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{\vrule\vbox{\hrule\hbox to 1ex{ \hfill}\kern 1 ex\hrule}\vrule}}

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\def\vf{\varphi}
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%\renewcommand{\refname}{Referenses}
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\ifx\tocauthor\undefined\def\tocauthor#1{\relax}\fi
\ifx\toctitle\undefined\def\toctitle#1{\relax}\fi
\begin{document}
\title{On the role of Lorentz-covariance in hybrid chiral quark bag
  models}
\tocauthor{I. Cherednikov}
\author{I. Cherednikov \thanks{e-mail address:
    igorch@goa.bog.msu.su } \\ {\small\em Department of Physics, Moscow State
University} \\ {\small\em 119899 Moscow, Russia} \and K. Sveshnikov
\thanks{e-mail address:  costa@bog.msu.su } \\ {\small\em Department of
Physics} \\ {\small\em and Institute for Theoretical Problems of
Microphysics,} \\ {\small\em Moscow State University} \\ {\small\em 119899
Moscow, Russia} }


\date{}

\maketitle

\begin{abstract}
The structure of surface terms for hybrid chiral quark bag models in
(1+1)D is studied by means of the method of covariant group
variables.  The analytical connection between fields from both
sides of the bag boundary following from the specific structure of
invariant dynamics is considered.  It is shown that in general case
such connection imposes essential constraints on the possible field
dynamics.
\end{abstract}

%\vspace{1cm}


%\vspace{1.5cm}


\section{Introduction}
In this paper we study the structure of surface terms in the (1+1)D
2-phase field theories with boundaries within the framework of the
covariant generalization of the Bogoliubov's collective coordinate
method.
The total action of a hybrid quark bag model \cite{Rho,Ho}
contains three terms
$$S_{total} = S_{in} +S_{out} +S_{on} \ . \eqno(1)$$
Here $S_{in} = \int \! dx \ \lagr _{F} (x) \tt (V)$ is the action of
the fermions confined inside the bag volume $V$ (the
phase of asymptotic freedom), $S_{out} = \int \! dx \ \lagr _{B} (x)
\tt  ({\bar V})$ is the effective non-linear theory (e.g. the
Skyrme model, \cite{Za}) which describes the meson cloud around the
bag (the hadronization phase).  Besides this, one needs to introduce
an additional surface action $S_{on} = \int \! dx \ \lagr _{FB} (x)
\d (\s)$, which describes the chiral quark-meson coupling
on the boundary surface $\s$ of the bag, while in the interior as
well as in the exterior of the bag the phases
are decoupled \cite{Rho,Ho}.

Although hybrid models show up a remarkable success in describing the
low-energy hadron physics \cite{Ho}, they suffer from the
principal problem of explicit breaking of the space-time symmetry.
Namely, the total action (1) contains explicit $\tt - $ and $\d - $
functions of space-time variables, which divide the bag space into
the interior and exterior parts.

In this paper we'll show that the
careful account of the relativistic covariance from the very
beginning leads to a number of conclusions concerning the
general treatment of hybrid models, that cannot be obtained in any
finite order of perturbative expansion  \cite{SC}.  The main tool is
to introduce the boundary between phases in the center-of-mass system
of the bag, rather than in the laboratory system as the standard
approach prescribes. Such a covariant division allows us to take
correctly into account any non-trivial classical object which has to
be included in the quantized theory.  The specific finite-difference
structure of the dynamics in the invariant c.m.s.  space \cite{S}
leads to surface terms and corresponding surface action in the form
of  contour integrals of fields, considered as analytical functions
of their complex spatial argument.  In turn, this circumstance
implies, that the boundary conditions on fields should be imposed on
these contours in the complex plane, rather than on the physical real
boundary between the phases. The most
important point here is that due to the requirement of analyticity of
fields these conditions are actually nothing else, but the condition
of analytical continuation between the phases. The direct consequence
of such analytical connection is the existence of chirally invariant
relation for the dynamics of fields in different phases, which is
consistent with bosonization procedure, but in more general situation
implies some strict predictions for the form of possible interaction
terms in both sides of the boundary.

