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\pagestyle{myheadings}  %% {Authors}{Title}
%\markboth{\small \em \quad Yu.A. Simonov and J.A. Tjon\hfill}{\hfill\small \em World-line representation in QCD\quad}

\begin{document}

\title{THE FEYNMAN-SCHWINGER  (WORLD-LINE)
%\\[1mm]
REPRESENTATION IN PERTURBATIVE QCD}
\author{Yu.A. Simonov$^{1}$ and J.A. Tjon$^{2}$\\
$^1$Jefferson Laboratory, Newport News, USA\\ and\\ 
Institute of Theoretical and Experimental Physics,
Moscow, Russia\\$^2$Institute for Theoretical Physics,University of
Utrecht,\\and\\KVI, University of Groningen,The Netherlands}

\maketitle
\abstracts{
   The proper time path integral representation is derived explicitly
   for an arbitrary $n$-point amplitude in QCD. In the standard
   perturbation theory the formalism allows to sum up the leading
   subseries, e.g. yielding double-logarithm Sudakov asymptotics for
   form factors. Correspondence with the standard perturbation theory is
   established and connection to the Bern-Kosower-Strassler method is
   illustrated.
}


\vspace*{0.8cm}

%\tableofcontents

\newpage 
\section*{Dedication.}
\addcontentsline{toc}{section} {\numberline{}Dedication}

During our studies in field theory we many times returned
to the set of methods and ideas, which was developed by
Misha Marinov. Here also belongs the path integral method in its 
different forms, with spin included in the quantum-mechanical
world-line version, which is described below. The excellent review
written by Misha on the subject which appeared in Physics 
Reports,\cite{0} was a most-read source on the subject at that time. 
  It is a great pleasure and honour for us to dedicate the paper to
     his memory.

\section{Introduction.}

   The present stage of development of field theory in general and of
   QCD in particular requires the exploiting of nonperturbative
   methods in addition to summing up perturbative series. This calls
   for specific methods where dependence on vacuum fields can be made
   simple and explicit. A good example is provided by the so-called
   Fock-Feynman-Schwinger Representation (FSR) based on the
   Fock-Schwinger proper time  and Feynman path integral
   formalism.\cite{1,2} For QED asymptotic estimates the FSR was
   exploited in Ref. 4.
%~\cite{3}. 
Later on this formalism was rederived 
   in Ref. 5
%~\cite{4}  
for scalar quarks in QCD and used in the framework of
   the stochastic background method in.\cite{5}

   More recently some modification of the FSR was suggested in Refs. 7,8.
%~\cite{6,7}.
    The one-loop perturbative amplitudes are especially
   convenient for FSR. These amplitudes were extensively studied in Refs. 9-11.
%~\cite{8}$^{\!-\,}$\cite{10}.
   Meanwhile the first extension of FSR to nonzero temperature field
   theory was done in.\cite{11,12} This formed
   the basis of a systematic study of the role of
   nonperturbative (NP) configurations in the temperature phase
   transitions.\cite{11,13}

   One of the most important advantages of the FSR is that it allows to
   reduce  physical amplitudes to weighted integrals of 
   averaged Wilson loops. Thus the fields (both perturbative and NP)
   enter only through Wilson loops. For the latter case one can apply
 the cluster expansion method,\cite{14} which allows to sum up a
   series of approximations directly in the exponent.
   As a result one can avoid the summation of Feynman diagrams to get
   the asymptotics of form factors.\cite{15}
   The role of FSR in the treatment of NP effects is more crucial. In
   this case one  can develop a powerful method of background
   perturbation theory\cite{16} treating the NP fields as a 
   background.\cite{17}

   In the present paper the main focus will be on perturbative QCD,
   with the aim of establishing the correspondence between the standard
   perturbative expansion and FSR based expansion, stressing the point that
   FSR allows to make exponentiation in a very simple way. Finally also 
   the relation of FSR to the popular Bern-Kosower-Strassler method 
   is discussed.
   

   \section{General form of FSR in QCD.}

Let us consider a scalar particle (e.g. Higgs boson) interacting
with the nonabelian vector potential, where the Euclidean Lagrangian
is given by
\be
L_\varphi = \frac12 |D_\mu \varphi|^2 + \frac12 m^2 |\varphi|^2
\equiv \frac12 |(\partial_\mu-ig
A_\mu) \varphi |^2 + \frac12 m^2 |\varphi|^2,
\label{1}
\ee
Using the Fock--Schwinger proper time representation 
the two-point Green's function of $\varphi$ can be written in the
quenched approximation as
\be
G(x,y) = (m^2-D^2_\mu)^{-1}_{xy} = \langle x | P \int^\infty_0 ds
e^{-s(m^2-D^2_\mu)}|y\rangle.
\label{2}
\ee
To obtain the FSR for $G$ a second step is needed.
As in Ref. 2
%~\cite{1} 
the matrix element in Eq. (\ref{2})
can be rewritten in the form of a path integral, treating $s$ as the
ordering parameter. Note the difference of the  integral (\ref{2})
from the case of the Abelian QED treated in Refs. 2,4,9:
%~\cite{1,3,8}:
$A_\mu$ in our case is the matrix operator $A_\mu(x)=
A_\mu^a(x)T^a$. It does not commute for different $x$. Hence
the ordering operator $P$ in Eq. (\ref{2}). The precise meaning of $P$
becomes more clear in the final form of a path integral
\be
G(x,y) =\int^\infty_0\!\! ds (Dz)_{xy}\, e^{-K} P \exp\left(ig \!\!\int^x_y
\!\!A_\mu (z) dz_\mu\right),
\label{3}
\ee
where $K= m^2s+ \frac14 \int^s_0 d\tau (\frac{dz_\mu}{d\tau} )^2$.
In Eq. (\ref{3}) the functional integral can be written as
\be 
(Dz)_{xy} \simeq \lim_{N\to \infty} \prod^N_{n=1}\!\int\!\!
\frac{d^4 z(n)}{(4\pi\varepsilon)^2}\! \int\!\! \frac{d^4p}{(2\pi)^4}
e^{ip \left(\sum^N_{n=1}\! z(n)-(x-y)\right)}
\label{4}
\ee
with $N\varepsilon =s$. 
The last integral in Eq. (\ref{4}) ensures that the path $z_\mu(\tau),
0\leq \tau\leq s$, starts at $z_\mu(0) = y_\mu$ and ends at
$z_\mu(s) =x_\mu$. The form of Eq. (\ref{3}) is the same as in the case
of QED except for the ordering operator $P$ which provides a
precise meaning to the integral of the noncommuting matrices
   $A_{\mu_1}(z_1), A_{\mu_2}(z_2)$ etc. In the case of QCD the forms
   (\ref{3}) and (\ref{4}) were introduced in Refs. 5,6.
%\cite{4,5}

The FSR, corresponding to a description in terms of particle
dynamics is equivalent to field theory, when all the
vacuum polarisation contributions are also included,\cite{4}
i.e.
\begin{eqnarray}
&&\sum_{N=0}^{\infty} \frac{1}{N!} \prod_{i=1}^{N} \int\! \frac{ds_i}{s_i}
\!\int\! (Dz_i)_{xx} \exp(-K)\; P\exp\left(ig\! \!\int\limits_y^x
\!\!A_{\mu}(z) dz_{\mu}\right) 
\nonumber\\[1mm]
&&= \int\! D\varphi \exp\left(\!-\!\!\int\! \! d^4 x L_{\varphi}(x)\right ).
\label{bss}
\end{eqnarray}
Both sides are equal to vacuum-vacuum transition amplitude in the presence
of the external nonabelian vector field and hence to each other.
For practical calculations proper regularization of the above equation has
to be done.

The field $A_{\mu}$ in  Eq. (\ref{1}) can be considered
as a classical external field or as a quantum one.
In the latter case the Green's functions $\lll A .. A \rrr$
induce nonlocal current-current interaction terms in the l.h.s. of
Eq. (\ref{bss}).
Such terms can also be generated by the presence of a $\varphi$-field potential,
$V(|\varphi|)$ in the r.h.s. of   Eq. (\ref{bss}). 

The advantage of FSR in this case follows from the very
clear space-time picture of the corresponding dynamics in terms of
particle trajectories.
This is especially important if the currents can be treated
as classical or static (for example, in the heavy quark case).
   The mentioned remark on usefulness of the FSR (\ref{3}) becomes 
clear when one
   considers the physical amplitude, e.g. the Green's function of the
   white state $tr(\varphi^+(x) \varphi(x))$ or its nonlocal version
   $tr[\varphi^+ (x) \Phi(x,y) \varphi(y)]$, where $\Phi(x,y)$ -- to be
   widely used in what follows -- is the parallel transporter along some
   arbitrary contour $C(x,y)$
   \be
   \Phi(x,y) =P\exp\left(ig\!\! \int^x_y \!\!A_\mu(z) dz_\mu\right).
   \label{5}
   \ee
   One has by standard rules
\begin{eqnarray}
 &&G_\varphi (x,y) =\left\langle
    tr\left[\varphi^+(x) \varphi(x)\right]\,
    tr \left[\varphi^+(y)
\varphi(y)\right]\right\rangle_A\nonumber\\[1mm]
 &&
     =\int^\infty_0\!\!ds_1\! \int^\infty_0\!\! ds_2
   (Dz)_{xy}(Dz')_{xy}\, e^{-K-K'} \left\langle W\right\rangle_A + \ldots
   \label{6}
\end{eqnarray}
   where dots stand for the disconnected part,
   $\langle G_\varphi (x,x) G_\varphi (y,y)\rangle_A$. We 
have used the fact that the propagator for the
   charge-conjugated field $\varphi^+$ is proportional to
$\Phi^{\dagger}(x,y) = \Phi(y,x)$.
%\be
%   \Phi_C(x,y) =P\exp (ig \int^x_y A_\mu^{(C)} (z) dz_\mu), ~~
%   A_\mu^{(C)}=-A_\mu^{(T)}
%   \label{7}
%   \ee
   Therefore the ordering $P$ must be inverted, $\Phi^{\dagger}(x,y)
   =P \exp (ig \int^y_x A_\mu (z) dz_\mu)$.
    Thus all dependence on $ A_\mu$ in $G_\varphi$ is reduced to the
     Wilson loop average
     \be
     \left\langle W\right\rangle_A=\left\langle tr \,P_C \exp ig \!\!\int_C
 \!\!  A_\mu (z) dz_\mu\right\rangle_A.
     \label{8}
     \ee
      Here $P_C$ is the ordering  around the closed loop
      $C$ passing through the points x and y, the loop
      being made of the paths $z_\mu(\tau)a$,
      $z'_\mu(\tau')$ and to be integrated over.

