%Dear colleagues
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%In this message I send my contribution to Proceedings of Q98.
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%Sincerely,
%Oleg Pavlovsky
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\begin{document}
\title{Regularization of self-energy of quark bag}

\tocauthor{O. V. Pavlovsky}
\author{
 O. V. Pavlovsky
 \thanks{e-mail address: pavlovsky@srdlan.npi.msu.su and
                         ovp@goa.bog.msu.su}
\\ {\small\em Physical Department, Moscow
 State University,}\\ {\small\em Moscow, 119899, Russia}
 }


\date{}

\maketitle

My report is devoted to the problem of constructing of the model of
quark bag on the basis of singular solutions of classical Yang-Mills
equations. These solutions had been obtained in works of F.A.Lunev
\cite{lun} in
gauge-invariant approach to Yang-Mills theory by using the analogy
between these solutions and black-hole solutions in general
relativity. These investigations ware reported in our oral
presentation on the previous conferences ''Quarks 96" \cite{q96}. The
idea to use these solutions as the basis of the confinement of quarks
is proposed in this report too. But on the way of realization of this
idea the very essential problem arises. The self-energy of the
gluon bag that confines the quark in the closed domain is infinite.
This problem seems to be very typical for the bag model, but in our
model this problem can be solved.

The basic assumption is that quarks in zero approximation move in a
certain effective YM potential that is a solution to classical YM
equations with singularity on the sphere. Our model obviously can be
derived from QCD by quantization in neighborhood of such singular
solution as zero approximation. Further correction can be also
obtained in a systematic way. Indeed let us consider ``partition
function" that is represented as path integral
$$
Z=\int\, DA\, D{\bar \Psi} D\Psi
exp\{i\int^T_0 dt\int d^3x(L_{YM}(A)+L_{ferm}
({\bar \Psi},\Psi,A))\}\eqno(1)
$$

We assume that the main contribution in this path integral is given
by trajectories close to classical solution with singularity on the
sphere. Applying stationary phase method one gets the follow
expression in zero approximation
$$
Z=\sum_{1\le k_s\le n_s}
\int\limits_{zero \atop modes}
e^{-i(E_{YM}({\cal R})+\sum_q \sum_s k_sE_s({\cal R},m_q))T} \eqno(2)
$$
where
\begin{itemize}
\item $E_{YM}({\cal R})$ is self-energy of the classical YM fields
configuration that has singularity on the sphere ;

\item $E_{s}({\cal R})$ are positive eigenvalues of Hamiltonian of
color Dirak particle $$ i\gamma^0{\vec \gamma}{\vec
\nabla}(A_{cl})-\gamma^0m_q ; $$

\item $n_s$ is doubled multiplicity of the eigenvalue $E_s$ ;

\item ${\cal R}$ is a radius of singularity.
\end{itemize}

Applying again stationary phase method, one  gets the formula to
determine the mass of ``hadron"
$$
M^{theor}=E_{YM}({\cal R}_0)+\sum_{quarks}E({\cal R}_0,m_q)
$$
where ${\cal R}_0$ is defined from an equation
$$
 \left.  {\partial \over \partial {\cal R}}
[E_{YM}({\cal R})+\sum_{quarks}E({\cal R}_0,m_q)]\right|_{{\cal R}
={\cal R}_0}=0
$$

 In this formula the quarks contribution $E({\cal R}_0,m_q)$ is
defined by a standard numerical procedure. Details of this calculation
you can find in our work \cite{lun-pav}.

But now I want to discuss the contribution to the mass formula from YM
fields. This is $E_{YM}({\cal R})$. The problem is that quantity
$E_{YM}({\cal R})$ is divergent due to singularity at $r={\cal R}$
and must be regularized.

But before this I wont shortly discuss the theory of singular
solution of YM equation.

