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% A.A.Abrikosov, jr., Instantons and Scalar Matter in a Closed Volume.
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\begin{document}
\hyphenation{pseu-do-par-ticle pseu-do-par-ticles}
%============================================================
\title{Instantons and Scalar Matter in a Closed Volume}
\tocauthor{A.~A.~Abrikosov,~Jr.}
\author{A.~A.~Abrikosov,~Jr.\thanks{E-mail: {\sc persik@vxitep.itep.ru}} \\
{\small\em 117 259, Moscow, ITEP.}}
\date{}
\maketitle
%============================================================
\begin{abstract}
The leading correction to the action of a small pseudoparticle interacting
with scalar matter in a closed volume is calculated. We discover
suppression of instantons which is proportional to the density of matter.
\end{abstract}

\section*{Introduction}
%============================================================
Among the unsettled questions of quantum chromodynamics is that about the
role of instantons \cite{BPST,Shifman}. After the cooling of the first
enthusiasm it was practically left till better times.  Phenomenological
studies, \cite{Schaefer/Shuryak}, indicate that instantons could be
extremely interesting for physics. However instanton models are missing
something important:  instantons do not provide confinement and there is
no consistent way to cut the divergence of the instanton density. The key
may lie in physics of large scale gauge fields that are responsible for
confinement.

In principle instantons could tell in the range between the asymptotic
freedom and confinement scales. The problem is whether they are
discernible or completely blurred by the background. This makes
interesting to include pseudoparticles into models of confinement. The
simplest of those is the famous MIT bag, \cite{MIT}, which secures quarks
in a closed volume. The only manifestation of confinement is that matter
and gauge fields are subject to boundary conditions.

I investigate interaction of pseudoparticles with matter in a closed
volume. This continues my previous work, \cite{q94}. For scalar quarks the
problem may be solved exactly%
\footnote{Boundary conditions break chiral invariance and greatly
complicate the problem for fermions.}, boundary terms being under control.
The confined matter suppresses instantons similarly to the unbounded quark
plasma, \cite{q-plasma}. This may indicate that the same happens with
fermions.

\section{Interaction of instantons with the bag.}
%============================================================
\subsection{Instantons in the bag model.}
%============================================================
Instantons look most natural in the framework of the 4-dimensional
euclidean field theory. We shall follow the tradition.  Let us consider a
hadron $N$. The state $\left| N\right\rangle $ is characterized by the quark
contents, spin, parity {\em etc.\/} Euclidean generating functional for
gauge fields and matter in a spherical bag is:
\begin{equation}
Z_N(\beta ,\,R_B)=\langle N,\,R_B|\int [d\bar{\phi}\,d\phi \,dA]\exp
-\int_0^\beta dx_4\int_Bd^3x\,(L+B^0)|N,\,R_B\rangle .  \label{Z_N/gen}
\end{equation}
The functional integral over $[d\bar{\phi}\,d\phi \,dA]$ keeps account of
gauge fields and matter described by the {\em QCD}-Lagrangian $L$.  We
shall disregard quantum fluctuations of the bag. In order to define it's
radius $R_B$ the energy as usual must be minimized at the end.

Following the analogy with quantum statistical physics we shall assume
that fields are periodic in $x_4$, the inverse temperature $\beta
\rightarrow \infty$ being the period. Conditions at the surface of the bag
will be specified later.

The generating functional (\ref{Z_N/gen}) is directly related to the free
energy $F = -\beta^{-1}\ln Z$. At zero temperature it is defined by the mass
of the hadron,
\begin{equation}
M_N = \lim_{\beta \rightarrow \infty} \frac 1\beta \min_{R_B}
\left|\ln Z_N(\beta,\, R_B)\right| .  \label{M_N/gen}
\end{equation}

To avoid standard problems of instanton physics let us focus on small
instantons in the dilute gas approximation. It is convenient to
characterize the density of instantons in the bag $d_N$ in terms of the
additional action, $\delta S_N$:
\begin{equation}
d_N(\rho,\, x,\, R_B)=d_0(\rho)\, \exp -\delta S_N(\rho,\, x,\, R_B).
\label{dltS_N/def}
\end{equation}
Here $\rho$ is the pseudoparticle radius and $d_0$ is the density in the
infinite space.

