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%************************   Abbreviations  ************************
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\begin{document}
\begin{titlepage}
\vspace{2cm}

\phantom{.}
\begin{center}
\large{THE NON-SINGLET
QCD EVOLUTION KERNELS
IMPROVED BY RENORMALON CHAIN CONTRIBUTIONS}\\[0.5cm]

S.~V.~Mikhailov
\footnote{E-mail: mikhs@thsun1.jinr.dubna.su}\\[0.5cm]

{\it Joint Institute for Nuclear Research,
Bogoliubov Laboratory of Theoretical Physics,\\
141980, Moscow Region, Dubna, Russia}\\[0.5cm]


\end{center}
\begin{abstract}
Closed expressions are presented for the contributions to QCD
non-singlet forward evolution kernels $P(z)$ for
the DGLAP equation and to $V(x,y)$ for
non-forward (ER-BL) evolution equation
for a certain class of diagrams which include renormalon chains.
Calculations are performed in covariant $\xi$-gauge, in the \MSbar scheme.
The assumption of ``naive nonabelianization" approximation for kernel
calculations is discussed, and a special choice of the gauge parameter
$\xi=-3$ is analyzed in this context.
Partial solutions to the ER-BL evolution equation are obtained.
\end{abstract}

\vspace {0.5cm}
PACS: 12.38.Cy, 12.38.-t, 13.60.Hb \\
Keywords: DGLAP, forward and nonforward evolution kernels, anomalous
dimensions, multiloop calculation

\vspace{8cm}
\begin{center}
{\sl To be published in the proceedings of XI International Seminar
 ``Quarks'98", Suzdal, Russia, May, 18-24, 1998}
\end{center}

\end{titlepage}

\section{Introduction}
Evolution kernels are the main ingredients of the well-known evolution
equations for parton distribution of DIS processes and for parton wave
functions in hard exclusive reactions.
These equations describe the dependence of parton distribution functions
and parton wave functions on the renormalization parameter $\mu^2$.
Here I  discuss the diagrammatic analysis and multiloop
calculation of the forward DGLAP evolution kernel $P(z)$ \cite{L75} and
non-forward Efremov-Radyushkin--Brodsky-Lepage (ER-BL) kernel $V(x,y)$
\cite{BL80} in a class of ``all-order"
approximation of the perturbative QCD.
The regular method of calculation and resummation of certain
classes of  diagrams for
these kernels has been suggested in ~\cite{M97}. These diagrams
include the chains of one-loop self-energy parts (renormalon chains) into
the one-loop diagrams (see Fig. 1).
Here the results for both the kinds of kernels (DGLAP and ER-BL),
obtained earlier in the framework of a scalar model in six dimensions
with the Lagrangian
 $\Ds L_{int} = g\sum^{N_f}_i \left( \psi^{*}_i \psi_i \varphi \right)
_{(6)}$
with the scalar ``quark" flavours ($\psi_i$) and
``gluon" ($\varphi$),
are extended to the non-singlet QCD kernels.
For the readers convenience
some important results of the paper
~\cite{M97} would be reminded.