For simplicity, let us consider the three dimensional $SU(2)$ YM
theory. There are two approaches to this theory: the first one is a
traditional approach in terms of potential $A_\mu$ of YM fields and
the second one is a gravity-like approach proposed by Lunev
\cite{lun}.  This approach operates with a second rank tensor
$$
G_{ij}=-{1 \over 2}\varepsilon_{ikl}\varepsilon_{jmn}\,tr\,
F^{kl}F^{mn}\eqno(1)
$$
where $F_{ij}$ is field strength tensor.
$G_{ij}$ is treated as metric of  some manifold ${\cal M}$.
Description of the ``dynamics" of this manifold is obsolutely
equivalent to description of YM fields in terms of potential. To make
the discussion more clear both of these approaches are used in
parallel way. It appears that the lagrangian and equation of motion
can be written in both approaches as in Tab.1.

\begin{table}[thb]
\begin{center}

 \begin{tabular}{lll}
% \hline
 & {\bf Approach I} &  {\bf Approach II } \\
 &                  &                     \\

% \hline
Lagrangian  &    &           \\
of the system  &
$ L=-\frac{1}{16g^2} tr(F_{ij}F_{ij})  $ &
$ L=-\frac{1}{16g^2}(\sqrt{G}R+\lambda g^{ij}G_{ij} ) $ \\
 &                  &                     \\
% \hline
Equation of motion &
$ D_i F_{ij}=0 $ &
$R^{ij}-\frac{1}{2}RG^{ij}=\lambda\frac{\sqrt{g}}{\sqrt{G}}g^{ij}$\\
 &                  &                     \\
Spherically &$ A_i^a=\varepsilon_{aij} {x^j \over r^2}(1-H(r))$&
$G_{rr}=A(r),\ G_{\theta \theta}=B(r),$\\
symmetrical ansatz & & $ G_{\phi \phi}=B(r)\sin^2\theta  $\\
 &                  &                     \\
Resulting equation & $r^2H''=H(H^2-1)$ &
After changing of variables \\
 &  & $ K=\sqrt{B} \quad H=K'/\sqrt{A} $ \\
 &  & one gets \\
 &  &$r^2H''=H(H^2-1)$  \\
 &                  &                     \\

Regularization part &
$\Delta L=-\frac{i\varepsilon^2}{12g^2}tr(F_{ij}F_{jk}F_{ki})$&
$\Delta L=-\frac{1}{16g^2}(-2\Lambda)\sqrt{G},$ \\
in lagrangian &  &   $\Lambda=\frac{1}{\varepsilon^2}$ \\
 &                  &                     \\
\end{tabular}

\end{center}
\caption{$R_{ij}$ is a Ricci tensor of manifold ${\cal M}$
corresponds with metric $G_{ij}$. $R$ is a scalar curvature.
There $g^{ij}$ is a metric of the Euclidean space.}
\end{table}



Substituting in these equations the spherically symmetrical ansatz one
gets absolutely identical equations. It is not surprising because
these approaches are absolutely equivalent.

These equations have singular on the  sphere solutions which have
  quasi-Schwarzschild behavior near the singularity $$ H(r) \sim
  {\sqrt{2} \over {1-r/{\cal R}}}\, , \, r\to {\cal R}-0 .  $$

The functional of classical energy of YM field
$$
E_{YM} [H(r)]=\frac{4\pi}{g^2} \int^R
\{ \frac{1}{2}(H')^2 + \frac{(H^2-1)^2}{4r^2} \} \to \infty
$$
is
divergent on this solution due to singularity of $A_{cl}$, and the
procedure of regularization must be suggested.

We suggest a very simple method to solve this problem. From
methodological point of view this method is a modification of
the well-known method of regularization by high covariant
  derivative \cite{slav}.

Instead of initial theory with lagrangian $L$, we would describe a
slightly-modified theory with lagrangian
$L^{\varepsilon}=L+\varepsilon \Delta L $. The modified theory
has to satisfy this obvious requirements:

1) If $\varepsilon$ tends to the zero, $L^{\varepsilon} $
 tends to L :
$ \qquad L^{\varepsilon}  \quad
\stackrel{\varepsilon \to 0}{\longrightarrow} \quad L $

2) $L^{\varepsilon}$ has a maximum symmetries of $L$

3) Theory with $L^{\varepsilon}$ set has a set of solutions
 $H_{\varepsilon}(r)$ that tend to the H(r) almost everywhere
if ${\varepsilon}$ tends to the zero.

4) Functional of classical energy of modified theory on the solutions
$H_{\varepsilon}(r)$ has to be finite for any nonzero $\varepsilon$.

5) The correction $\Delta L$ must have the same order by inverse $g$
that has initial lagrangian $L$.