The function $\delta S_N$ is of crucial importance. It is well known that
the integral density of pseudoparticles
$D_0=\int \rho ^{-5}\,d\rho \,d_0(\rho )$ diverges. A growth of $\delta
S_N$ could cut the divergence and make the model quantitatively
consistent.

Suppose that this is the case. Then the statistical sum of the dilute
instanton gas in the hadron is easy to calculate:
\begin{equation}
Z_N^I(R_B) = Z_N^0(R_B)\, \left\{\exp \left[\int_0^\beta dx_4\, \int_B
\frac{d^3x\, d\rho}{\rho^5}\, d_N(\rho,\, x,\, R_B) \right] \right\}^2;
\end{equation}
(the square of the exponential takes into account anti-instantons).

Comparing this with the definition (\ref{Z_N/gen}) of $Z_N$ we see that the
contribution of instantons may be included into the bag constant:
\begin{equation}
B^{eff} = B^0 - 2\int \frac{d\rho}{\rho^5}\, d_N(\rho,\, x,\, R_B).
\label{B^I/def}
\end{equation}
We see that $B^{eff}$ is smaller than $B^0$ by the total density of
topological fluctuations.  The outward pressure of pseudoparticles
$B^{eff} - B^0$ depends on structure and varies from hadron to hadron.

\subsection{Additional action of an instanton in the bag.}
%===========================================================
Now we shall discuss the additional action $\delta S_N$ of a
pseudoparticle in the bag. The correction to the action $\delta
S_N$ may be split into summands $\delta S$ according to their origin:
\begin{equation}
\delta S_N = \delta S_{cl}+\delta S_d+\delta S_q+\delta S_C.
\end{equation}
\begin{itemize}
\item  The first addend
$\delta S_{cl} = \left. S_{cl}\right|_B - \left. S_{cl}\right|_0$
allows for deformations of the instanton. This takes into account
interactions of the classical field with the walls. An estimate for
$\delta S_{cl}$ is given by the part of the action of the original
instanton gained outside the bag. For a small pseudoparticle located at
the center of the bag this gives:
\begin{equation}
\delta S_{cl} \sim
\int_{x > R_B}d^4x\,\frac{\left. F_{\mu\nu}^2\right|_0}4 \approx
\frac{3\rho^4}{R_B^4} S_{cl}, \qquad S_{cl} = {8 \pi^2 \over g^2}.
\label{dltS_cl} \end{equation}
\item  The second piece
$\delta S_d = B\,\delta V_4$
is associated with the swelling $\delta V_4$ of the bag around a
pseudoparticle. It must be of the same order:
\begin{equation}
\delta S_d = B^0 \delta V_4 \sim \delta S_{cl}
\sim \frac{\rho^4}{R_B^4}S_{cl}.
\end{equation}
Note that the collective effect of the instanton gas may be significant
even if each of individual $\delta V_4$'s is small.
\item The third term describes interaction of the pseudoparticle with
quarks in the closed volume:
\begin{eqnarray}
\delta S_q & = &
-\ln \frac{\langle N|\int [d\bar{\phi}\,d\phi \,d\delta A]\,
\exp -\int_0^\beta dx_4\int_Bd^3x\,L^I|N\rangle _B}%
{\langle 0|\int [d \bar{\phi}\,d\phi \,d\delta A]
\exp -\int_0^\beta dx_4\int_Bd^3x\,L^I|0\rangle _B}  \nonumber \\ &&
+\ln \frac{\langle N|\int [d\bar{\phi}\,d\phi \,dA]\,
\exp -\int_0^\beta dx_4\int_Bd^3x\,L^0\,|N\rangle _B}%
{\langle 0|\int [d\bar{\phi}\,d\phi\,dA]
\exp -\int_0^\beta dx_4\int_Bd^3x\,L^0|0\rangle _B}.  \label{dltS_q/def}
\end{eqnarray}
The notations  $\left\langle  N\left| \ldots \right| N\right\rangle _B$ and
$\left\langle 0\left| \ldots \right| 0\right\rangle _B$ stand for bag  with
and without quarks respectively. We shall show that $\delta S_q$ is of the
first order in $\rho ^2/R_B^2$ and plays the dominant role for small
instantons.
\item The last is the
Casimir correction $\delta S_C$ coming from the
quantization of gauge fields in the closed volume. Certainly it could stop
the growth of $g^2(\rho )$ at $\rho \sim R_B$ and cut off the divergence
of the density. However for small pseudoparticles the Casimir correction
appears mainly because of deformations of the quantum modes. The latter
are of the same value as those of the classical field.  Thus one could
expect that it is of the same order,
\begin{equation}
\delta S_C \sim {\rho^4 \over R_B^4} S_{cl}, \quad (?).
\end{equation}
\end{itemize}