The FSR can also be used to describe the quark and gluon propagation.
Similar to the QED case, 
      the fermion (quark) Green's function in the presence of an
      Euclidean external gluonic field can be written as
\begin{eqnarray}
     \label{9}
 &&     G_q(x,y)= \langle \psi(x) \bar \psi(y)\rangle_q=\langle x
      |(m_q+\hat D)^{-1}|y\rangle 
\nonumber
\\[1mm]
&&     = \langle x|(m_q-\hat D)(m^2_q-\hat D^2)^{-1}|y\rangle
\nonumber
\\
&&  =  (m_q-\hat D)\!\int^\infty_0\!\! ds (Dz)_{xy} e^{-K} \Phi_\sigma(x,y)\,,
\end{eqnarray}
      where $\Phi_\sigma$ is the same as was introduced in Ref. 2
%~\cite{1}
      \be
      \Phi_\sigma (x,y) = P_A\exp\left(ig \int^x_y A_\mu
      dz_\mu\right)\,P_F 
      \exp \left(g\int^s_0 d\tau \sigma_{\mu\nu} F_{\mu\nu}\right)
      \label{10}
      \ee
      and $\sigma_{\mu\nu}=\frac{1}{4i}
      (\gamma_\mu\gamma_\nu-\gamma_\nu\gamma_\mu)$, while $K$ and
      ($Dz)_{xy}$ are defined in Eqs. (\ref{3}) and (\ref{4}). Note that
      operators $P_A, P_F$ in Eq. (10) preserve the proper ordering of
      matrices $A_\mu$ and $\sigma_{\mu\nu}  F_{\mu\nu}$
      respectively. Explicit examples are considered below.

      Finally we turn to the case of FSR for the valence gluon
      propagating in the background nonabelian field. 
Here we only quote the result for the gluon Green's function in the background
      Feynman gauge.\cite{5,17} We have
      \be
      G_{\mu\nu} (x,y) =
      \langle x|(D^2_\lambda \delta_{\mu\nu}-
      2ig F_{\mu\nu})^{-1}
      |y\rangle
      \label{11}
      \ee
      Proceeding in the same way as for quarks, one obtains the FSR
      for the gluon Green's function
      \be
      G_{\mu\nu} (x,y) =
      \int^{\infty}_0 ds (Dz)_{xy} e^{-K_0}\Phi_{\mu\nu}(x,y),
      \label{12}
      \ee
      where we have defined
\begin{eqnarray}
 K_0&\!\!=&\!\!\frac14 \int^\infty_0 \left
(\frac{dz_\mu}{d\tau}\right)^2
      d\tau, \nonumber\\[1mm]
\Phi_{\mu\nu} (x,y) &\!\!\!\!=&\!\!\!\!\left [P_A\exp
\left(ig\!\!\int^x_y\!\! A_\lambda
      dz_\lambda\right)P_F
\exp \left(2g\!\!\int^s_0\!\! d\tau F_{\sigma\rho}
      (z(\tau))\right)\right]_{\mu\nu}\! .
      \label{13}
\end{eqnarray}
      Now in the same way as is done above for scalars in
Eq.  (\ref{6}), one may consider a Green's function,
corresponding to the physical
      transition amplitude from a white state of $q_1, \bar q_2$ to
      another white state consisting of $q_3,\bar q_4$. It is given by
      \be
      G^\Gamma_{q\bar q} (x,y) =
      \langle G_{q} (x,y)
       \Gamma G_{\bar q} (x,y)\Gamma-
       G_{ q} (x,x)
              \Gamma G_{\bar q} (y,y)\Gamma\rangle_A,
              \label{14}
              \ee
where $\Gamma$ describes the interaction between the $q,\bar{q}$ pair in the
meson. The first term on the r.h.s. of Eq. (\ref{14}) can be
              reduced to the same form as in Eq. (\ref{6}) but with the
              Wilson loop containing ordered insertions of the
              operators $\sigma_{\mu\nu} F_{\mu\nu}$ (cf. Eq.~(\ref{10})).



\section{Perturbation theory in the framework  of FSR.
Identities and partial summation.}

In this section we  discuss in detail how the usual results of
perturbation theory follow from FSR. It  is useful to establish such
a general  connection between the perturbation series (Feynman
diagram technique) and FSR.
At the same time the FSR
presents a unique possibility to sum up Feynman diagrams in a very
simple way, where the final result of the summation is written in an
exponentiated way.\cite{15,17} This method will be discussed  in
the next section.

Consider the FSR for the  quark Green's function. According to
(\ref{9}), the 2-nd order of  perturbative expansion of
Eq. (\ref{10}) can be written as
%\begin{eqnarray}
$$G_q(x,y)= (m_q-\hat D) \!\int^\infty_0\!\!  ds 
\!\int^\infty_0 \!\!  d\tau_1\!
\int^{\infty_1}_0\!\!  d\tau_2\, e^{-K} (Dz)_{xu} d^4u (Dz)_{uv}d^4v(Dz)_{vy}
$$
%\nonumber\\[1mm]
%&&\!\! 
\be
\times
 \left(ig A_\mu(u)\dot u_\mu+g\sigma_{\mu\nu} F_{\mu\nu} (u)\right)
\left(ig A_\nu(v)\dot v_\nu+ g\sigma_{\lambda\sigma}
F_{\lambda\sigma} (v)\right),
\label{15}
%\end{eqnarray}
\ee
 where we have used the identities
%\begin{eqnarray}
\be
%&&
(Dz)_{xy} = (Dz)_{xu(\tau_1)} d^4u(\tau_1)(Dz)_{u(\tau_1)v(\tau_2)} 
d^4 v(\tau_2) (Dz)_{v(\tau_2)y},
 \label{16}\\[1mm]
\ee
%&&
\be 
\int^\infty_0\!\!\!\!\!\!ds\!\!\int^s_0 \!\!\!\!d\tau_1\!\!\int^{\tau_1}_0
\!\!\!\!\!\!d\tau_2
\,f(s,\tau_1,\tau_2)=
\int^\infty_0\!\!\!\!\!\!ds\!\!\int^\infty_0\!\!\!\!
\!\!d\tau_1\!\!\int^\infty_0
\!\!\!\!\!\!d\tau_2
\,f(s+\tau_1+ \tau_2, \tau_1+\tau_2,\tau_2).
 \label{17}
%\end{eqnarray}
\ee
One can also expand only in the color magnetic moment interaction ($\sigma
F$). This is useful when the spin-dependent interaction can be
treated perturbatively, as it is in most cases for mesons and
baryons (exclusions are Goldstone bosons and nucleons,
where spin interaction is very important and interconnected with
chiral dynamics). In this case one obtains
to the second order in ($\sigma F$)
%\begin{eqnarray}
$$
G_q^{(2)}(x,y) = i(m_q-\hat
D)\!\!\int^\infty_0\!\!ds\!\!\int^\infty_0\!\! d\tau_1\!\!\int^\infty_0
\!\!d\tau_2\, e^{-m^2_q(s+\tau_1+\tau_2)-K_0-K_1-K_2} $$
%\nonumber\\[1mm]
%&\times&   
\be
(Dz)_{xu}\Phi (x,u) g(\sigma F(u)) d^4 u(Dz)_{uv}
\Phi(u,v) g(\sigma F(v)) d^4v(Dz)_{vy}. \label{18} 
%\end{eqnarray}
\ee
In another way it can be written as
$$ G^{(2)}_q(x,y) =i(m_q-\hat
D) (m^2_q-D^2_\mu)^{-1}_{xu} d^4u~g(\sigma
F(u))(m^2_q-D^2_\mu)^{-1}_{uv} d^4v $$ 
\be 
\times g(\sigma
F(v)) (m^2_q-D^2_\mu)^{-1}_{vy}. \label{19}
\ee
 Here $( m^2_q-D^2_\mu)^{-1}$ is the Green's function of a scalar
 quark in the external gluonic field $A_\mu$.
 This type of expansion is useful also for the study of small-$x$
 behavior of static potential, since
 the correlator $\lan\sigma F(u) \sigma F(v)\ran$ plays an
 important role there.