It tern out that it is possible to find a correction that satisfies
  these requirements.  You can see this in the last section of table
  1.  This correction isn't surprising, in gravity-like approach this
  correction is the well-known $\Lambda$-part.  This one has a
  $\frac{1}{g^2}$ order since this term can be expressed the
following form
  $$ tr(D_i F_{jk} \times D_i F_{jk})-tr(D_i F_{ik}
\times D_J F_{jk})= -4i \, tr(F_{ij}F_{jk}F_{ki}) $$

Let us variate the new modified lagrangian $L^{\varepsilon}$ and
substitute the spherically symmetrical ansatz in the results one
gets the following expression
$$
(1-\frac{\varepsilon^2}{r^2}(H_{\varepsilon}^2-1))r^2 H_{\varepsilon}''=
H_{\varepsilon}(H_{\varepsilon}^2-1)+\frac{\varepsilon^2}{r^2}
(rH_{\varepsilon}')^2 H_{\varepsilon}
-2\frac{\varepsilon^2}{r^2} rH_{\varepsilon}'(H_{\varepsilon}^2-1)
$$

If $\varepsilon$ tends to zero this equation becomes a Wu-Yang
equation that has singular on the sphere solutions.
But this equation has solutions suggested to contributions 3) and
4) but nonsingular on the sphere. In point $r=R$ these solutions have
asymptotic
$$
H_{\varepsilon}(r) \sim \sqrt{1+R^2/\varepsilon^2} +
C(1-r/R)^{2/3}+ \dots
$$
where $C$ is a constant.

If we substitute these solutions in new functional of classical energy
$$
E_{YM}^\varepsilon [H(r)]=\frac{4\pi}{g^2} \int^R
\{ \frac{1}{2}(1-\frac{\varepsilon^2}{r^2}(H^2-1))(H')^2 +
\frac{(H^2-1)^2}{4r^2} \}
$$
we get a finite result. But if $\varepsilon$ tends to zero, result
tends to infinity. This is a principal point of our contruction. On
the set of solutions $ \{ H_\varepsilon (r) \}$ the functional of
classical energy is a function of $\varepsilon$. Getting the
asymptotic at $\varepsilon \to 0$ of this function is very nontrivial
operation but result is following
$$
E_{YM}^{\varepsilon} [ H_{\varepsilon} (r) ]
\stackrel{\varepsilon \to 0}{\longrightarrow}
\frac{1}{g^2}(constant_1 + O(\varepsilon))
$$

Now if we suggest that $g$ is
$$
g=\frac{constant_2}{\varepsilon^2} + O(\frac{1}{\varepsilon})
$$
one gets the finite expression for the $E_{YM}$
$$
E_{YM}^{\varepsilon} [ H_{\varepsilon} (r) ]
=B_{phys}R^3 + O(\varepsilon)
$$

Now as we expect in the previous works \cite{q96,lun-pav} the
regularized self-energy of the bag gives a polynomial contribution in
mass formula and is proportional to the volume of bag. In order to
achieve the best agreement with the experimental data we choose the
following volum for the constant $B$. Our result is in agreement with
experiment data with accuracy 3-7 per cent for all hadron mass except
those of light pseudoscalar mesons. This accuracy is a maximal
possible accuracy for any constituent quark model in which
interaction between the quark isn't taken into account.


\begin{thebibliography}{99}

 \bibitem{lun}
{\it F. A. Lunev.} Phys.Lett.B. 1992. V.295. P.99.;
Theor. Math. Phys. 1993. V.94. P.66.;
Phys.Lett.B. 1993. V.311. P.273.

 \bibitem{q96}
{\it F.A.Lunev, O.V.Pavlovsky.}
`` Reformulation of QCD in the language of general relativity, black
holl-like solutions of Yang-Mills equations, and hadron mass
spectrum.": in Proceedings of the 9th International Seminar ``Quark's
  96" , valume II, pp. 68

 \bibitem{lun-pav}
{\it F.A.Lunev, O.V.Pavlovsky.} Theor. Math. Phys. 1998. V.8.

 \bibitem{slav}
{\it A.A.Slavnov.} Theor. Math. Phys. 1972. V.13. P.1062.



 \end{thebibliography}


\end{document}