The insertion of the chain into ``gluon" line (``chain-1" in \cite{M97})
of the diagram in Fig.1 a,b and resummation over all bubbles lead to the
transformation of the one-loop kernel ( see, \eg, \cite{MR86})
$aP_0(z)= a\z \equiv a(1-z)$
into the kernel $P^{(1)}(z; A)$
\ba \la{IntA}
\Ds aP_0(z)= a\z \stackrel{chain-1}{\longrightarrow} P^{(1)}(z; A) = a\z
\left[ (z)^{-A}(1-A) \frac{\gamma_{\varphi}(0)}{\gamma_{\varphi}(A)} \right];
~\mbox{where}~A=a N_f \gamma_{\varphi}(0), ~a=\frac{g^2}{(4\pi)^{3}}.
\ea
Here, $\gamma_{ \psi(\varphi)}(\varepsilon)$
are the one-loop coefficients of the anomalous dimensions of
quark (gluon at $N_f=1$) fields in D-dimension ($D=6-2\varepsilon$)
discussed in \cite{M97}; for the scalar model
$\gamma_{\psi}(\varepsilon)=
\gamma_{\varphi}(\varepsilon)=B(2-\varepsilon, 2-\varepsilon)
C(\varepsilon)$,
and $C(\varepsilon)$ is a scheme-dependent factor corresponding
to a certain choice of an \MSbar--like scheme.
The argument $A$ of the function
$\gamma_{\varphi}(A)$ in (\ref{IntA}) is
the standard anomalous dimension (AD) of a gluon field. So, one can
conclude that the ``all-order" result in (\ref{IntA}) is completely
determined by the single quark bubble diagram.
The resummation of this ``chain-1" subseries into an analytic function
in $A$ shouldn't be taken by surprise. Really, the considered problem
can be connected
with the calculation of large $N_f$ asymptotics of the AD's
in order of $1/N_f$. An approach was suggested by A. Vasil'ev and
collaborators at the beginning of 80' ~\cite{VPH81} to calculate
the renormalization-group functions in this limit,
they used the conformal properties of the theory
at the critical point $g=g_c$ corresponding to the non-trivial
zero $g_c$ of the D-dimensional $\beta$-function. This approach has been
extended by J. Gracey for calculation of the AD's of
the composite operators of
DIS in QCD in any order $n$ of PT, ~\cite{Gr94}.
I have used another approach,
which is close to \cite{P-M-P84};
contrary to the large $N_f$ asymptotic method, it does not appeal
to the value of parameters $N_fT_R$,
$C_A/2$ or $C_F$, associated in QCD with different kinds of loops.
To illustrate this feature, let us consider the insertions of
chains of one-loop self-energy parts into the ``quark" line of
diagram Fig.1a (``chain-2" in \cite{M97}).
Contributions of these diagrams, calculated in the framework of the
above scalar model,
 do not contain the parameter
$N_f$ and can be summarized into the kernel $P^{(2)}(z; B)$ \cite{M97}
\ba \la{IntB}
\Ds aP_0(z)= a\z \stackrel{chain-2}{\longrightarrow} P^{(2)}(z; B) = a\z
\left(1+B \frac{d}{dB} \right)
\left[ (\bar z)^{-B} \frac{\gamma_{\psi}(0)}{\gamma_{\psi}(B)} \right];
~\mbox{where}~B=a \gamma_{\psi}(0),
\ea
according to the same approach.
Following this way, the ``improved''
QCD kernel  $P^{(1)}(z; A)$ has been obtained in \cite{M97} for the
case of quark or gluon bubble chain insertions in the Feynman gauge
and in \cite{MS98} for general case of the mixed insertions
in $\xi$-- gauge.

In this talk, we present the QCD results similar to Eq.(\ref{IntA}),
for each type of
diagrams appearing in the covariant $\xi$-- gauge for the
DGLAP non-singlet kernel $P(z; A)$. The analytic properties
of the function $P(z; A)$ in variable $A$ are analyzed. The assumption
of ``Naive Nonabelianization'' (NNA) approximation \cite{BrGr95} for
the kernel calculation \cite{GK97} is discussed and its deficiency
is demonstrated. The ER-BL evolution kernel $V(x,y)$ is obtained in the
same approximation as the DGLAP kernel, by using the exact relations
between $P$ and $V$ kernels  \cite{MR85,M97} for a class
of ``triangular diagrams'' in Fig. 1. The considered class
of diagrams represents the leading $N_f$ contributions to both
kinds of kernels. The partial solutions for the ER-BL equation are
derived (compare with \cite{GK97, BeMul97}).

\section{Triangular diagrams for the DGLAP evolution kernel}
Here, the results of the bubble chain resummation for QCD
diagrams in Fig.1 a,b,c for the DGLAP kernel are discussed.
These classes of diagrams generate contributions
$\sim \as \left(\as \ln[1/z] \right)^n$
in any order $n$ of PT.
Based on the resummation method of Ref.~\cite{M97} in the QCD version,
one can derive the kernels $P^{(1a,b,c)}$
(corresponding to the diagrams in Fig.1) in the covariant
$\xi-$gauge\footnote{The gauge parameter $\xi$ is defined via the
gluon propagator in lowest order $\Ds i D_{\mu \nu}(k^2) =
\frac{-i \delta^{ab}}{k^2+i\epsilon}
\left(g_{\mu \nu} + (\xi-1)\frac{k_{\mu}k_{\mu}}{k^2} \right)$}
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\caption{The diagrams in figs. 1a -- 1c are the ``triangular"
 diagrams for the QCD DGLAP kernel;
 dashed line for gluons,  solid line for quarks;
 black circle denotes the sum of all kinds of the one-loop insertions
 (dashed circle), both quark and gluon (ghost) or mixed chains;
 MC denotes the mirror--conjugate diagram; 1d is an example of diagram
 for the non-forward ER-BL kernel.}
\label{fig:diagr}
\end{figure}
\vspace*{-8mm}