\section{Lorentz-covariance in two-phase theories}

To avoid unnecessary complications we study the bag model in (1+1)D,
the generalization to higher dimensions can be performed directly.
The non-trivial classical objects which commute with any
unitary transformation of the Poincare group are the
bag surface, which is an explicit $c$-number function of the
space-time variables, and the external soliton-like solution. For a
consistent treatment of such objects in the quantum theory  we apply
the method of covariant group variables \cite{S} using the QFT
analogies of the relativistic Newton-Wigner center-of-mass coordinate
operators \cite{NW}.  These operators enable the bag with the surface
and meson cloud to move as a whole.  Let the (1+1)D Poincare
algebra be $$[H,P]=0, \ \ [L,P]=i\h H, \ \ [L,H]=i\h P \ , \eqno(2)$$
where $\h$ is the effective Plank constant, then the NW coordinate is
given by $$ q={1 \over 2} \{ H^{-1},L \}_{+} \ . \eqno(3)$$ By
definition the operator $q$ plays the role of the c.m.s.  coordinate
and so enters the fields arguments only in the combination $x-q(t)$
\cite{Sem}.  It is convenient to use the following substitution $$ x
\to \x(x,t) \ , \eqno(4)$$ where $$
\x(x,t)=Hx-Pt-L=H^{1/2}(x-q(t))H^{1/2} \ . \eqno(5)$$ Recall, that the
replacement (4) follows also from the direct evaluation of the
secular terms in the perturbation series around static classical
solution \cite{Sem}.  The dimensionless coordinate $\x(x,t)$ behaves
like a real covariant scalar field under the Poincare group
transformations, so any $c$-number function of $\x$ will also be a
covariant scalar.  This feature is the starting point for  the
Lorentz-covariant description of 2-phase field theories with
nontrivial boundaries for configurations, which are static in the
c.m.s. of the bag in the leading approximation.

Firstly we consider the surface terms for the scalar
field, described by the Lagrangian density
$$\lagr = \1/2 \pd_\m \vf \pd^\m \vf - U (\vf) \ , \eqno(6)$$
which is stationary in the c.m.s.
The corresponding action in terms of the invariant c.m.s.
variable $\vf(\x)$ is obtained by means of evaluating of matrix
element between the states with fixed energy-momentum and using the
expansion $1= \int \!  d\x \ |\x\> \<\x| $ \cite{SSC}.  One finds
$$S [\vf]  = \int \!  { d\x \over M} \ \left\{ {M^2 \over 4\h^2
} \( \[ \vf_+ - \vf \]^2  + \[ \vf_- - \vf \]^2 \) - U(\vf) \right\}\
, \eqno(7)$$ where $\vf_{\pm} = \vf (\x \pm i\h)$.  The variation of
this action with respect to the field $\vf$ gives $$ \d S = \int \!
{ d\x \over M} \ \d \vf \ \left\{ {M^2 \over \h^2} [2 \vf - \vf_+ -
\vf_- ] - U'(\vf) \right\} + $$ $$ + {M \over 2 \h^2} \ \int \!  d\x
\ \Big\{ \d \vf_+ [\vf_+ - \vf ]  + \d \vf_- [\vf_- - \vf ] +  \d \vf
[\vf_+ + \vf_- - 2 \vf ] \Big\}\ . \eqno(8)$$ The first term in (8)
is nothing else, but the correct field equation of motion $\dalam \vf
+ U{'}(\vf) =0$, written in terms of the invariant field, while the
second one is the surface term $\d S_{\s} [\vf]$. Consider now the
scalar field $\f$ defined in a region with a boundary, e.g.  the
segment $[-\infty , \x_0]$.  Due to the analyticity of $\f (z)$ in
the strip $|\hbox{Im} \ z | \leq \h$, what is shown in \cite{SSC} to
be crucial for the fields to have a smooth classical limit, we can
express the surface term as a contour integral over the imaginary
segment in the complex plane $$ \d S_\s [\f]= {M \over 2 \h^2 }
\ \Big( \int \limits_{\x_0}^{\x_0+i\h} \! dz \ \d \f \[ \f - \f_- \]+
\int \limits_{\x_0}^{\x_0-i\h} \! dz \ \d \f \[ \f - \f_+ \] \Big)\ .
\eqno(9)$$ In the classical limit $\h \to 0$ one finds $ \d S_\s [\f]
\to - M \f'(\x_0) \d \f $, i.e.  the surface term, which arises from
the variation of the kinetic term of the classical action $ S_{cl}=
-\1/2 \int \nolimits^{x_0} \!  dx \ \f'^2(x) $, provided $\x \to Mx$.