The analysis indicates that for small pseudoparticles interaction with
matter is more important than deformations. The question of the Casimir
correction may require additional investigation.

\section{Interaction with scalar quarks.}
%===========================================================
Let us return to the bag model. Let the quark Hamiltonian and states be
$\hat{H}$, $\left| \alpha,\,a,\,f\right\rangle _B$.  The multi-index
$\alpha$ includes the principal quantum number, angular momentum {\em
etc.,\/} while $a$ and $f$ stand for colour and flavour.  The wave
functions $\phi^{\alpha ,\,a,\,f}(\vec x)$ satisfy lineary boundary
conditions.  Quarks are created and destroyed by operators $\hat a_{\alpha
,\,a,\,f}^\dagger$, $\hat a_{\alpha ,\,a,\,f}$ with usual commutation
relations:
\begin{equation}
\left[ \hat{a}_{\alpha ,\,a,\,f},\,
\hat{a}_{\beta ,\,b,\,g}^{\dagger}\right] =
\delta _{ab}\,\delta _{\alpha \beta }\,\delta _{fg}.
\label{a-comm}
\end{equation}
The role of the vacuum in the model belongs to the empty bag
$\left|0\right\rangle _B$.  A hadron $\left| N\right\rangle _B$ is
produced by simultaneous action of quark operators. Baryons are
characterized by colour singlet vertices $N^{\alpha \beta \gamma \ldots}$
(we suppress colour and flavour indices):
\begin{equation}
\left| N\right\rangle _B=N^{\alpha \beta \gamma  \ldots}\,
\hat{a}_\alpha^{\dagger }\,\hat{a}_\beta ^{\dagger }\,
\hat{a}_\gamma ^{\dagger } \ldots \left|0 \right\rangle _B;\qquad
\left\langle N\right| _B=N_{\alpha \beta \gamma \ldots}^* \,
\left\langle 0\right|_B \hat a_\alpha \,\hat a_\beta \,
\hat a_\gamma \ldots . \label{N_op}
\end{equation}

It is possible to define the Heisenberg creation and annihilation
operators of quarks in the coordinate representation:
$$
\hat{\phi}_{a,\, f}^{\dagger }(\vec{x},\, t)=
\sum_\alpha e^{\hat{H}t}\,\hat a_{\alpha,\,a,\,f}^\dagger\, e^{-\hat{H}t}\,
\phi _{\alpha ,\,\,f}^{*}; \quad
$$
\begin{equation}
\hat{\phi}^{a,\, f}(\vec{x},\, t)=
\sum_\alpha e^{\hat{H}t}\, \hat{a}_{\alpha ,\,a,\,f} \, e^{-\hat{H}t}\,
\phi^{\alpha ,\,\,f};
\label{psi/def}
\end{equation}
where $\phi^{\alpha,\, f} (\vec x)$ and $\phi^*_{\alpha,\, f} (\vec x)$
are wave functions of the corresponding states. The propagation function
of quarks in the bag is defined as:
\begin{equation}
\Delta_f(\vec{x},t;\,\,\vec{y},\,t^{\prime })=
\left\langle {0}\right| {\bf T}\, \hat{\phi}^f(\vec{x},\,t)\,
\hat{\phi}_f^{\dagger}(\vec{y},\, t^{\prime}) \left| {0}\right\rangle_B;
\label{Dlt_op/def}
\end{equation}
where $\bf T$ stands for time ordering.