 However, in establishing the general connection between perturbative
 expansion for Green's functions in FSR and expansions of
 exponential $\Phi_\sigma$ in Eq. (\ref{10}), one encounters a
 technical difficulty since the coupling constant $g$ enters in three
 different ways in FSR:
%\begin{enumerate}
%\item 

1. in the factor $(m_q-\hat D)$ in front of the integral in Eq. (\ref{9})

%\item 
2. in the parallel transporter (the first exponential in Eq. (\ref{10}))
%\item

3. in the  exponential of $g(\sigma F)$.
%\end{enumerate}

Therefore it is useful to compare the two expansions in the operator form, the
standard one
  $$
  (m+\hat D)^{-1}= (m+\hat \partial - ig \hat A)^{-1}=(m+\hat
  \partial)^{-1}+ (m+\hat \partial)^{-1} ig \hat A (m+\hat
  \partial)^{-1}+
  $$
  \be
  + (m+\hat \partial)^{-1} ig \hat A(m+\hat \partial)^{-1} ig \hat
  A(m+\hat \partial )^{-1}+...
  \label{20}
  \ee
  and the FSR 
  \be
  (m+\hat D)^{-1} = (m-\hat D) (m^2-\partial^2)^{-1}
  \sum^\infty_{n=o}(\delta(m^2-\partial^2)^{-1})^n,
  \label{21}
  \ee
  where we have introduced 
  \be
  \delta=-ig (\hat A\hat \partial+\hat \partial \hat A) - g^2 \hat
  A^2\equiv \hat D^2-\partial^2.
  \label{22}
  \ee
To see how the expansion (\ref{21}) works, using $\hat D=\hat \partial -ig \hat A$
Eq. (\ref{21}) becomes
$$
  (m+\hat D)^{-1}=[(m+\hat \partial)^{-1} +ig \hat A
  (m^2-\partial^2)^{-1}]\sum^\infty_{n=0}
  [\delta(m^2-\partial^2)^{-1}]^n
  $$
Separating out the first term we may rewrite this as
\be
 (m+\hat D)^{-1}=(m+\hat \partial)^{-1} + (m+\hat \partial)^{-1} ig \hat
  A(m-\hat D) (m^2-\partial^2)^{-1}\sum^\infty_{n=0}
     [\delta(m^2-\partial^2)^{-1}]^n
\label{23}
\ee

%  \be
%  =(m+\hat \partial)^{-1}+ (m+\hat\partial)^{-1} ig \hat A(m+\hat D)^{-1}.
%     \label{23}
%     \ee
The last three factors in Eq. (\ref{23}) are the
same as occurring in Eq. (\ref{21}). As a consequence the formal iteration of 
the resulting equation for the Greens' function
reproduces the same series as in Eq. (\ref{20}), showing the equivalence of the
two expansions.

It is important to note that each term in the expansion in powers of
$\delta$, after
 transforming the operator form of Eq. (\ref{21}) into the integral
 form of FSR, becomes an expansion of the  exponential
 $\Phi_\sigma$ in Eq. (\ref{10}) in powers of $g$.
 The second order term
 of this expansion was written down before in Eq. (\ref{15}).

It is our purpose now to establish the connection
between the expansion (\ref{21}), (\ref{23}) and the
expansion of the exponential $\Phi_\sigma$ in 
Eq. (\ref{10}) in the quark propagator (\ref{9}).
One can start with term linear in $\hat A$ and write
(for the Abelian case see Appendix B of Ref. 19)
%~\cite{18})
 \be 
G_q^{(1)} = ig \int G^{(0)}_q(x,z(\tau_1)) d^4
z\frac{\xi_\mu(n)}{\varepsilon}
 \bar A_\mu(\tau_1) G_q^{(0)}(z(\tau_1), y),
\label{24}
\ee
where the notation is clear from the general representation of $G_q$,
given by Eq. (\ref{9})
\be
G_q(x,y)
 =\int^\infty_0 ds e^{-sm^2_q} \prod^N_{n=1}
\frac{d^4\xi(n)}{(4\pi\varepsilon)^2}
\exp\left[-\sum^N_{n=1} \frac{\xi^2(n)}{4\varepsilon} \right ]
\Phi_\sigma (\bar A,\xi) 
\label{25} 
\ee 
with 
$\xi (n) = z(n) - z(n-1),~~ \bar A_\mu (n) =\frac12[A_\mu(z(n))+A_\mu(z(n-1))]$ 
and 
\be 
\Phi_\sigma (\bar A, \xi) =P\exp \{ ig \sum^N_{n=1} \bar
A_\mu(n) \xi_\mu(n) + g\sum^n_{n=1} \sigma_{\mu\nu}
F_{\mu\nu}(z(n))\varepsilon \}.
\label{26} 
\ee 
Representing $\xi(n)$ in Eq. (\ref{24}) as $\frac12(\xi_\mu(L)+\xi_\mu(R))$,
where
 $\xi_\mu(L)$ refers to the integral over $\xi_\mu$ in
 $G_q^{(0)}$
to the left of $\xi_\mu$ in Eq. (\ref{24}) and $\xi_\mu(R)$ to the
integral in $G_q^{(0)}$ standing to the right of $\xi_\mu$, we
obtain 
\be 
\int\xi_\mu(n)\frac{d^4\xi(n)}{(4\pi\varepsilon)^2}
e^{ip\xi-\frac{\xi^2}{4\varepsilon}}=-i\frac{\partial}{\partial
p_\mu}  e^{-ip^2\varepsilon}=2 ip_\mu\varepsilon
e^{-p^2\varepsilon}. \label{27}
\ee
Thus Eq. (\ref{24}) in momentum space becomes 
\be
G_q^{(1)}=-gG^{(0)}_q(q)\lan q|p_\mu A_\mu+A_\mu p_\mu|q'\ran
G^{(0)}_q(q') 
\label{28} 
\ee
In a similar way the second order term from the  coinciding
arguments yields
\be 
G^{(2)}_q(coinc) = -g^2\int G^{(0)}_q(x,z) A^2_\mu(z) d^4 z
G_q^{(0)}(z,y). 
\label{29} 
\ee 
Finally, the first order expansion
of the term $\sigma_{\mu\nu} F_{\mu\nu}$ in Eq. (\ref{10}) yields the
remaining missing component of the combination $\delta$, Eq. (\ref{22}), 
which can be  rewritten as 
\be 
\delta=-ig (A_\mu\partial_\mu+\partial_\mu
A_\mu)-g^2 A^2_\mu+g\sigma_{\mu\nu} F_{\mu\nu}. 
\label{30} 
\ee
Hence the second term in the expansion (\ref{21}) 
\be 
(m+\hat D)^{-1}= (m-\hat D) (m^2-\partial^2)^{-1}+(m-\hat
D)(m^2-\partial^2)^{-1}\delta(m^2-\partial^2)^{-1}+... 
\label{31}
\ee 
is exactly reproduced by the expansion of the FSR (\ref{9}),
where in the first exponential $ \Phi_\sigma$ in Eq. (\ref{9}) one
keeps terms of the first and second order, $O(gA_\mu)$ and
$O((gA_\mu)^2)$, while in the second exponential one keeps only
the first order term $O(g\sigma_{\mu\nu}F_{\mu\nu})$. It is easy
to see that this rule can be generalized to higher orders of
the expansion in $\delta$ in Eq. (\ref{21}) as well.

\section{Summing up leading perturbative contributions in FSR.
Sudakov asymptotics.}

Consider now the n-point Green's function with external momenta $p_i$
at the i-th vertex 
 \be
G(p_1,...p_n)=<J_1(p_1)...J_n(p_n)>, J_i(x)=\psi(x)^{\dag}\Gamma_i\psi(x)
\label{32} 
\ee 
Insertion of Eq. (\ref{9}) into Eq. (\ref{32}) for the
one--fermion loop yields 
%\be 
\begin{eqnarray}
G(p_1,...p_n)&&=<tr\prod^n_{i=1}\Gamma_i(m_i-\hat D_i)\int^\infty_0
ds_i
\nonumber
\\
&&\times
(Dz^{(i)})_{x^{(i)},x^{(i-1)}}
e^{-K_i}\Phi^{(i)}_\sigma
e^{ip^{(i)} x^{(i)}}dx^{(i)}>_A 
\label{33}
\end{eqnarray}
% \ee
 We shall disregard in what
follows the factors $\Gamma_i(m_i-\hat D_i)$ since we shall be
interested only in the exponentiated contributions. Performing the
$dx^{(i)}$ integrals, one obtains
 $$ G\to \bar
G_n\delta(\sum^n_{i=1}p_i)(2\pi)^4,
$$
 where
 \be
  \bar
G_n=\int\frac{d^4q}{(2\pi)^4}\prod^n_{i=1}
ds_i\prod^N_{k=1}\frac{d\xi^{(i)}(k)}{(4\pi\varepsilon)^2}
e^{iq^{(i)}\sum_k\xi^{(i)}(k)}e^{-K_i}< \Gamma_i(m_i-\hat
D_i)W_\sigma>
\label{34}
\ee
with
$$ <W_\sigma>=<\prod^n_{i=1}\Phi_\sigma^{(i)}>_A,
$$
where $q^{(i)}$ is the momemtum of the fermion loop going from the
i-th to the (i+1)-th vertex.
The integral $d^4q$ denotes the integral over one of $q^{(i)}$, all
others being expressed through it and all $p_i$.

We note that $<W_\sigma>$ is a gauge invariant quantity summing
all the perturbative exchanges  inside the fixed Wilson contour,
defined by the set $\{\xi^{(i)}(k)\}$. In addition to the usual
Wilson (charge) vertices, there are also magnetic moment vertices
$\sigma F$, hence the notation $<W_\sigma>$.

We concentrate now on the contribution of the $A_\mu$ field in
(\ref{34}), yielding the dominant contribution in the asymptotics
(the reader is referred  for the discussion of the $\sigma
F$ term to the Appendix of Ref. 16.
%~\cite{15}).

The  crucial step for what follows is the use of the cluster
expansion method,\cite{14} which yields for $<W_\sigma>\to
<W>$ \be <W>\equiv \exp \{\sum^\infty_{r=1}\frac{(ig)^r}{r!}
\sum_{k_i}\xi_{\mu_1}(k_1)\xi_{\mu_2} (k_2) ... \xi_{\mu_r}
(k_r)\ll A_{\mu_1} (z_{k_1})... A_{\mu_r} (z_{k_r})\gg\}
\label{35} \ee

Here double brackets denote cumulants.\cite{14} The lowest order
contribution (in the exponent) can be expressed through the photon (gluon)
propagator. In the Feynman gauge it is (the gauge is irrelevant
since $<W>$ is gauge invariant) 
\be 
<A_{\mu}(z)A_\nu(z')>=
\frac{\delta_{\mu\nu}C_2(f)\hat 1}{4\pi^2 (z-z')^2}. 
\label{36}
\ee
 Here
$C_2(f)$ is the quadratic Casimir operator for the fundamental
representation, $\hat 1$ is the unit color matrix. For QED one
should replace $C_2\hat 1\to 1$.