\ba \la{Pqcd_a}
P^{(1a)}(z; A)& =& \as C_F 2\z \cdot (1-A)^2 z^{-A}
\frac{\gamma_g(0)}{\gamma_g(A)} -
\as C_F\cdot \delta(1-z) \left(\frac1{(1-A)}
\frac{\gamma_g(0)}{\gamma_g(A)} -\xi \right), \\
 \la{Pqcd_b}
P^{(1b)}(z; A) &=& \as C_F 2 \cdot \left(\frac{2 z^{1-A}}{1-z}
\frac{\gamma_g(0)}{\gamma_g(A)} \right)_{+},  \\
 \la{Pqcd_c}
P^{(1c)}(z; A) &=& \as C_F  \cdot  \delta(1-z)
\left(\frac{A(3-2A)}{(2-A)(1-A)}\frac{\gamma_g(0)}{\gamma_g(A)}
- \xi \right),
\ea
where $\Ds \as=\frac{\alpha_s}{4\pi}$,
~$\Ds C_F= (N_c^2-1)/2N_c$,~$C_A=N_c$
and $\Ds T_R=\frac1{2}$ are the Casimirs of SU($N_c$) group,
and $A=-\as \gamma_g(0)$.
The function $\gamma_g(\varepsilon)$ is the one-loop coefficient of the
anomalous dimension of gluon field in D-dimension, here
$D=4-2\varepsilon$. In other words, it is the
coefficient $Z_1(\varepsilon)$ of a
simple pole in the expansion of the gluon field renormalization constant
$Z$, that includes both its finite part and all the powers
of the $\varepsilon$-expansion.
The function $\gamma_g(\varepsilon)$ thus defined is an analytic function in
the variable $\varepsilon$ by construction.
Equations (\ref{Pqcd_a}) -- (\ref{Pqcd_c}) are valid for any kind
of insertions, \ie, $\gamma_g = \gamma_g^{(q)}$ for the quark loop,
$\gamma_g = \gamma_g^{(g)}$
for the gluon (ghost) loop, or for their sum
$$
\gamma_g(A, \xi) = \gamma_g^{(q)}(A) + \gamma_g^{(g)}(A, \xi);
$$
when both kinds of insertions are taken into account.
The sum of contributions (\ref{Pqcd_a}),
(\ref{Pqcd_b}),
(\ref{Pqcd_c})  results in $P^{(1)}(z; A, \xi)$ which has the
expected ``plus form"
\ba \la{Psum}
\Ds P^{(1)}(z; A, \xi) &=&  \as C_F 2 \cdot
\left[\z z^{-A}(1-A)^2 + \frac{2 z^{1-A}}{1-z} \right]_{+}
\frac{\gamma_g(0, \xi)}{\gamma_g(A, \xi)},\\
\Ds \as P_{0}(z) &=&  \as C_F 2 \cdot
\left[\z ~~~+ ~~~\frac{2 z}{1-z} \right]_{+},
\ea
here, for comparison with (\ref{Psum}), the one-loop result $\as P_0(z)$
is written down, the latter can be
obtained as the limit $P^{(1)}(z; A \to 0, \xi)$.
Note that in (\ref{Psum}) the $\delta(1-z)$ - terms are exactly
accumulated in the form of the $[\ldots]_{+}$ prescription,
 and the $\xi$ - terms successfully cancel.
This is due to the evident current conservation for the case of quark
bubble insertions, including the gluon bubbles into consideration
merely modifies
the effective AD, $\gamma_g(A, \xi) \to \gamma_g^{(q)}(A)$,
conserving the structure of the result (\ref{Psum}), see ~\cite{M97,MS98}.
Substituting  the well-known
expressions for $\gamma_g(\varepsilon)$ from the quark or gluon (ghost)
 loops (see, \eg, \cite{IZ})
\ba \la{Piq}
\gamma_g^{(q)}(\varepsilon) &=& -8 N_f T_R B(D/2,D/2) C(\varepsilon),\\
\gamma_g^{(g)}(\varepsilon, \xi)& =&  \frac{C_A}{2} B(D/2-1,D/2-1) \left(
\left(\frac{3D-2}{D-1}\right) +\right. \nn \\
&&  ~~~~~~~~\left.(1-\xi)(D-3) +
 \left(\frac{1-\xi}{2} \right)^2 \varepsilon \right) C(\varepsilon),
\ea
into the general formulae (\ref{Pqcd_a}) -- (\ref{Pqcd_c}),
and (\ref{Psum})
one can obtain $P^{(1)}(z; A, \xi)$ for both the quark and
the gluon loop insertions simultaneously. Here, the coefficient
$C(\varepsilon)=\Gamma(1-\varepsilon)\Gamma(1+\varepsilon)$ implies a
certain choice of the \MSbar scheme where every loop integral is
multiplied by the scheme factor $\Gamma(D/2-1)(\mu^2/4\pi)^{\varepsilon}$.
