.
\nn \\ & & \mbox{}
\left. \qquad \qquad \qquad \qquad
- v^\dagger(- \vec p_i, \s, f_i ) i \c^0 
v(- \vec p_i, \s, d_i ) \right]
\nn \\ & = & \mbox{}
\sum_{f=u}^d \sum_{i=1}^3   
2 m_f \int_{S(A,f,i)} \!\! \frac {d^3p}{p^0} .
\eea
In terms of the ratio \( r = 4 m_f/(g \l^8_{ii} | \vec A_8 |) \),
this order parameter is
\bea
\langle {\thalf} ( \bar u u + \bar d d ) \rangle 
\!\!& = & \!\!\! \sum_{f=u}^d \! \sum_{i=1}^3
\frac{\pi}{4} m_f \left( g \l^8_{ii} | \vec A_8 | \right)^2
\! \int_{1-\sqrt{1-r^2}}^{1+\sqrt{1-r^2}} 
\!\!\!\!\!\!\!\!\!\! dx
\left( \frac{x^2}{\sqrt{x^2 + r^2}} 
- \frac{x}{2} \sqrt{x^2 + r^2}
\right) \nn\\
\!\!& = & \mbox{}
\!\!\! \sum_{f=u}^d \sum_{i=1}^3
\frac{\pi}{4} m_f \left( g \l^8_{ii} | \vec A_8 | \right)^2
\nn\\
& & 
\left[ \left( \frac{x}{2} - \frac{x^2+r^2}{6} \right) \sqrt{x^2+r^2} 
- \frac{r^2}{2} {\rm arcsinh} \left( \frac{x}{r} \right)
\right]_{1-\sqrt{1-r^2}}^{1+\sqrt{1-r^2}}.
\eea
\par
For small $r$ this condensate or order parameter is
\beq
\langle {\thalf} ( \bar u u + \bar d d ) \rangle
\simeq \sum_{f=u}^d \sum_{i=1}^3 
\frac{\pi}{4} m_f 
\left( g \l^8_{ii} | \vec A_8 | \right)^2 
\left[ \frac{2}{3} + 
\frac{r^2}{2} \left( \ln \left( \frac{r}{4} \right) 
- \frac{1}{2} \right) \right].
\eeq
In the limit \( r \to 0 \) and
summed over colors (and over $u$ and $d$), it is
\beq
\langle {\thalf} ( \bar u u + \bar d d ) \rangle
\simeq \frac{\pi}{3} ( m_u + m_d )
\left( g | \vec A_8 | \right)^2.
\eeq
\par
We may use this formula
and Weinberg's relation
(Eq.(19.4.46) of \cite{Weinberg})
\beq
\langle {\thalf} ( \bar u u + \bar d d ) \rangle
\simeq \frac{m_\pi^2 F_\pi^2}{4(m_u + m_d )}
\eeq
in which \( F_\pi \simeq 184 \, \)MeV 
is the pion decay constant,
to estimate the effective magnitude \( \langle g | \vec A_8 | \rangle \)
of the gauge field in the physical vacuum of \emph{QCD} as
\beq
\langle g | \vec A_8 | \rangle \simeq
\sqrt{ \frac{3}{4\pi} }
\, \frac{ m_\pi F_\pi}{( m_u + m_d )}.
\label {gA}
\eeq
In the \(\overline{\rm MS}\) scheme 
at a renormalization scale \( \mu = 1 \,\)GeV,
the mass range
\( 3 \, \)MeV \( < ( m_u + m_d )/2 < 8 \, \)MeV
of the Particle Data Group~\cite{PDG}
implies that
\( 770 \, \)MeV \( < \langle g | \vec A_8 | \rangle < 2070 \, \)MeV\@.
Since \( \mu = 1 \,\)GeV is somewhat high
for hadronic physics, the low end of the range,
\( \langle g | \vec A_8 | \rangle \simeq 800 \, \)MeV,
may be more reliable.  With this estimate
of \( \langle g | \vec A_8 | \rangle \),
a drop of only 1\% in the hadronic quark-antiquark
condensate would produce a confining pressure
of the order of \(p \simeq (1 {\rm GeV})^4 \).

\section{The Range of the Strong Force}
Let us now specialize to the component \( | \Omega_A \rangle \)
of the \emph{QCD} vacuum
that has a constant gauge field \(A_8^1\)
pointing in direction 1 of space 
and direction 8 of color space.
In this component
the mean value of the hamiltonian of the gauge fields 
contains a mass term for the fields \( A_b^m \)
for \(m \ne 1 \) 
\beq
\sum_{a,b,c,m} {\thalf} f_{a8b} f_{a8c} \langle A_8^1 \rangle^2
A_b^m A_c^m
= \sum_{b,c,m} {\thalf} M_{bc}^2 A_b^m A_c^m
\eeq
in which the mass matrix is
\beq
M_{bc}^2 =
\langle g A_8^1 \rangle^2 
\sum_a f_{a8b} f_{a8c} 
= \langle g A_8^1 \rangle^2 
\left( T_8^2 \right)_{bc} 
\eeq
where the \(8\times8\) matrix \(T_8\)
is a generator 
of the group \emph{SU(3)} in the adjoint representation.
Since the matrix \(T_8\) is hermitian, 
the eigenvalues of the mass matrix
\( M_{bc}^2 \) are all non-negative.