\section{Boundary conditions for the chiral bag}

Now we apply the results obtained above to the hybrid bag model.
The covariant fermion field is given by \cite{S}
$$ \p(x,t)=A \ \c (\x(x,t))\ ,  \eqno(10)$$ where $ A=
\exp(\a \tt /2) \ , \ \tt = \arcth P/H $ --- the total rapidity
operator, and  $\a=\g^0\g^1 $ is a Dirac matrix.  The additional
factor $A$ provides the transformation of the laboratory field
$\p(x,t)$ according to the spinor representation under the Lorentz
transformations.  Therefore $\c (\x)$ is the invariant fermion field
in the c.m.s. of the bag.  The variation of the kinetic part of the
fermionic action $ i \bp \hat{\pd} \p $ in terms of the invariant
variables $\c(\x)$ with respect to $\c$, provided by the validity of
the Dirac equation of motion inside the bag, which is now the segment
$[-\x_0 , \x_0]$, yields the fermionic surface term, which can also
be presented as a sum of contour integrals over the boundary segments
\cite{SC}.  Therefore the surface action, which has to be included
in the total action of boson and fermion field, should be taken in
the form $S_{on} = S_{on}(\x_0)+S_{on}(-\x_0)$ (taking into
account the chiral invariance, the renormalizability and the correct
limit $\h \to 0$), where $$ S_{on}(\x)= { 1 \over 2 i\h} \Big\{ \int
\limits_{\x}^{\x+i\h} \!  dz \ e^{-i\l(\f+\f_-)/2} \tc_R \c_L -  \int
\limits_{\x}^{\x-i\h} \!   dz \ e^{i\l(\f+\f_+)/2} \tc_L \c_R \Big\}\
, \eqno(11)$$ where  $\f_{\pm}=\f(z \pm i\h) \ , \ \g_5=\g^1\g^0 \ ,
\ \c_{R,L}$ are the right- and left-hand chiral fermions, $ \g_5
\c_{R,L}= \pm \c_{R,L} \ $, $\l$ is the constant of the chiral
coupling and the factor $1/2i\h$ provides the correct classical
limit.  The tilded fermion fields $\tc$ denote the analytical
continuation of Dirac conjugated fields $\bar \c$ from the real axis
in the complex $\x-$space.

The variation of the total action of the bag (1), which
contains the fermion, the boson and the fermion-boson surface parts,
provided by the validity of the equations of motion in the interior
as well as exterior sector, yields the following boundary conditions
$$ (\g^0-\g^1) \c_- =i e^{-i\l(\f+\f_-)/2} (1-\g_5) \c
\eqno(12)$$ for fermions, and $$ 2M (\f - \f_-) = \h\l \[
e^{i\l(\f+\f_-)/2} \( \tc_L \c_R \)_- - e^{-i\l(\f+\f_-)/2} \tc_R
\c_L \]  \eqno(13)$$ for bosons on the segment $z \in
[\x_0,\x_0+i\h]$.  The conditions on the segment $z \in
[\x_0,\x_0-i\h]$ can be obtained via the Schwartz reflection
\cite{SC}.  In order to include the  contribution of the sea quarks
as well as valence ones, one must take the vacuum expectations of
bilinear fermionic forms in the r.h.s. of (13).

The eqs. (12,13) are the chirally invariant
nonlocal boundary conditions which connect the fields from
inside and outside the bag on the complex segments $[\x_0, \x_0 \pm
i\h]$.  Due to the uniqueness of the analytical
continuation the relations (12,13), which were originally determined
on the finite segments $[\x_0, \x_0\pm i\h]$ in the complex
$\x$-plane, should now be extended to the whole area of
the field analyticity,  what imposes strong constraints on the
possible dynamics of fields inside as well as outside the bag.


To demonstrate the role of these constraints, let the boson
self-interaction be $U(\f)$ and $\f(\x)$
be the odd topological soliton, which to the leading order in $\h$
obeys the equation $-\f'' + U'(\f)=0 $ ($x=\x/M$).  The generic
fermion dynamics is determined by the potential $V(A,B)$, where
$A=\bc \c, \ B=-i\bc \g_5 \c$.  Then we find from the boundary
conditions (12,13) the following connection between fermionic and
bosonic potential for the given energy $E$ of a fermionic mode inside
the bag $$ V(A,B)= U \( {1 \over \l} \ \arctan { A \over B} \) + E
\sqrt{C} - {\l^2 \over 8} C \ , \eqno(14)$$ where $C=A^2+B^2$ is
actually the Thirring interaction of fermionic currents $(\bc \g^\m
\c)^{2}$.  Note, that this non-linear chirally 
invariant connection cannot be obtained in any finite order of the 
perturbative expansion in powers of coupling constant $\l$.  
Nevertheless, (14) reflects the situation correctly, what can be 
easily checked via (1+1)D bosonization \cite{Col}, what yields $ 
V(\bc, \c) \to U(\f) + \hbox{Const} $.  In other words, in terms of 
bosonization the dynamics of fermions described by (14) and the 
bosonic self-interaction $U(\f)$ are equivalent.