  Eq. (\ref{35}) represents the perturbative sum in the exponent,
  which by itself is an important advantage, since each term of
  perturbative expansion is already exponentiated. This property of
  exponentiation is well known for the Wilson loop,
   without using the powerful cluster expansion technique.
In particular, the static
 $Q\bar Q$ potential exponentiates and it can be defined through the
 Wilson loop as follows
 \be
 V(R)=-\lim_{T\to \infty} \frac{1}{T} \ln W(R,T).
 \label{37}
 \ee

In our case of possibly light quarks with masses $m_i$ one should
carry out an additional integration over quark trajectories (i.e. over
$d\xi^{(i)}(k)$). In this section we consider the
asymptotics of the amplitude (\ref{34}) and therefore use
the stationary point analysis for trajectories $\{\xi^{(i)}(k)\}$.
Instead in the next section another method will be exploited,
which was used for partial summation of perturbative diagrams 
in Refs. 9-11.
%~\cite{8}$^{\!-\,}$\cite{10}. 
In that section we shall also show how to
modify this method to include also nonperturbative contributions.

In what follows we confine ourselves to the lowest contribution
(\ref{36}) in Eq. (\ref{35}) and show that it yields the double
logarithmic asymptotics.
First of all one can persuade oneself that the approximation
(\ref{36}) yields in Eq. (\ref{34}) all diagrams with exchanges of
photon/gluon lines between fermion lines,  all orderings of lines
included. For QCD this means the following: all orderings, i.e.
all intersection of gluon lines in space-time are included, except
that the color ordering of operators $t^a$ is kept fixed. Since
the commutator of   any two generators $t^a$ is subleading at large
$N_c$, it means that Eq. (\ref{36}) sums up all exchanges including
intersections of gluonic lines in the leading $N_c$ approximation

Our next point is the integration over $d\xi(k)$ in Eq. (\ref{34}) which is
Gaussian in the main term $K_i$, defining the measure of
integration. Therefore we can carry it out by expanding the  exponent in
Eq. (\ref{34}) around the stable fixed point $\bar \xi$, which is obtained by
differentiating the exponent in (\ref{34}) with respect to
$\xi^{(i)}(k)$. One has
$$
\bar \xi^{(i)}(k)=2\varepsilon_i \{iq^{(i)}-
\frac{g^2C_2(f)}{4\pi^2}\sum_{j,k'} \frac{\bar \xi^{(j)}
(k')}{(\bar z^{(i)}(k)-\bar z^j(k'))^2}+
$$
\be +\frac{2g^2C_2(f)}{4\pi^2}\sum_{j,k'} \sum_{m\geq k}\frac{(
\bar \xi^{(i)} (m)\bar \xi^{(j)}(n'))(\bar z^{(i)}(m)-\bar
z^{(j)}(n'))}{(\bar z^{(i)}(m)-\bar z^{(j)}(n'))^4}\}+ 0(g^4)
\label{38}
 \ee
 Here e.g. $\bar
z^{(i)}(k)=\sum^i_{j=1}\sum^k_{\nu=1}\bar \xi^{(j)}(\nu)$, where
we have chosen as the origin the coordinate $x^{(1)}$ of the first
vertex, and all other coordinates are calculated using the
connection $x^{(i)}-x^{(i-1)}=\sum^N_{k=1}\xi^{(i)}(k)$ with the
cyclic condition $x^{(n+1)}=x^{(1)}$.

One can solve the system of equations (\ref{38}) iteratively  expanding
in powers of $g^2$. The first two terms are given in (\ref{38}), where
one should replace $\bar \xi^{(i)}$ inside the curly brackets by
$2i\varepsilon_i q^{(i)}$. If one represents the exponential
appearing in Eq. (\ref{34}) after insertion of Eq. (\ref{35}) 
as $exp (-f(\xi, q))$, then one can write
% \be 
\begin{eqnarray}
f(\xi,q) &=&
\sum_{i,k}\frac{(\xi^{(i)}(k))^2}{4\varepsilon_i}
-i\sum_{i,k} q^{(i)} \xi^{(i)} (k)
\nonumber\\
&&-\frac{g^2C_2(f)}{8\pi^2}
\sum_{i,j,kk'}\frac{\xi^i(k) \xi^j(k')}{(z^i(k)-z^j(k'))^2}+0(g^4).
\label{39}
\end{eqnarray}
The Gaussian integration in (\ref{34}) finally yields
 \be \bar G_n\sim
\int\frac{d^4q}{(2\pi)^4} \prod^n_{i=1} ds_i e^{-f(\bar
\xi,q)-\frac{1}{2} tr \ln \varphi},
\label{40}
\ee 
where the matrix $\varphi$ is 
\be
\varphi^{ij}_{kn} = \frac{1}{2} \frac{\partial^2}{\partial
\xi^{(i)}(k) \partial\xi^{(j)} (n)}f(\xi, q)\biggl | _{\xi=\bar
\xi}.
\label{41}
 \ee

The most important  for what follows is the term $f(\bar \xi,q)$
which can be written as (at this point we reestablish Minkowskian
metric)
\be f(\bar \xi, q)
=\sum^n_{i=1}s_i(q^{(i)})^2+\frac{g^2C_2 (f)}{8\pi^2}
\sum_{ij}\int^{s_i}_0 \int^{s_j}_0\frac{d\tau_id\tau_j(q^{(i)}
q^{(j)})}{(\tau_iq^{(i)}-\tau_jq^{(j)} -\Delta_{ij})^2},
\label{42}
 \ee
 where
we have defined $\tau_i=k\varepsilon_i$, and \be
\Delta_{ij}=\sum^{j-1}_{k=i}s_kq^{(k)}, i<j.
\label{43}
 \ee

The integral in the last term on the r.h.s. of Eq. (\ref{42}) can be written
as
 $s_is_j(q^{(i)}q^{(j)})I_{ij}(s,q)$, where
 \be
 I_{ij}(s,q)=\int^1_0\int^1_0\frac{d\alpha
 d\beta}{(\alpha s_iq^{(i)}-\beta
 s_jq^{(j)}-\Delta_{ij})^2}.
 \label{44}
 \ee
 The diagonal terms, $I_{ij},$ with $i=j$ do not contribute to the
 asymptotics and contain only  selfenergy divergencies, which are
 of no interest to us in what follows. Therefore we shall consider
only the nondiagonal terms with $i\neq j$.

 Let us first study the term with $i=j-1$ ("the dressed
 vertex contribution") and $\Delta_{i,i+1}=s_iq^{(i)}$.
%, see Fig. 1.
 Then Eq. (\ref{44}) is reduced to the form which will be studied
 below
% \be
 $$I_i\equiv I_{i,i+1}(s,q)=\int^1_0\int^1_0
 \frac{d\alpha
 d\beta}{(\alpha s_iq^{(i)}+\beta
 s_jq^{(j)})^2}
$$
\be
=\int^1_0\int^1_0
 \frac{d\alpha
 d\beta}{(a^2\alpha^2+\beta^2b^2+2\alpha\beta(ab))}
 \label{45}
 \ee
 with $a=s_iq^{(i)}, b=s_jq^{(j)}, j=i+1$.
 As it stands the integral (\ref{45}) diverges at small $\alpha,\beta$ (or
 at small $\tau_i,\tau_j$ in Eq. (\ref{42})). The origin of this divergence
 becomes physically clear, when one expresses the distance $z^{(i)}$
 from the vertex position,
 (we go over to the Minkowskian space--time)
 \be
 z^{(i)}=2q^{(i)}\tau_i,~~~z^{(j)}=2q^{(j)}\tau_j.
 \label{46}
 \ee
 The quasiclassical motion (\ref{46}) cannot be true for small
 $\tau_i$,
 when quantum fluctuations wash out the straight--line trajectories.
 The lower limit $\tau_{min}$ can be obtained from
  the quantum uncertainty principle
 \be
\Delta z\Delta q\sim (z^{(i)}-z^{(j)})(q^{(i)}-q^{(j)})\sim 1
\label{47}
\ee
Furthermore, we shall be interested in the kinematical region, where
 \be
|q^{(i)}q^{(j)}|\gg(q^{(i)})^2, (q^{(j)})^2.
\label{48}
\ee
 The value of $\tau_{min}$ then
is found from (\ref{47}) to be \be \tau_{min}\sim
\frac{1}{2|q^{(i)}q^{(j)}|}.
\label{49}
 \ee
 Using Eq. (\ref{49}) one can easily
calculate the integral (\ref{45}), since the term $2\alpha\beta(ab)$ in
the denominator of the integrand in Eq. (\ref{45}) always dominates. The
result is 
\be
I_i=\frac{1}{2s_is_{i+1}(q^{(i)}q^{(i+1)})}
ln(2(q^{(i)}q^{(i+1)})s_i)ln(2(q^{(i)}q^{(i+1)})
s_{i+1}).
\label{50}
 \ee
 The integration of the general term $I_{ij}$ with
$j\neq i-1,i+1$ can be done using  the expressions for the Spence
functions.
 However in the
general case the lower limit $\tau_{min}$ is inessential and the
double logarithmic situation does not occur unless there is a
large ratio, $|\frac{(q_iq_k)}{(q_lq_m)}|\gg 1$.

We start with the open triangle, 
%Fig. 1, 
corresponding to the Sudakov vertex function asymptotics,
i.e. when the fermion loop is not closed. We have 
\be \bar G_3 = (-i\hat q
+m)^{-1} \Gamma(q,q')(-i\hat q'+m)^{-1}.
\label{51}
 \ee
  In this case there is no
integration over $d^4q$ in Eq. (\ref{34}) and only one integral $I_{12}$
is present in Eqs. (\ref{44}) and (\ref{50}) (we disregard as before the
selfinteracting pieces $I_{ii}$, which do not contribute to the
asymptotics).