The \emph{QCD} vacuum (\ref {qcd vacuum})
is an integral over slowly varying gauge
fields and their correlated condensates.
In this vacuum \( | \Omega \rangle \),
every space component \(A_c^i\) 
of the gauge field acquires a non-zero mean value. 
Thus gluons are massive in the vacuum of \emph{QCD},
and according to the estimate (\ref{gA}),
the mass of the gluon is in the range of hundreds of MeV. 
This mass explains why the strong force is of short range
and why strong van der Waals forces are not
seen in low-energy hadronic scattering
even though gluons are massless in perturbation theory.
But to the extent that color-electric
fields sweep away the quark-antiquark condensate,
gluons are massless inside hadrons,
particularly near the quarks.

\section{Summary and Conclusions}
We have seen that a vacuum component
\( | \Omega_A \rangle \)
consisting of a coherent state
of a slowly varying gauge field \( \vec A_c \)
and an associated quark-antiquark condensate
(\ref{cond})
possesses a negative energy density \( \rho_\Omega \)
of the order of \(  - (g | \vec A_c |)^4 \).
The difference between \( \rho_\Omega \) 
and the energy density \( \rho_h \) inside hadrons
confines quarks.
The \( q \bar q \) condensate 
in the component \( | \Omega_A \rangle \)
leads to a spontaneous breaking of chiral
symmetry with an order parameter 
\( \langle ( \bar u u + \bar d d )/2 \rangle \)
that agrees with soft-pion physics if the effective
strength \( g | \vec A_c | \) of the gauge field
is of the order of 800 MeV.
If the gauge field \( \vec A_c \) points
in the direction 1,
then some of the gauge fields \( A_a^m \)
for \( m \ne 1 \) become massive. 

The real vacuum is an integral over 
all such components \( |\Omega_A \rangle \)\@.
In the temporal gauge, the gauge field \( A_a^0 \)
is absent, and the vacuum is an 
integral (\ref{qcd vacuum}) over all gauge transformations
\( \omega(\vec x) \)
of the image \( | \Omega_A^\omega \rangle \)
of a state like the component \( | \Omega_A \rangle \)
under the gauge transformation \( \omega(\vec x) \).
This integral removes any breaking of rotational
invariance associated with the uniform field \( A_8^1 \).
The mean value of any gauge-invariant operator,
for instance the quark hamiltonian \( H_q \),
is a double integral
\beq
\langle \Omega | H_q | \Omega \rangle
= \int \! D \vec A_a \int \! D \vec A_a^\prime 
\, \langle \Omega_A | H_q | \Omega_{A^\prime} \rangle 
\label {mean value}
\eeq
in which most of the off-diagonal terms
are very small.  In the vacuum \( | \Omega \rangle \),
all the gluons acquire masses in the range of
hundreds of MeV.

Of course, the actual energy density
of the vacuum is small and non-negative.
But by using normal ordering,
we have been ignoring the zero-point energies 
of the fields.
Zero-point energies augment \( \rho_\Omega \)
by a positive or negative 
energy density that is quartically divergent
unless the number of Fermi fields
is equal to the number of Bose fields
and is quadratically divergent unless the super-trace
\( \sum (-1)^{2j} m^2_j \)
of the squared masses of all particles vanishes.
The large negative energy density \( \rho_\Omega \)
may make it possible to cancel a large positive
energy density due to the breaking of supersymmetry.





\section*{Acknowledgements}
I should like to thank L.~Krauss
for a discussion of chiral symmetry,
S.~Weinberg for e-mail about symmetry breaking,
and H.~Bryant, M.~Gell-Mann, M.~Gold,
G.~Herling, M.~Price, and G.~Stephenson
for helpful comments.

\begin{thebibliography}{99}
\bibitem{Gribov} The possibility that the lightness
of the $u$ and $d$ quarks might play a key role
in quark confinement was first suggested by
V.~N. Gribov, {\sl Phys. Scr.\/} T15 (1987) 164;
                {\sl Phys.\ Lett.\ B\/} 194 (1987) 119.
\bibitem{CahillHerling} For an early version
of this work in the context of lattice gauge theory,
see K.~Cahill and G.~Herling, {\sl Nucl.\ Phys.\ B\/}
(Proc.\ Suppl., in press), .
\bibitem{Wilczek} The quark-antiquark condensates
considered here should not be confused with the
diquark condensates studied by M.~Alford,
K.~Rajagopal, and F.~Wilczek, 
and  and by T.~Sch\"{a}fer and F.~Wilczek,
.
\bibitem{Glauber} R.~J. Glauber, {\sl Phys.~Rev.\/ \rm 130} (1963) 2529; 
131 (1963) 2766.
\bibitem{Weinberg} The notation, \emph{mutatis mutandis,}
is that of S.~Weinberg, {\sl The Quantum Theory of Fields\/},
Vol. I and II (Camb.\ Univ.\ Press, 1995--1998).
\bibitem{PDG} Particle Data Group,
{\sl Eur.\ Phys.\ J.\ C\/} 3 (1998) 341.

%\bibitem{kcrg1} K.~Cahill and R.~J. Glauber,
%{\sl Phys.~Rev.\/ \rm 177} (1969) 1857, 1882. 
%\bibitem{kc2} K.~Cahill, {\sl Phys.~Rev.\/ \rm 180}
%(1969) 1239, 1244. 
\end{thebibliography}
\end{document}