The renormalization scheme dependence of $P^{(1)}(z; A)$
is accumulated by the factor $C(\varepsilon)$
\footnote{For another popular definition of a minimal scheme, when a
scheme factor is chosen as
$\exp(c \cdot \varepsilon), ~c=-\gamma_E+ \ldots$ instead of
 $\Gamma(D/2-1)$, the coefficient $C(\varepsilon)$ does not contain
any scheme ``traces''
in final expressions for the renormalization-group functions.}.
Of course, the final result (\ref{Psum}) will be gauge-dependent
in virtue of
the evident gauge dependence of the gluon loop contribution
$\gamma_g^{(g)}(\varepsilon, \xi)$, in this case, \eg,
\ba \la{A}
A(\xi)=-\as \gamma_g(0, \xi) = -\as
\left( \gamma_g^{(g)}(0, \xi) +  \gamma_g^{(q)}(0)\right)= -\as
\left[\left(\frac5{3}+\frac{(1-\xi)}{2}\right)C_A
 -  \frac4{3} N_f T_R \right],
\ea
is the contribution to the one-loop renormalization of the gluon
field. The positions of zeros of the function $\gamma_g(A, \xi)$ in $A$,
which manifest the poles of $P(z; A, \xi)$, also depend on $\xi$. The
kernel  $P^{(1)}(z; A, \xi)$ became gauge-invariant in the case when only
the quark insertions are involved, \ie, ~$\gamma_g^{(q)} \to \gamma_g$;
~$\Ds A=A^{(q)}=-\as \gamma_g^{(q)}(0) = \as \frac{4}{3} T_R N_f$, and
$P^{(1)}(z; A^{(q)}) \to P^{(1)}(z; A, \xi)$ as it
was presented in \cite{M97}. It is instructive to consider this case in
detail. To this end, let us choose the common factor
$\gamma_g^{(q)}(0) /\gamma_g^{(q)}(A)$ in formula (\ref{Psum}) for
the crude measure of modification of the kernel in comparison
with the one-loop result $\as P_0(z)$. Considering the curve of this
factor in the argument $A$ in Fig.2, one may conclude:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       %%%%%%%    %%%%%        %%%%%%    %%%%%   %     %
%       %      %  %             %     %  %     %   %   %
%       %      %  %             %     %  %     %    % %
%       %%%%%%%    %%%%%        %%%%%%   %     %     %
%       %               %       %     %  %     %    % %
%       %               %       %     %  %     %   %   %
%       %         %%%%%%        %%%%%%    %%%%%   %     %
%
%       By Jean Orloff
%       Comments & suggestions by e-mail: ORLOFF@surya11.cern.ch
%       No modification of this file allowed if not e-sent to me.
%
% WHAT IS IT:
% psbox is a set of machine-independent TeX macros to
% 1) allow (Encapsulated) PostScript figure inclusion in all versions
%    of TeX (Plain, LaTeX) on all machines using a PostScript printer
% 2) facilitate the communication (e-mail, ftp, ...) of all the files
%    (text, macros, figs) needed to reproduce a TeX document by grouping
%    them together into a single, TeXable file.
% For more info, get the file pub/TeX/psbox/PSBOXALL.TEX by anonymous
%     ftp from cs.nyu.edu(=128.122.140.24)
%
% History:
%  1.34  \readfilename=final fix for all filename scans; try \psforptips
%  1.33: corrects \psnewinput for LaTeX (still fails if fname=a{b}c)
%  1.32: corrects \psfordvialw and adds .TEX to PSBOXALL(!)
%  1.31: adds \psfordvialw(?)
%  1.30: adds \splitfile & \joinfiles for multi-file management
%  1.24: fix error handling & add \psonlyboxes
%  1.22: makes \drawingBox \global for use in Phyzzx
%  1.21: accepts %%BoundingBox: (atend)
%  1.20: tries to add \psfordvitps for the TeXPS package.