Inserting Eq. (\ref{50}) into Eq. (\ref{42}) and
integration over $ds_1ds_2$ in Eq. (\ref{40}) yields the leading
contribution, where in Eq. (\ref{50}) the arguments $s_i$ are taken to
be $s_i = \frac{1}{q_i^2}$.
For the case of QED $(C_2\equiv 1)$ we find
\be 
\Gamma(q,q')\sim \exp (-\frac{\alpha}{2\pi}\ln
\frac{2|qq'|}{q^2} \ln \frac{2|qq'|}{(q')^{2}}),
\label{52}
\ee
which coincides with the known Sudakov asymptotics.

We turn now to the case of QCD, where the basic  triangle diagram
is closed due to color gauge invariance and try to find out
whether the kinematical region (\ref{48}) plays an important role in the
integral over $d^4q$ in Eq. (\ref{40}).

In the general case, when all $q_i$ are unconstrained and
expressed through three external momenta $p_1,p_2,p_3$ and one
integration variable, the region (\ref{48}), yielding double logarithmic
asymptotics (DLA) (\ref{50}), is suppressed due to large values of
$f(\bar \xi, q)$ in the exponent. As a result the integral over
$dq$ does not lead to the DLA form for $\bar G_3$.

The situation changes however, if one considers instead of   $\bar
G_3$ the form factor, i.e. when the pole terms are factored out
from the vertices 2 and 3
%, see Fig. 2, 
and the vertex functions appear
there. To simplify matter, one can consider for the form factor the
same representation (\ref{40}). Under the integral one has the
vertex functions $\psi_i(k_i),i=1,3$, where
$$
k_1=
q^{(1)}+q^{(3)}-p^{(1)}\frac{(q^{(1)}+q^{(3)})p^{(1)}}{(p^{(1)})^2},
$$
\be k_3=
q^{(2)}+q^{(3)}-p^{(3)}\frac{(q^{(2)}+q^{(3)})p^{(3)}}{(p^{(3)})^2}.
\label{53}
 \ee
The definition (\ref{53}) yields in the c.m. system of particle 1
or 3 the
 familiar relative momentum of two emitted fermions. The presence of
 $\psi_i$ imposes a restriction on the momenta $q_i$, namely
 \be
 k^2_1, k^2_3 \la \kappa^2
 \label{54}
 \ee
 where $\kappa^2$ is some hadronic scale.

  Let us define in the Breit system the momenta $$p^{(1)},
  p^{(2)}, p^{(3)}= (p_0,-\frac{\vec Q}{2}),(0,\vec Q), (p_0,
  \frac{\vec Q}{2}),$$ where $\vec Q^2\gg \kappa^2$ and
  $p_0^2=M^2+\frac{\vec Q^2}{4}$.
   One can then easily see, that Eq. (\ref{54}) constrains the region of integration
   over $dq\equiv d^4q_1$ to the region $|\vec q |\sim
   \kappa, |q_0-p_0|\sim \kappa$, and the conditions (\ref{48}) are
   satisfied.
   Hence in this case one recovers the Sudakov asymptotics (\ref{52}), where
   \be
   |qq'|\to |q^{(1)}q^{(2)}|\approx \frac{\vec Q^2}{4},
   q^2\sim q^{\prime 2}\sim \kappa^2,~~
   \alpha\to \alpha_sC_2.
   \label{55}
   \ee

\section{Explicit path integration in  FSR. Connection to the
Bern-Kosower-Strassler method.}

 The path integral (\ref{9}) can be performed in a direct
 way. After the expansion of Eq. (\ref{10}) in powers of $A_\mu$ and
 $F_{\mu\nu}$ and exploiting Fourier transform for the latter one
 obtains Gaussian integrals which can be easily done.

This procedure is similar to  the method introduced in Refs. 9-11
%\cite{8}$^{\!-\,}$\cite{10}
for one-loop diagrams and the effective action in
QED and QCD. To this end we extend the method  of Refs. 9-11
%~\cite{8}$^{\!-\,}$\cite{10}
to the case of meson and glueball Green's  functions. We will show, that
 the method used in Refs. 9-11
%~\cite{8}$^{\!-\,}$\cite{10}  
can be simplified and generalized.

 Let us start with three typical expressions in the order of
 increasing complexity. We may consider the one-loop effective action
  \be
  \Gamma\{A\} =\int^\infty_0 \frac{ds}{s} \xi(s)
  e^{-m^2s-\frac14\int^s_0\dot z^2_\mu d\tau} (Dz)_{xx} tr W(A_\mu),
  \label{56}
  \ee
  where $\xi(s)$ is a regularization. Other expressions, which may be studied,
are the heavy-light quark-antiquark Green's function
  \be
  G_{HL} (Q) =\int^\infty_0 ds
  e^{-m^2s-\frac14\sum^N_{n=1}\frac{\xi^2(n)}{\varepsilon}-iQ\sum^N_{n=1}
  \xi(n)} \prod^N_{n=1}
  \frac{d^4\xi(n)}{(4\pi\varepsilon)^2} \lan W(A_\mu)\ran.
  \label{57}
  \ee
or the Green's function for the arbitrary mass quark-antiquark 
system
$$ G(Q)
=\int^\infty_0 ds_1\int^\infty_0 ds_2 \frac{d^4p}{(2\pi)^4}
e^{-ip_\mu\sum^{N_1}_{n_1=1} \xi^{(1)}_\mu(n_1) -i(Q_\mu-p_\mu)
\sum^{N_2}_{n_2=1} \xi^{(2)}_\mu(n_2)}$$
 \be
\times e^{-m^2_1s_1-m^2_2s_2}\prod^{N_1}_{n_1=1}
\frac{d^4\xi^{(1)}(n_1)}{(4\pi\varepsilon_1)^2} \prod^{N_2}_{n_2=1}
\frac{d^4\xi^{(2)}(n_2)}{(4\pi\varepsilon_2 )^2}  \lan tr
W(A_\mu)\ran .
\label{58} 
\ee 
Here we have introduced  $N_1 \varepsilon_1=s_1, N_2 \varepsilon_2=s_2$.

Let us consider the expression (\ref{57}), since Eq. (\ref{56}) can be
managed easily in the same way.
To proceed one expands in (Eq. \ref{57}) $\lan W(A_\mu)\ran$ in powers
of $A_\mu$ and uses the Fourier transform $\tilde A_\mu{(k)}$ for
the latter. To the second order in $g$ one obtains
$$
G_{HL} (Q) -G^{(0)}_{HL} (Q) =-g^2\int^\infty_0
dse^{-m^2s-iQ\sum\xi(n)-\frac14\sum\frac{\xi^2(n)}{\varepsilon}}
$$
$$\times \prod^N_{n=1}\frac{d\xi(n)}{(4\pi\varepsilon)^2}\int^s_0\dot
z_\mu(t_1)dt_1\int^{t_1}_0 \dot z_\nu(t_2) dt_2\lan\tilde
A_\mu(k_1) \tilde A_\nu (k_2)\ran
$$
 \be
\times\frac{ d^4k^{(1)}
d^4k^{(2)}}{(2\pi)^8}
e^{ik^{(1)}_\mu z_\mu(t_1)+ ik^{(2)}_\mu z_\mu(t_2)} .
 \label{59} 
\ee
Defining similarly to Ref. 11
%~\cite{10}
\be \sum^2_{i=1} ik^{(i)} z(t_i) = i \int^s_0 J_\mu (\tau) z_\mu(\tau)
d\tau,~~ J_\mu(\tau) =k_\mu^{(1)} \delta(\tau-t_1) + k_\mu^{(2)}
\delta(\tau-t_2),
 \label{60} \ee
one arrives at the integral
 \be
I(Q,J)\equiv \prod^N_{i=1} \frac{d^4\xi(i)}{(4\pi\varepsilon)^2}
\exp\{-iQ\sum_n\xi(n)-\frac{1}{4\varepsilon}
\sum_n\xi^2(n)+i\varepsilon\sum^N_{k=1} \xi_\mu(k) \sum^N_{n=k}
J_\mu(n)\},
\label{61} 
\ee 
where we have used that $z_\mu(t)=
\sum^n_{k=1} \xi_\mu(k)$ and interchanged the order of summation
in $\int J_\mu(\tau) z_\mu (\tau) d\tau=\sum J_\mu(n) z_\mu (n)
\varepsilon.$

The Gaussian integration in Eq. (\ref{61}) can be performed trivially and one
obtains 
$$ I(Q,J) =\exp\{ -\varepsilon^3\sum^N_{k=1} \sum^N_{n=k}
\sum^N_{n'=k} J_\mu(n) J_\mu(n') - Q^2s+\varepsilon^2
2Q_\mu\sum^N_{k=1}\sum^N_{n=k} J_\mu(n)\}$$ 
\be 
=\exp
\{-\int^s_0d\tau(\int^s_\tau J_\mu(\tau') d\tau')^2- Q^2s+ 2Q_\mu
\int^s_0d\tau\int^s_\tau J_\mu(\tau') d\tau'\}.
 \label{62} 
\ee
Substituting (\ref{60}) into (\ref{62}) one finally gets 
\be
I(Q,J) =\exp\{-Q^2s+2Q_\mu(k_\mu^{(1)} t_1+k_\mu^{(2)} t_2)
-\sum_{i,j=1,2} k_\mu^{(i)} k^{(j)}_\mu g_B(t_i,t_j)\},
\label{63}
\ee 
where
\be 
g_B(t_i, t_j)=Min (t_i, t_j).
\label{64} 
\ee
In the derivation of Eq. (\ref{63}) one could apply the string vertex
technique  of Refs. 9-11.
%~\cite{8}$^{\!-\,}$\cite{10}. 
A corresponding method for the
Dirac propagator was used in Ref. 20.
%~\cite{19}.