\section{Fermion charge of the bag}

Now we show the role of such analytical connection between the
exterior and interior regions calculating the total fermion number of
the bag in terms of the invariant fields.  It is well known that in a
hybrid chiral bag the total fermion number is shared between the
interior fermions and the topological soliton of the meson theory,
where it corresponds to the topological charge \cite{Rho}. In the
standard approach for the bag with a static boundary this fact arises
as the result of the special regularization of the secondary
quantized fermionic current $j^{\m}(x)=\bp (x)\g^\m \p(x) $ and the
local boundary conditions \cite{Rho}.  The rate of the charge leakage
through the surface appears to be proportional to the time derivative
of the scalar field $\f$ on the surface.   Let us consider now the
total fermion number in terms of invariant fields. The vector current
$j^\m$ reads $$j^{\m} = {1 \over 2M} \ \big[ \bc (\g^\m-\e^{\m\n}
\g_\n ) \c_+ K_+ + \bc (\g^\m + \e^{\m\n} \g_\n ) \c_- K_-
\big] \ , \eqno(15)$$ where $K_{\pm} = H \pm P$.  Note, that this
current is a well-defined operator due to the imaginary shifts of the
arguments.  The calculation of the total
fermion number $Q_F= \int \!  dx \ j^0 (x) $ in terms of invariant
fields is described in detail in \cite{SC}.  The final result is
$$Q_F[\c]= \int \!  {d\x \over 2M} \ \[ \bc (\g^0+\g^1) \c_+ +\bc
(\g^0-\g^1) \c_- \] \ . \eqno(16)$$ As in (13), the average of the
bilinear fermionic forms over corresponding vacuum state is implied.

For a  2-phase system under consideration the integration in (18) has
to be performed over the fermionic region, i.  e.  over the segment
$[-\x_0,\x_0]$.
Using the boundary conditions (13), analytically
continued to the interior sector of the bag, we immediately find  the
total fermion number in terms of the boson field only $$ Q_F= {1
\over i\h \l} \ \int \limits_{-\x_0}^{+\x_0} \! d\x \ (\f_+-\f_-)\ .
\eqno(17)$$ The limit $\h \to 0$ gives $$ Q_F= {1 \over i\h \l} \
\int \limits_{-\x_0}^{+\x_0} \! d\x \ 2i\h \ \pd_\x \f = { 2 \over \l
} \ \[ \f(\x_0)-\f(-\x_0)\] \ , \eqno(18)$$ what coincides with the
expression for the vector anomaly of the standard approach, obtained
via special regularization of the fermionic current \cite{Rho}.  Let
us emphasize,  that in this approach the vector anomaly appears to be
the purely quantum mechanical effect caused by the finite-difference
structure of the dynamics of invariant fields and the boundary
conditions, rather than by second quantization.


To conclude, we have studied the generic structure of surface terms
and boundary conditions for the hybrid chiral quark bag models in
(1+1)D within a covariant treatment of the center-of-mass motion. It
is shown that the correct account of the relativistic covariance for
the field theories in the closed space leads to  rather strong
constraints on the dynamics compared to the standard
approach.  These constraints are trivial in the
case of the exactly equivalent (in terms of bosonization) theories,
such as the sin-Gordon and Thirring models, due to validity of the
Cheshire Cat principle \cite{Rho}, but in a more general situation
might lead to strong limitations on the possible field dynamics in
2-phase field theories with a boundary.


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\bibitem{Ho} H. Hosaka and O. Toki, Phys. Reports 277 (1996) 65.
\bibitem{Za} I. Zahed, G. E. Brown, Phys. Reports 142 (1986) 1.
\bibitem{SC} K. Sveshnikov and I. Cherednikov, Phys. At. Nucl. N4
(1999), in print.
\bibitem{S} K.Sveshnikov, Phys. Lett. A136 (1989) 1.
\bibitem{NW} T.Newton and E.Wigner, Rev.  Mod.  Phys.  21 (1949) 400.
\bibitem{Sem} G.Semenoff, H.Matsumoto and H.Umezawa, J.  Math.  Phys.
22 (1981) 2208.
\bibitem{SSC} K.Sveshnikov, P.Silaev and I.Cherednikov, Mod.
Phys.Lett.  A12 (1997) 465.
\bibitem{Col} S. Coleman, Phys. Rev. D11 (1975) 2088; S.
Mandelstam, ibid. 3026
\end{thebibliography}


\end{document}