%  1.10: adds \psforoztex, error handling...
%2345678 1 2345678 2 2345678 3 2345678 4 2345678 5 2345678 6 2345678 7 23456789
%
% Checking version no to avoid multiple loadings
\def\temp{1.34}%
\let\tempp=\relax
\expandafter\ifx\csname psboxversion\endcsname\relax
  \message{PSBOX(\temp) loading}%
\else
    \ifdim\temp cm>\psboxversion cm
      \message{PSBOX(\temp) loading}%
    \else
      \message{PSBOX(\psboxversion) is already loaded: I won't load
        PSBOX(\temp)!}%
      \let\temp=\psboxversion
      \let\tempp=\endinput
    \fi
\fi
\tempp
\let\psboxversion=\temp
\catcode`\@=11
% Every macro likes a little privacy...
%
%Trying to tame the variety of \special commands for Postscript: the
%  universal internal command \PSspeci@l##1##2 takes ##1 to be the
%  filename and ##2 to be the integer scale factor*1000 (as for usual
%   TeX \scale commands)
%
\def\psfortextures{%     For TeXtures on the Macintosh
%-----------------
\def\PSspeci@l##1##2{%
\special{illustration ##1\space scaled ##2}%
}}%
\def\psfordvitops{%      For the DVItoPS converter on IBM mainframes
%----------------
\def\PSspeci@l##1##2{%
\special{dvitops: import ##1\space \the\drawingwd \the\drawinght}%
}}%
\def\psfordvips{%      For DVIPS converter on VAX, UNIX and PC's
%--------------
\def\PSspeci@l##1##2{%
%    \special{/@scaleunit 1000 def}% never read dox without trying!
\d@my=0.1bp \d@mx=\drawingwd \divide\d@mx by\d@my% BUG! for large \drawingwd
\special{PSfile=##1\space llx=\psllx\space lly=\pslly\space%
urx=\psurx\space ury=\psury\space rwi=\number\d@mx
}}}%
\def\psforoztex{%        For the OzTeX shareware on the Macintosh
%--------------
\def\PSspeci@l##1##2{%
\special{##1 \space
      ##2 1000 div dup scale
      \number-\psllx\space \number-\pslly\space translate
}}}%
\def\psfordvitps{%       From the UNIX TeXPS package, vers.>3.12
%---------------
% Convert a dimension into the number \psn@sp (in scaled points)
\def\psdimt@n@sp##1{\d@mx=##1\relax\edef\psn@sp{\number\d@mx}}
\def\PSspeci@l##1##2{%
% psfig.psr contains the def of "startTexFig": if you can locate it
% and include the correct pathname, it should work
\special{dvitps: Include0 "psfig.psr"}% contains def of "startTexFig"
\psdimt@n@sp{\drawingwd}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\drawinght}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psllx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\pslly bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psurx bp}
\special{dvitps: Literal "\psn@sp\space"}
\psdimt@n@sp{\psury bp}
\special{dvitps: Literal "\psn@sp\space startTexFig\space"}
\special{dvitps: Include1 "##1"}
\special{dvitps: Literal "endTexFig\space"}
}}%
\def\psfordvialw{%   Try for dvialw, a UNIX public domain
%---------------
\def\PSspeci@l##1##2{
\special{language "PostScript",
position = "bottom left",
literal "  \psllx\space \pslly\space translate
  ##2 1000 div dup scale
  -\psllx\space -\pslly\space translate",
include "##1"}
}}%
\def\psforptips{%   For MS-DOS; LUOMA@brandeis.bitnet
%---------------
\def\PSspeci@l##1##2{{
\d@mx=\psurx bp
\advance \d@mx by -\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\incm{\d@mx}
\let\tmpx\dimincm
\d@my=\psury bp
\advance \d@my by -\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\incm{\d@my}
\let\tmpy\dimincm
\d@mx=-\psllx bp