It does not take long to see that $\delta S_q$, (\ref{dltS_q/def}), may be
expressed in terms of matrix elements of the propagator with and without
instanton.
\begin{equation}
\delta S_q = -\ln \frac{N_{\alpha \beta \gamma \ldots}^* \,
\Delta_{\alpha^\prime}^\alpha (\beta)^I\,
\Delta _{\beta^\prime}^\beta (\beta)^I\,
\Delta _{\gamma^\prime}^\gamma (\beta )^I \ldots\,
N^{\alpha^\prime \beta^\prime \gamma^\prime \ldots}}%
{N_{\alpha \beta \gamma \ldots}^* \,
\Delta_{\alpha^\prime}^\alpha (\beta)^0 \,
\Delta_{\beta^\prime}^\beta (\beta)^0 \,
\Delta_{\gamma^\prime}^\gamma (\beta)^0 \ldots \,
N^{\alpha^\prime \beta^\prime \gamma^\prime \ldots}}.
\label{dltS_q/ME}
\end{equation}
The matrix element $\Delta_{\alpha^\prime}^\alpha (\beta)$ is defined as
follows:
\begin{equation}
\Delta _{\beta ,\,b}^{\alpha ,\,a}(\beta)=
\left\langle {0}\right|
\hat{a}_{\alpha ,\,a}(\beta)\, a_{\beta ,\,b}^{\dagger }(0)
\left| {0}\right\rangle _B = \int_B d^3 x\, d^3 y\,
\phi _{\beta ,\,b}^{*}(\vec{x})
\Delta (\vec x,\, \beta;\, \vec y,\, 0)\,
\phi^{\alpha ,\,a}(\vec{y}). \label{ME/def}
\end{equation}

Expression (\ref{dltS_q/ME}) simplifies if corrections are small.
Expansion of the matrix element in $\rho ^2/R_B^2$ has the form (up to
irrelevant terms):
\begin{equation}
\Delta _{{\alpha }^{\prime },\,a^{\prime }}^{\alpha ,\,a}(\beta )^I\,
=\Delta_{{\alpha }^{\prime },\,a^{\prime }}^{\alpha ,\,a}(\beta )^0
\left( 1+\frac{\rho^2}{R_B^2}\epsilon _{\alpha ,\,a}(\beta ,\,R_B)\right)
+ \ldots ;  \label{Dlt/exp}
\end{equation}
the nonperturbed propagator being diagonal,
$\Delta _{{\alpha }^{\prime },\,a^{\prime }}^{\alpha ,\,a}(\beta )^0
\propto \delta _{\alpha ^{\prime }}^\alpha \,\delta _{a^\prime}^a$.

In order to calculate the sum in (\ref{dltS_q/ME}) we recall that the
instanton is embedded into a $SU(2)$ subgroup of the gauge group and only
two of $\epsilon$'s are not zero. The vertex $N$ is colour singlet, {\em
i.~e.\/} $N\propto \varepsilon^{abc\ldots}$ and summation gives:
\begin{equation}
\delta S_q = -\frac2{N_c} \sum_{\alpha =1}^{N_c} \epsilon_\alpha
\frac{\rho^2}{R_B^2} .
\label{dltS_q/scal} \end{equation}
Thus $\delta S_q$ may be expressed in terms of the instanton correction to
the quark propagator. An interesting feature is that $\delta S_q \propto
N^{-1}_c$ and matter does not influence instantons in the large-$N_c$
limit.