So far we have neglected the factors $\dot z_\mu(t_1)\dot
z_\nu(t_2)$ in (\ref{59}). To take them into account one could
exploit the trick suggested in Ref. 11.
%~\cite{10}. 
Instead we shall use
below another approach, based on the following relation ( we
denote $I_{\mu\nu} (Q,J)$ the integral (\ref{61}) with $\dot z_\mu
\dot z_\nu$ taken into account). Noting that  $\dot z_\mu(t) =
\frac{\xi_\mu (k)}{\varepsilon}$  we obtain
\begin{eqnarray}
I_{\mu\nu}(Q,J)
&\!\!=&\!\!\prod^N_{n=1}\frac{d^4\xi(n)}{(4\pi\varepsilon)^2}\,
\frac{\xi_\mu(k_1)}{\varepsilon}\, \frac{\xi_\nu(k_2)}{\varepsilon}
\nonumber\\
&\!\!\times&\!\!\exp\left[\sum^N_{k=1}\left(i\varepsilon\,\xi_\mu(k)
j_\mu(k) -iQ\,\xi(k) -\frac{\xi^2(k)}{4\varepsilon}\right)
\right],
\label{65} 
\end{eqnarray}
where
$j_{\mu}(k) \equiv \sum^N_{n=k}J_\mu(n)$. Hence one can make a
simple connection 
\be 
I_{\mu\nu} (Q,J) =\left
(\frac{1}{i\varepsilon^2}\frac{\partial}{\partial
j_{\mu}(k_1)}\right ) \left (
\frac{1}{i\varepsilon^2}\frac{\partial}{\partial
j_{\nu}(k_2)}\right ) I(Q,J).
\label{66} 
\ee 
Computing
(\ref{66}) one should take into account that $t_2\leq t_1$.
In doing so one obtains 
\be 
\dot z_\mu^{(1)} \to -
2i(Q_\mu-k_\mu^{(1)});~~ \dot z_\mu^{(2)} \to -
2i(Q_\mu-k_\mu^{(1)}-k^{(2)}_\mu). 
\label{67} 
\ee 
The $HL$
Green's function to the second order $O(g^2)$ has the form
$$
G_{HL}(Q) -G_{HL}^{(0)}(Q)=4g^2\int^\infty_0 dse^{-(m^2+Q^2)s}
\frac{d^4k^{(1)} d^4k^{(2)}}{(2\pi)^8} \lan \tilde A_\mu(k^{(1)})
\tilde A_\nu(k^{(2)})\ran
$$
$$
\times \int^s_0dt_1 \int^{t_1}_0 dt_2(Q_\mu-k^{(1)}_\mu)
(Q_\nu-k^{(2)}_\nu-k_\nu^{(1)})e^{2Q_\mu(k^{(1)}_\mu t_1+k^{(2)}_\mu
t_2)}
$$
\be 
\times\exp [-(k^{(1)})^2t_1-(k^{(2)})^2t_2-2k^{(1)}_\mu k^{(2)}_\mu
t_2]. 
\label{68} 
\ee
In some cases another form is convenient with integrals over $dt_1
dt_2$ taken explicitly,
$$
G_{HL}(Q) -G_{HL}^{(0)}(Q)=4g^2\int \frac{d^4k^{(1)}
d^4k^{(2)}}{(2\pi)^8} \frac{\lan \tilde A_\mu(k^{(1)}) \tilde
A_\nu(k^{(2)})\ran}{2Qk^{(2)}-(k^{(2)})^2-2k^{(1)}k^{(2)}}
$$
$$
\times
\left(Q_\mu\!-\!
k_\mu^{(1)}\!\right)\!\left(Q_\nu-k_\nu^{(2)}-k_\nu^{(1)}\!\right)\left
\{
\frac{1}{2Q(k^{(1)}\!+k^{(2)})-\!(k^{(1)}+k^{(2)})^2}\!
\left[-\frac{1}{m^2+Q^2}\right.\right.$$ 
\be\left.\left.+
\frac{1}{m^2+(Q\!-\!k^{(1)}\!-\!k^{(2)})^2}\right]\!-
\frac{1}{2Qk^{(1)}\!-\!(k^{(1)})^2}\left
[-\frac{1}{m^2\!+\!Q^2}+\frac{1}{m^2\!+\!(Q\!-\!k^{(1)})^2}\right]\right\}.
\label{69} 
\ee

Eqs. (\ref{68}) and (\ref{69}) contain the general result for the
contribution of the second order correlator to $G_{HL}(Q)$. One
can further specify the correlator $\lan\tilde
A_\mu(k^{(1)})\tilde A_\nu(k^{(2)})\ran$. For example, for the
perturbative propagator in the Feynman gauge one has 
\be
\lan\tilde A_\mu(k^{(1)})\tilde
A_\nu(k^{(2)})\ran=\frac{4\pi\delta_{\mu\nu}}{(k^{(1)}-k^{(2)})^2},
\label{69'} 
\ee 
while for the nonperturbative correlator in the
Balitsky gauge one obtains, using results from\,\cite{20}
$$
\lan\tilde A_\mu(k)\tilde A_\nu(k')\ran = \frac{\sigma
e^{-(k_4-k'_4)^2T_g^2}}{\sqrt{\pi} T_g}\int^1_0 dt\int^1_0dt'
\alpha_\mu (t) \alpha_\nu(t')
$$
$$\times (\delta_{\mu\nu} \delta_{ik} -\delta_{i\nu}\delta_{k\mu})
\left (i\frac{\partial}{\partial k_i}i\frac{\partial}{\partial
k'_k}\right) \exp \{ -[(\alpha+\tilde t^{\prime
2})\vek^2+(\alpha+\tilde t^2)\vek^{\prime 2}+
$$
\be
+ 2\tilde t  \tilde t'(\vek\cdot \vek')][4\alpha(\alpha+\tilde
t^2+\tilde t^{\prime 2})]^{-1}\}
\cdot [\alpha(\alpha+\tilde
t^2+\tilde t^{\prime 2})]^{-3/2}.
 \label{69.a}
 \ee
  Here we have
introduced above the regularizing factor $\exp
(-\alpha(\vez^2+\vew^2))$ and used the notation
$$ \tilde t=\frac{t}{2T_g}, \tilde t' =\frac{ t'}{2T_g};~~
\alpha_\mu(t):\alpha_4(t)=1,~ \alpha_i(t)=t,i=1,2,3.$$
It is assumed that the limit $\alpha\to 0$ is taken at the end of
the calculations.

 Inserting Eq. (\ref{69.a}) into Eqs. (\ref{68}) and (\ref{69})
one obtains the lowest order nonperturbative contribution to
$G_{HL}$, which can be studied for all values of $Q$. In
particular, one can expand the result in powers of $1/Q^2$ to
compare with the standard OPE result, where the expansion starts with
$\frac {tr(F_{\mu\nu}(0))^2}{Q4}$. The kernel (\ref{69.a}) is
nonlocal and generates a series in powers of ($QT_g)^2$.
See Ref. 22
% ~\cite{21} 
for further details.

The equations (\ref{59}-\ref{69}) obtained above are valid in the
second order of expansion of $\lan tr W(A_\mu)\ran$ in $gA_\mu$.
The method used above can easily be generalized to higher orders.
Indeed, writing
%\begin{eqnarray}
$$
\lan tr W(A_\mu)\ran = \sum^\infty_{n=0} (ig)^n \left \lan tr \left \{
\int^s_0A_{\mu_1} (k^{(1)}) \dot z_{\mu_1}(t_1) dt_1
\int^{t_1}_0A_{\mu_2} (k^{(2)}) \dot z_{\mu_2}(t_2) dt_2
\right . \right .
$$
%\nonumber\\&&
\be
 \left . \left . \ldots
\int^{t_{n-1}}_0A_{\mu_n} (k^{(n)}) \dot z_{\mu_n}(t_n) dt_n
 e^{i\sum^n_{m=1}k^{(m)}z(t_m) } \prod^n_{i=1} \frac{ d^4k^{(i)}}{(2\pi)^4}
\right\}\right\ran,
 \label{72}
\ee
% \end{eqnarray}
one can define similarly to Eq. (\ref{60}) for the $n$-th term in
Eq. (\ref{72}) 
\be 
J^{(n)}_\mu(\tau) = \sum^n_{i=1} k_\mu^{(i)} \delta (\tau-t_i).
\label{73} 
\ee
Doing the integrals over $\prod^N_{k=1} d\xi(k)$ as before (cf. Eq.
(\ref{61})) one obtains the following result, generalizing
Eq. (\ref{62}), 
\be 
I^{(n)}(Q, J) =\exp\left \{ -\int^s_0\!\! d\tau\left
(\int^s_\tau j_\mu(\tau') d\tau'\right)^2\!\!-Q^2s+2Q_\mu\int^s_0
\!\!d\tau \int^s_\tau \!\!J_\mu(\tau') d\tau'\right\}. \label{74} \ee
Inclusion of the terms $\dot z_{\mu_1} (t_1) ... \dot
z_{\mu_n}(t_n)$ can be done in the same way as in Eq. (\ref{66}) and
the generalization of $I^{(n)}(Q, J)$ to
$I^{(n)}_{\mu_1...\mu_n}(Q,J)$ is straightforward. In this way one
obtains the general form of the expansion of Green's functions
(\ref{56}) -(\ref{58}) to all orders in $g$. In
contrast to the results of Refs. 9-11
%~\cite{8}$^{\!-\,}$\cite{10} 
one keeps not only 
perturbative but also nonperturbative contributions.

Although we have used here the FSR method to study the case
of scalar quarks, it can also also be applied for quarks with spin starting
from Eq. (\ref{18}). We have shown in the present paper how to use the FSR 
to treat perturbative QCD, establishing in particular the correspondence between
the standard and FSR perturbation expansion.