\section{Scalar correlator in the instanton field.}
%===========================================================
Let $\phi$ be a massless scalar field in fundamental representation of the
$SU(2)$ gauge group. The action of the matter and the gauge field is:
\begin{equation}
S=-\int_{B_4} d^4x\,
\left[\phi ^{\dagger }\nabla _\mu ^2 \phi +
\frac1{4g^2}(\partial _\mu A_\nu ^a-\partial _\nu A_\mu ^a
+\epsilon^{abc} A_\mu ^b A_\nu ^c)^2 \right ].
\label{s_matter+s_gauge} \end{equation}
Vector potential is denoted by $A_\mu ^a$ with $\mu =1, \ldots,\, 4$ and
$a=1,\, 2,\, 3$ being the Lorentz and group indices respectively. The
covariant derivative in fundamental representation is $\nabla_\mu =
\partial_\mu -\frac{i\tau^a} 2A_\mu^a$ where $\tau^a$ are Pauli matrices.
We shall impose the linear Dirichlet boundary condition, {\em i.~e.\/} put
$\phi$ to zero at the boundary:
\begin{equation}
\phi(x) = 0\quad {\rm for}\quad x \in \partial B_3.
\label{3d-bound} \end{equation}
The object under study will be the correlation function%
\footnote{Surprisingly in a limited space it does not automatically
coincide with the propagator.}.
\begin{equation}
\Delta (x,\,y) = \left. \frac \delta{\delta J^\dagger (x)}
\frac \delta{\delta J(y)}\right|_{J^{\dagger }=J=0}
\frac 1{V_4} \ln \int DA_\mu ^a\,D\phi \,D\phi^{\dagger}
\exp (-S + J^{\dagger }\phi + \phi^\dagger J);
\label{Dlt_def}
\end{equation}

The classical field $A_\mu ^a$ is an instanton of a radius $\rho $ in
singular gauge:
\begin{equation}
A_\mu ^{a}(x) = \frac{2\rho^2 \bar\eta_{\mu\nu}^a (x-z_0)_\nu}%
{(x-z_0)^2((x-z_0)^2+\rho ^2)} =
-\bar{\eta}_{\mu \nu }^a\partial _\nu \ln \Pi (x-z_0).  \label{inst}
\end{equation}
Here $\Pi (x-z_0) = 1+\rho^2/(x-z_0)^2$ ($z_0$ is the instanton center)
and $\bar{\eta}_{\mu \nu }^a$ is the 'tHooft's $\bar{\eta}$-symbol.
At the main order in $\rho^2 / R_B^2$ we may neglect deformations of the
instanton.

Instanton field can not be taken into account perturbatively.
Fortunately for the chosen boundary conditions the problem can be solved
exactly.

It is convenient to make use of the two sets of $2\times 2$ matrices,
(latin indices stand for the ``spatial'' dimensions):
\begin{equation}
\tau _\mu =(\tau _i,i), \qquad \tau _\mu ^{\dagger }=(\tau _i,-i).
\end{equation}
Those are closely related to the $\eta$-symbols:
\begin{equation}
\tau_\mu ^{\dagger }\tau_\nu =\delta _{\mu \nu }+i\eta _{\mu \nu }^a\tau_a;
\qquad
\tau_\mu \tau _\nu ^{\dagger }=
\delta _{\mu \nu }+i\bar{\eta}_{\mu \nu }^a\tau _a.  \label{tau}
\end{equation}
Convolutions of \mbox{$\tau$-matrices} with 4-vectors are denoted by hats:
\begin{equation}
\hat{x}=x_\mu \tau _\mu ; \qquad
\hat{x}^{\dagger }=x_\mu \tau _\mu ^{\dagger};
\end{equation}

The covariant d'Alembert operator in the instanton field, Eq.~(\ref{inst}),
can be converted to the form:
\begin{equation}
\nabla_\mu ^2=\Pi^{\frac 12}(x)\, \hat{\partial}\frac 1{\Pi (x)}\,
\hat{\partial}^\dagger \Pi^{\frac 12}(x) = D D^\dagger; \label{nasq}
\end{equation}
where we have introduced linear differential operators $D$ and
$D^{\dagger }$:
\begin{equation}
D(x)= \Pi^{\frac 12}(x)\hat{\partial}\frac 1{\Pi^{\frac 12}(x)};\quad
D^\dagger (x) = \frac 1{\Pi^{\frac 12}(x)}\hat{\partial}^\dagger
\Pi^{\frac 12}(x).  \label{D/D_dag}
\end{equation}

With the help of the representation (\ref{nasq}) we shall immediately
obtain for the correlator (\ref{Dlt_def}):
\begin{equation}
\Delta(x,\, y) = \left. -\frac \delta {\delta J^\dagger(x)}
\frac \delta{\delta J(y)}Z\right|_{J^\dagger=J=0} = -\int_0^\beta dz_4
\int_{B_3} d^3z\, D^{\dagger\, -1}(x,\, z)\, D^{-1}(z,\, y).
\label{Dlt_gen} \end{equation}

The integral operators $D^{-1}$ and $D^{\dagger\, -1}$ are defined as follows
(mind the order):
\begin{equation}
D^{\dagger\, -1}(x,\, y)\, \overleftarrow D {\vphantom D}^\dagger =
\overrightarrow D\, D^{-1}(x,\, y) = {\cal I} (x,\, y) ;
\label{D/D_dag_inv}
\end{equation}
where ${\cal I}(x,\, y)$ is the unity integral operator which fulfills the
Dirichlet boundary conditions, (\ref{3d-bound}). It is not a problem to
check that,
\begin{mathletters}
\begin{eqnarray}
D^{-1}(x,\, y) =\Pi^{\frac 12}(x)\, \overrightarrow{\hat\partial}
\vphantom{\hat \partial}^\dagger
\left. \frac 1{\partial^2}(x,\, y)\right|_B\, \Pi^{- \frac12}(y); \\
D^{\dagger\, -1}(x,\, y) = \Pi^{-\frac 12}(x) \left.
\frac1{\partial^2} (x,\, y)\right|_B\, \overleftarrow{\hat\partial}
\Pi^{\frac 12}(y). \end{eqnarray} \end{mathletters}

Substituting these formulae into the equation (\ref{Dlt_gen}) we obtain the
general expression for the scalar correlator:
\begin{equation}
\left.\Delta ^I(x,\, y)\right|_B = - \frac 1{\sqrt{\Pi(x) \Pi(y)}}
\int^\beta_0 dz_4 \int_{B_3} d^3z
\left. \frac 1{\partial^2}(x,\, z)\right|_B\, \hat{\partial}\Pi (z)
\left. \hat\partial^\dagger \frac 1{\partial^2}(z,\, y)\right|_B.
\label{Dlt_int}
\end{equation}
Note that since the free Green function $\left. \partial^{-2} \right|_B$
satisfies the boundary condition the correlator (\ref {Dlt_int}) also is
zero at the boundary.

Suppose that $x$, $y$ are far from the instanton. Retaining only the
leading diagonal terms we obtain after some algebra:
\begin{equation}
\left.\Delta(x,\, y)\right|_B = -\left. \frac1{\partial^2}(x,\, y)\right|_B
- 2\pi^2 \rho^2\, \left.\frac 1{\partial^2}(x,\, z_0)\right| _B
\left.\frac 1{\partial^2}(z_0,\, y)\right|_B + \ldots
\label{Dlt/far}
\end{equation}
The same expression would describe scattering by a $\delta$-potential.
Hence interaction with a small pseudoparticle can be modelled by adding to
the action the point-like vertex:
\begin{equation}
S_{\rm matter} \Rightarrow S_{\rm matter}(z_0) = -\int_{B_4}d^4x\,
\left[\phi^{\dagger}(x)\, \partial^2\phi (x) - 2\pi^2\rho^2\,
\left|\phi(x)\right|^2\, \delta^4(x-z_0)\right];  \label{i_vert}
\end{equation}
At the lowest order instanton feels only the density of matter near the
center.