%\newpage
%\section*{References}
\addcontentsline{toc}{section} {\numberline{}References}

\begin{thebibliography}{99}
\bibitem{0}
M.S. Marinov, Phys. Rept. {\bf 60C} 1 (1980).
\bibitem{1}
R.P. Feynman, Phys. Rev. {\bf 80} 440 (1950); ibid {\bf 84} 108 (1951).
\bibitem{2}
V.A. Fock, Izvestya Akad. Nauk USSR, OMEN, 1937, p.557; J. Schwinger,
      Phys. Rev. {\bf 82} 664 (1951).
     \bibitem{3} G.A. Milkhin and E.S. Fradkin, ZhETF {\bf 45} 1926 (1963); 
E.S. Fradkin, Trudy FIAN, {\bf 29} 7 (1965).
     \bibitem{4} M.B. Halpern, A. Jevicki and P. Senjovic, Phys. Rev.
     {\bf D16} 2474 (1977);
K. Bardakci and S. Samuel, Phys.Rev. {\bf D18} 2849 (1978);
R. Brandt et al, Phys. Rev. {\bf D19} 1153 (1979); 
S. Samuel, Nucl. Phys. {\bf B149} 517 (1979);
     J. Ishida and A. Hosoya, Progr. Theor. Phys. {\bf 62} 544 (1979).
 \bibitem{5} Yu.A. Simonov, Nucl. Phys. {\bf B307} 512 (1988).
     \bibitem{6} A.I. Karanikas and C.N. Ktorides, Phys. Lett.  {\bf
     B275} 403 (1992); Phys. Rev. {\bf D52} 58883 (1995).
     \bibitem{7} A.I. Karanikas, C.N. Ktorides and N.G. Stefanis, Phys.
      Rev. {\bf D52} 5898 (1995).
      \bibitem{8} Z. Bern and D.A. Kosower, Phys. Rev. Lett. {\bf 66}
      1669 (1991); Nucl. Phys. {\bf B379} 451 (1992).
      \bibitem{9}Z. Bern and D.C. Dunbar, Nucl. Phys. {\bf B379} 562 (1992).
      \bibitem{10} M.J. Strassler, Nucl. Phys. {\bf B385} 145 (1992).
      \bibitem{11} Yu.A. Simonov, JETP Lett {\bf 54} 249 (1991);
ibid. {\bf 55} (1992) 627; Phys.  At. Nucl. {\bf 58} 309 (1995).
\bibitem{12} 
Yu.A. Simonov, In "Varenna 1995, Selected
      topics in nonperturbative QCD", p.319.
\bibitem{13}
      H.G. Dosch, H.-J. Pirner and Yu.A. Simonov, Phys. Lett. {\bf B349}
      335 (1995); E.L. Gubankova and Yu.A. Simonov, Phys.
      Lett. {\bf 360} 93 (1995);
      N.O. Agasyan, in preparation.
\bibitem{14} 
N.G. van Kampen, Phys. Rept. {\bf C24} 171 (1976).
\bibitem{15} 
Yu.A. Simonov, Phys. Lett. {\bf 464 } 265 (1999).
\bibitem{16}
 B.S. De Witt, Phys. Rev. {\bf 162} 1195 1239 (1967);\\
 J. Honerkamp, Nucl. Phys. {\bf B48} 269 (1972);\\
 G.'t Hooft, Nucl. Phys. {\bf B62} 444 (1973); Lectures
at Karpacz, in : Acta Univ. Wratislaviensis {\bf 368} 345 (1976);\\
L.F. Abbot, Nucl. Phys. {\bf B185} 189 (1981).

\bibitem{17} 
Yu.A. Simonov, Phys. At Nucl. {\bf 58} 107 (1995); 
JETP Lett.  {\bf 75} 525 (1993);\\
Yu.A. Simonov, in: Lecture Notes in Physics v.479,
p. 139; ed. H. Latal and W. Schwinger, Springer, 1996.
      
\bibitem{18} Yu.A. Simonov and J.A. Tjon, Ann. Phys. {\bf 228} 1 (1993).

\bibitem{19} 
M. Henneaux and C. Teitelboim, Ann. Phys. {\bf 143} 127 (1982);\\
   V.Ya. Fainberg, A.Va. Marshakov, Nucl.Phys. {\bf B306} 659 (1988).
\bibitem{20} 
Yu.A. Simonov, Phys. At. Nucl. {\bf 60} 2069 (1997), ;\\
Yu.A. Simonov and J.A. Tjon, Phys. Rev. {\bf D62} 014501 (2000).

\bibitem{21} 
V.I. Shevchenko and Yu.A. Simonov, .
\end{thebibliography}
\end{document}


%====================================================================%
%                  SPROCL.TEX     27-Feb-1995                        %
% This latex file rewritten from various sources for use in the      %
% preparation of the standard proceedings Volume, latest version     %
% by Susan Hezlet with acknowledgments to Lukas Nellen.              %
% Some changes are due to David Cassel.                              %
%====================================================================%

\documentstyle[sprocl]{article}

%\input{psfig}
\bibliographystyle{unsrt}    % for BibTeX - sorted numerical labels by order of
                             % first citation. 

% A useful Journal macro
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}

% Some useful journal names
\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} A}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.}  B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}
% Some other macros used in the sample text
\def\st{\scriptstyle}
\def\sst{\scriptscriptstyle}
\def\mco{\multicolumn}
\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
\def\kb{\bar{K^0}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\CPbar{\hbox{{\rm CP}\hskip-1.80em{/}}}%temp replacement due to no font


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                %
%    BEGINNING OF TEXT                           %
%                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title{INSTRUCTIONS FOR PRODUCING A CAMERA-READY MANUSCRIPT
USING WORLD SCIENTIFIC PUBLISHING STYLE FILES}

\author{ A.B. AUTHOR, C.D. AUTHOR }

\address{World Scientific Publishing Co, 1060 Main Street,
River Edge,\\ NJ 07661, USA}

\author{ A.N. OTHER }

\address{Department of Physics, Theoretical Physics, 1 Keble Road,\\
Oxford OX1 3NP, England}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle\abstracts{
This is where the abstract should be placed. It should consist of one paragraph
and give a concise summary of the material in the article below.
Replace the title, authors, and addresses within the curly brackets
with your own title, authors, and addresses; please use 
capital letters for the title and the authors. You may have as many authors and
addresses as you wish. It's preferable not to use footnotes in the abstract 
or the title; the
acknowledgments for funding bodies etc. are placed in a separate section at
the end of the text.}
  
\section{Guidelines}
\subsection{Producing the Hard Copy}\label{subsec:prod}

The hard copy may be printed using the advice given in the file
{\em splread.1st}, which is repeated in this section. You should have
three files in total.\footnote{Downloadable at {\sf http://www.wspc.demon.co.uk/Styles/procs/procsread1st.html}.}

\noindent {\em splread.1st} --- the preliminary guide.

\noindent {\em sprocl.sty} --- the style file that provides the higher
level latex commands for the proceedings. Don't change these parameters.

\noindent {\em sprocl.tex} --- the main text. You can delete our sample
text and replace it with your own contribution to the volume, however we
recommend keeping an initial version of the file for reference.
Strip off any mail headers and then latex the tex file.
The command for latexing is {\sf latex sprocl}, do this twice to
sort out the cross-referencing.

If you wish to use some other form of word-processor, some guidelines are
given in Sec.~\ref{subsec:wpp} below.
These files will work with standard latex 2.09. If there is an abbreviation
defined in the new definitions at the top of the file {\em sprocl.tex} that
conflicts with one of your own macros, then
delete the appropriate command and revert to longhand. Failing that, please
consult your local texpert to check for other conflicting macros that may
be unique to your computer system.
Page numbers are included at the bottom of the page for your guidance. Do not
worry about the final pagination of the volume which will be done after you
submit the paper. 

\subsection{Using Other Word-Processing Packages}\label{subsec:wpp}

If you want to use some other form of word-processor to construct your
output, and are using the final hard copy version of these files as guidelines;
then please follow the style given here for headings, table and figure captions,
and the footnote and citation marks.  For this size of volume, the final page
dimensions will be 8.5 by 6 inches (21.5 by 15.25 cm)
however you should submit the copy on
standard A4 paper. The
text area, which includes the page numbers should be 7 by 4.7 inches
(17.75 by 12 cm).
The text should be in 10pt roman 
for the title, section heads and
the body of the text, and 9pt for the authors' names and addresses.
Please use capitals for the title and authors, bold face
for the title and headings, and italics for the subheadings. 
The abstract, footnotes, figure and 
table captions should be in 8pt. 

It's also important to reproduce the spacing
of the text and headings as shown here. Text should be slightly
more than single-spaced; use
a baselineskip (which is the average distance from the base of one line of
text to the base of an adjacent line) of 18 pts and 14 pts for footnotes.
Headings should be separated from the text preceding it by a baselineskip
of about 40 pts and use a baselineskip of about 28 pts for the text following
a subheading and 32 pts for the text following a heading.

Paragraphs should have a first line indented by about 0.25in (6mm) except 
where the
paragraph is preceded by a heading and the abstract should be indented on
both sides by 0.25in (6mm) from the main body of the text.

\subsection{Headings and Text and Equations}

Please preserve the style of the
headings, text fonts and line spacing to provide a
uniform style for the proceedings volume. 

Equations should be centered and numbered consecutively, as in
Eq.~\ref{eq:murnf}, and the {\em eqnarray} environment may be used to
split equations into several lines, for example in Eq.~\ref{eq:sp},
or to align several equations.
An alternative method is given in Eq.~\ref{eq:spa} for long sets of
equations where only one referencing equation number is wanted.