\section{Matrix elements in the spherical bag.}
%===========================================================
Correlation function can be expressed in terms of 3-dimensional harmonics.
Let $\phi_n(\vec x)$, $\phi_n(\vec y)^\dagger$ be eigenfunctions of the
$3d$-Laplace operator $\partial_i^2$ that fulfill the boundary condition
(\ref{3d-bound}):
\begin{equation}
- \stackrel\rightarrow \partial \!\! \vphantom{\partial}^2_i
\phi_n(\vec x) = E_n^2\, \phi_n(\vec x); \qquad
- \phi_n (\vec y)^\dagger \, \stackrel\leftarrow\partial \!\!
\vphantom{\partial}^2_i = E_n^2\, \phi_n(\vec y)^\dagger.
\end{equation}
The scalar propagator in the $3+1$-cylinder without pseudoparticles is:
\begin{equation}
\left. -\frac 1{\partial^2}(x,\, y) \right|_{B_3} =
\sum\limits_{n=1}^\infty \frac{\phi_n(\vec x)\, \phi_n(\vec y)^\dagger}%
{2E_n} \exp -E_n |x_4 - y_4| .  \label{dlt-free}
\end{equation}

Substituting this into (\ref{Dlt/far}) we obtain for the diagonal matrix
element, (\ref{ME/def}):
\begin{equation}
\left\langle n,\, \beta,\, b\left| \Delta^I(z_0,\, \beta)\right|
n,\, \alpha,\, a\right\rangle =
\frac{\exp -E_n \beta}{2E_n} \left(1- \frac{\pi^2 \rho^2}{E_n}\,
\left| \phi_n(\vec z_0) \right|^2 \right) \delta^\alpha_\beta \delta^a_b.
\label{m/e_full_nn}
\end{equation}
This formula contains all the information about interaction of the
instanton with scalar quarks in the leading approximation. With the help
of (\ref{dltS_q/scal}) we obtain for a ``baryon'' with $N_c$ scalar
quarks:
\begin{equation}
\delta S_q = \frac{2 \pi^2 \rho^2}{N_c} \sum_{\alpha=1}^{N_c}
\frac{\left| \phi_\alpha (\vec z_0) \right|^2}{E_\alpha}.
\label{dltS_q/gen}
\end{equation}
The sign of $\delta S_q$ corresponds to the suppression of pseudoparticles
by matter.

Symmetrical wave functions that satisfy the boundary condition
(\ref{3d-bound}) are
\begin{equation}
\varphi_n(\vec{z})=\frac 1{\sqrt{2\pi R_B}|\vec z|}\sin \pi n
\frac {|\vec z|}{R_B},\qquad E_n=\frac{\pi n}{R_B}.
\end{equation}
Inserting these into the formula (\ref {m/e_full_nn}) we get for diagonal
matrix elements,
\begin{equation}
\left\langle n,\, \beta,\, b\left| \Delta^I(z^0,\, \beta)\right|
n,\, \alpha,\, a\right\rangle_{{\rm sph}} = \frac{e^{-E_n \beta}}{2E_n}
\left(1 - \frac{\rho ^2}{2n(\vec z_0)^2}
\sin^2 \pi n \frac{|\vec z_0|} {R_B} \right)\,
\delta^\alpha_\beta \delta^a_b. \label{m/e_sph_nn}
\end{equation}

The correction is of the order $\rho ^2/R_B^2$ and reaches the maximum at
$\vec z_0 = 0$:
\begin{equation}
\delta S_q \leq \lim_{z\rightarrow 0} \frac2{N_c} \sum_{\alpha=1}^{N_c}
\frac{\rho ^2}{2n_\alpha z^2} \sin^2 \pi n_\alpha \frac z{R_B} =
\frac{\pi^2}{N_c} \sum_{\alpha=1}^{N_c} n_\alpha \frac{\rho^2}{R_B^2}.
\label{S_q/sph}
\end{equation}
This justifies discarding the higher powers of $\rho ^2/R_B^2$. The
question is what values of $\rho $ are allowed? It is natural to claim
that the correction should not exceed the main term. Introducing the
average value $\bar n = N_c^{-1} \sum_{\alpha=1}^{N_c} n_\alpha$ we find:
\begin{equation}
\frac{\rho ^2}{R_B^2} \leq \frac 1{\pi^2 \bar n} \approx 0.1\, {\bar n}^{-1}
\quad {\rm and\quad }\frac\rho{R_B} \leq \frac 1\pi \sqrt{\frac 1{\bar n}}
\approx 0.32 \,{\bar n}^{-\frac 12}, \label{rho/R_mx_sph}
\end{equation}
This is in accord with the previous assumption that $\rho ^2/R_B^2\ll 1 $.

\section*{Conclusion.}
%===========================================================
The analysis proves that scalar matter suppresses pseudoparticles. The
leading effect depends only on the density and the energy of matter. The
same suppression takes place in the quark plasma, \cite{q-plasma}. It is
highly probable that the present result is universal and is not specific
to the bag model.

The interaction may be modelled by a point-like term in the Lagrangian.
Instantons are swept towards the boundary where the quark density is
less. Thus the model indicates spatial separation of the internal
asymptotically free, intermediate instanton and external confinement
domains.

The derivation was mathematically exact and limitations come from
physical assumptions. Neglecting deformations of the instanton and of the
bag is allowed unless (see Eqs.~\ref{dltS_cl},~\ref{S_q/sph}),
$\delta S_{cl} \ge \delta S_q$. This leads to the condition
\begin{equation}
{\rho^2 \over R_B^2} \le g^2 \ll 1,
\end{equation}
that is more restrictive than (\ref{rho/R_mx_sph}). Our work is a step
towards and may offer a hint at what happens in the more realistic case
of spinor quarks.

I would like to thank the organizers for the financial support that made
possible my participation. The work was done with partial support of the
RFBR grant 97-02-16131.

\begin{thebibliography}{99}
%----------- References ----------------------------------
\def\journal#1#2#3#4{{\it #1}{\bf#2},\ (#3),\ {#4}}
%
%     Specific journals:
\def\NPB{\journal{Nucl.\ Phys.\ \mbox{\bf B}}}
\def\PLB{\journal{Phys.\ Lett.\ \mbox{\bf B}}}
\def\PRD{\journal{Phys.\ Rev.\ \mbox{\bf D}}}
\def\JETPL{\journal{JETP\ Lett.\ }}
\def\PJETP{\journal{Pis'ma\  Zh.E.T.P.\ }}
\def\RMP{\journal{Rev.\ Mod.\ Phys.\ }}
%===================================================
\bibitem{BPST} A.~A.~Belavin, A.~M.~Polyakov, A.~S.~Schwartz,
Yu.~S.~Tyupkin, \PLB{59}{1975}{85}.
\bibitem{Shifman} ``Instantons in gauge theories'', ed. by
M.~A.~Shifman,  World Scientific, Singapore, 1994.
\bibitem{Schaefer/Shuryak} T.~Sch\"afer, E.~V.~Shuryak, \RMP{70}{1998}{323}.
\bibitem{MIT} A.~Chodos, R.~L.~Jaffe, C.~B.~Thorn, and V.~F.~Weisskopf,
\PRD{9}{1974}{3471}.
\bibitem{q94} A.~A.~Abrikosov, Jr., \PJETP{63}{1996}{833}.
\bibitem{q-plasma}  A.~A.~Abrikosov,~Jr., \NPB{218}{1983}{459}.
\end{thebibliography}
\end{document}

With compliments, AAA.