In latex, it is simplest to give the equation a label, as in 
Eq.~\ref{eq:murnf}
where we have used {\em $\backslash$label\{eq:murnf\}} to identify the
equation. You can then use the reference {\em $\backslash$ref\{eq:murnf\}} 
when citing the equation in the
text which will avoid the need to manually renumber equations due to
later changes. (Look at
the source file for some examples of this.)

The same method can be used for referring to sections and subsections.

\subsection{Tables}

The tables are designed to have a uniform style throughout the proceedings
volume. It doesn't matter how you choose to place the inner
lines of the table, but we would prefer the border lines to be of the style
shown in Table~\ref{tab:exp}.
 The top and bottom horizontal
lines should be single (using {\em $\backslash$hline}), and
there should be single vertical lines on the perimeter,
(using {\em $\backslash$begin\{tabular\}\{$|...|$\}}).
 For the inner lines of the table, it looks better if they are
kept to a minimum. We've chosen a more complicated example purely as
an illustration of what is possible.

The caption heading for a table should be placed at the top of the table.

\begin{table}[t]
\caption{Experimental Data bearing on $\Gamma(K \ra \pi \pi \gamma)$
for the $\ko_S, \ko_L$ and $K^-$ mesons.\label{tab:exp}}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{|c|c|c|l|}
\hline
& & & \\
&
$\Gamma(\pi^- \pi^0)\; s^{-1}$ &
$\Gamma(\pi^- \pi^0 \gamma)\; s^{-1}$ &
\\ \hline
\mco{2}{|c|}{Process for Decay} & & \\
\cline{1-2}
$K^-$ &
$1.711 \times 10^7$ &
\begin{minipage}{1in}
$2.22 \times 10^4$ \\ (DE $ 1.46 \times 10^3)$ 
\end{minipage} &
\begin{minipage}{1.5in}
No (IB)-E1 interference seen but data shows excess events relative to IB over
$E^{\ast}_{\gamma} = 80$ to $100MeV$
\end{minipage} \\ 
& & &  \\ \hline
\end{tabular}
\end{center}
\end{table}


\subsection{Figures}

If you wish to `embed' a postscript figure in the file, then remove the \% mark
from the declaration of the postscript figure within the figure description
and change the filename to an appropriate one.
Also remove the comment mark from the {\em input psfig}
command at the top
of the file. You may need to play around with this as different computer
systems appear to use different commands.

Next adjust the
scaling of the figure until it's correctly positioned,
and remove the declarations of the
lines and any anomalous spacing.

If you prefer to use some other method then it's very important to
leave the correct amount of vertical space in the figure declaration
to accomodate your figure (remove the lines and change the vspace in the
example.) Send the hard copy figures on separate pages with clear
instructions to match them to the correct space in the final hard copy
text. Please ensure the final hard copy figure is correctly scaled to
fit the space available (this ensures the figure is legible.)

The caption heading for a figure should be placed below the figure.

\subsection{Limitations on the Placement of Tables,
Equations and Figures}\label{sec:plac}

Very large figures and tables should be placed on a page by themselves. One
can use the instruction {\em $\backslash$begin\{figure\}$[$p$]$} or
{\em $\backslash$begin\{table\}$[$p$]$}
to position these, and they will appear on a separate page devoted to
figures and tables. We would recommend making any necessary
adjustments to the layout of the figures and tables
only in the final draft. It is also simplest to sort out line and
page breaks in the last stages.

\subsection{Acknowledgments, Appendices, Footnotes and the Bibliography}
If you wish to have 
acknowledgments to funding bodies etc., these may be placed in a separate
section at the end of the text, before the Appendices. This should not
be numbered so use {\em $\backslash$section$\ast$\{Acknowledgments\}}.

It's preferable to have no appendices in a brief article, but if more
than one is necessary then simply copy the 
{\em $\backslash$section$\ast$\{Appendix\}}
heading and type in Appendix A, Appendix B etc. between the brackets.

Footnotes are denoted by a letter superscript 
in the text,\footnote{Just like this one.} and references
are denoted by a number superscript.
We have used {\em $\backslash$bibitem} to produce the bibliography.
Citations in the text use the labels defined in the bibitem declaration,
for example, the first paper by Jarlskog~\cite{ja} is cited using the command
{\em $\backslash$cite\{ja\}}.

If you more commonly use the method of square brackets in the line of text
for citation than the superscript method,
please note that you need  to adjust the punctuation
so that the citation command appears after the punctuation mark.
Two examples of multiple citations that you may find useful are given
here: for citing several consecutive references,\cite{ja}$^{\!-\,}$\cite{bd} and
for citing two non-consecutive references.\cite{ja}$^{\!,\,}$\cite{bd}

\subsection{Final Manuscript}

The final hard copy that you send must be absolutely clean and unfolded.
It will be printed directly without any further editing. Use a printer
that has a good resolution (300 dots per inch or higher). There should
not be any corrections made on the printed pages, nor should adhesive
tape cover any lettering. Photocopies are not acceptable.

The manuscript will not be reduced or enlarged when filmed so please ensure 
that indices and other small pieces of text are legible.

\section{Sample Text }

The following may be (and has been) described as `dangerously irrelevant'
physics. The Lorentz-invariant phase space integral for
a general n-body decay from a particle with momentum $P$
and mass $M$ is given by:
\begin{equation}
I((P - k_i)^2, m^2_i, M) = \frac{1}{(2 \pi)^5}\!
\int\!\frac{d^3 k_i}{2 \omega_i} \! \delta^4(P - k_i).
\label{eq:murnf}
\end{equation}
The only experiment on $K^{\pm} \ra \pi^{\pm} \pi^0 \gamma$ since 1976
is that of Bolotov {\it et al}.~\cite{bu}
        There are two
necessary conditions required for any acceptable
parametrization of the
quark mixing matrix. The first is that the matrix must be unitary, and the
second is that it should contain a CP violating phase $\delta$.
 In Sec.~\ref{subsec:wpp} the connection between invariants (of
form similar to J) and unitarity relations
will be examined further for the more general $ n \times n $ case.
The reason is that such a matrix is not a faithful representation of the group,
i.e. it does not cover all of the parameter space available.
\begin{equation}
\begin{array}{rcl}
\bf{K} & = &  Im[V_{j, \alpha} {V_{j,\alpha + 1}}^*
{V_{j + 1,\alpha }}^* V_{j + 1, \alpha + 1} ] \\
       &   & + Im[V_{k, \alpha + 2} {V_{k,\alpha + 3}}^*
{V_{k + 1,\alpha + 2 }}^* V_{k + 1, \alpha + 3} ]  \\
       &   & + Im[V_{j + 2, \beta} {V_{j + 2,\beta + 1}}^*
{V_{j + 3,\beta }}^* V_{j + 3, \beta + 1} ]  \\
       &   & + Im[V_{k + 2, \beta + 2} {V_{k + 2,\beta + 3}}^*
{V_{k + 3,\beta + 2 }}^* V_{k + 3, \beta + 3}] \\
& & \\
\bf{M} & = &  Im[{V_{j, \alpha}}^* V_{j,\alpha + 1}
V_{j + 1,\alpha } {V_{j + 1, \alpha + 1}}^* ]  \\
       &   & + Im[V_{k, \alpha + 2} {V_{k,\alpha + 3}}^*
{V_{k + 1,\alpha + 2 }}^* V_{k + 1, \alpha + 3} ]  \\
       &   & + Im[{V_{j + 2, \beta}}^* V_{j + 2,\beta + 1}
V_{j + 3,\beta } {V_{j + 3, \beta + 1}}^* ]  \\
       &   & + Im[V_{k + 2, \beta + 2} {V_{k + 2,\beta + 3}}^*
{V_{k + 3,\beta + 2 }}^* V_{k + 3, \beta + 3}],
\\ & &
\end{array}\label{eq:spa}
\end{equation}
where $ k = j$ or $j+1$ and $\beta = \alpha$ or $\alpha+1$, but if
$k = j + 1$, then $\beta \neq \alpha + 1$ and similarly, if
$\beta = \alpha + 1$ then $ k \neq j + 1$.\footnote{An example of a 
matrix which has elements
containing the phase variable $e^{i \delta}$ to second order, i.e.
elements with a
phase variable $e^{2i \delta}$ is given at the end of this section.}
   There are only 162 quark mixing matrices using these parameters
which are
to first order in the phase variable $e^{i \delta}$ as is the case for
the Jarlskog parametrizations, and for which J is not identically
zero.
It should be noted that these are physically identical and
form just one true parametrization.
\bea
T & = & Im[V_{11} {V_{12}}^* {V_{21}}^* V_{22}]  \nonumber \\
&  & + Im[V_{12} {V_{13}}^* {V_{22}}^* V_{23}]   \nonumber \\
&  & - Im[V_{33} {V_{31}}^* {V_{13}}^* V_{11}]. 
\label{eq:sp}
\eea


\begin{figure}
\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\vskip 2.5cm
\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%\psfig{figure=filename.ps,height=1.5in}
\caption{Radiative (off-shell, off-page and out-to-lunch) SUSY Higglets.
\label{fig:radish}}
\end{figure}

\section*{Acknowledgments}
This is where one places acknowledgments for funding bodies etc.
Note that there are no section numbers for the Acknowledgments, Appendix
or References.

\section*{Appendix}
 We can insert an appendix here and place equations so that they are
given numbers such as Eq.~\ref{eq:app}.
\be
x = y.
\label{eq:app}
\ee
\section*{References}
\begin{thebibliography}{99}
\bibitem{ja}C Jarlskog in {\em CP Violation}, ed. C Jarlskog
(World Scientific, Singapore, 1988).

\bibitem{ma}L. Maiani, \Journal{\PLB}{62}{183}{1976}.

\bibitem{bu}J.D. Bjorken and I. Dunietz, \Journal{\PRD}{36}{2109}{1987}.

\bibitem{bd}C.D. Buchanan {\it et al.}, \Journal{\PRD}{45}{4088}{1992}.

\end{thebibliography}

\end{document}

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% End of sprocl.tex  %
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