\divide \d@mx by 1000\multiply\d@mx by \xscale
\d@my=-\pslly bp
\divide \d@my by 1000\multiply\d@my by \xscale
\at(\d@mx;\d@my){\special{ps:##1 x=\tmpx, y=\tmpy}}
}}}%
\def\psonlyboxes{%     Draft-like behaviour if none of the others works
%---------------
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1})}}\hss}}}}
}}%
\def\psloc@lerr#1{%
\let\savedPSspeci@l=\PSspeci@l%
\def\PSspeci@l##1##2{%
\at(0cm;0cm){\boxit{\vbox to\drawinght
  {\vss\hbox to\drawingwd{\at(0cm;0cm){\hbox{({\tt##1}) #1}}\hss}}}}
\let\PSspeci@l=\savedPSspeci@l% restore normal output for other figs!
}}%
%\def\psfor...  add your own!
%
% Some common defs
%
\newread\pst@mpin
\newdimen\drawinght\newdimen\drawingwd
\newdimen\psxoffset\newdimen\psyoffset
\newbox\drawingBox
\newcount\xscale \newcount\yscale \newdimen\pscm\pscm=1cm
\newdimen\d@mx \newdimen\d@my
\newdimen\pswdincr \newdimen\pshtincr
\let\ps@nnotation=\relax
{\catcode`\|=0 |catcode`|\=12 |catcode`|%=12 |catcode`~=12
|catcode`#=12 |catcode`*=14
|xdef|backslashother{\}*
|xdef|percentother{%}*
|xdef|tildeother{~}*
|xdef|sharpother{#}*
}%
% useful to display special chars in \tt; fails for \,#,%
\def\R@moveMeaningHeader#1:->{}%
\def\uncatcode#1{%
\edef#1{\expandafter\R@moveMeaningHeader\meaning#1}}%
%
\def\execute#1{#1}% NOT stupid: cs in #1 are then identified BEFORE execution
\def\psm@keother#1{\catcode`#112\relax}% borrowed from latex
\def\executeinspecs#1{%
\execute{\begingroup\let\do\psm@keother\dospecials\catcode`\^^M=9#1\endgroup}}%
\def\@mpty{}%
% \if\matchin#1#2<=> \iftrue if #1 contains #2, <=>\iffalse otherwise:
% \if\matchexpin: idem, but #1 & #2 are first fully expanded (no \if
% inside!)
% \tmpa & \tmpb contain what's before and after the occurence of #2
\def\matchexpin#1#2{
  \fi%
%\message{(#1>#2)}
  \edef\tmpb{{#2}}%
  \expandafter\makem@tchtmp\tmpb%
  \edef\tmpa{#1}\edef\tmpb{#2}%
  \expandafter\expandafter\expandafter\m@tchtmp\expandafter\tmpa\tmpb\endm@tch%
  \if\match%
}%
\def\matchin#1#2{%
  \fi%
  \makem@tchtmp{#2}%
  \m@tchtmp#1#2\endm@tch%
  \if\match%
}%
\def\makem@tchtmp#1{\def\m@tchtmp##1#1##2\endm@tch{%
  \def\tmpa{##1}\def\tmpb{##2}\let\m@tchtmp=\relax%
  \ifx\tmpb\@mpty\def\match{YN}%
  \else\def\match{YY}\fi%
}}%
% converts any dimen in cm, with 1E-4 cm precision
\def\incm#1{{\psxoffset=1cm\d@my=#1
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\number\d@mx.}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \edef\dimincm{\dimincm\number\d@mx}
  \advance\d@my by -\number\d@mx cm
  \multiply\d@my by 100
 \d@mx=\d@my
  \divide\d@mx by \psxoffset
  \xdef\dimincm{\dimincm\number\d@mx}
}}%
%
%  \ReadPSize{PSfilename} reads the dimensions of a PostScript drawing
%      and stores it in \drawinght(wd)
\newif\ifNotB@undingBox
\newhelp\PShelp{Proceed: you'll have a 5cm square blank box instead of
your graphics (Jean Orloff).}%
\def\s@tsize#1 #2 #3 #4\@ndsize{
  \def\psllx{#1}\def\pslly{#2}%
  \def\psurx{#3}\def\psury{#4}%  needed by a crazyness of dvips!
  \ifx\psurx\@mpty\NotB@undingBoxtrue% this is not a valid one!
  \else
    \drawinght=#4bp\advance\drawinght by-#2bp
    \drawingwd=#3bp\advance\drawingwd by-#1bp
%  !Units related by crazy factors as bp/pt=72.27/72 should be BANNED!
  \fi
  }%
\def\sc@nBBline#1:#2\@ndBBline{\edef\p@rameter{#1}\edef\v@lue{#2}}%
\def\g@bblefirstblank#1#2:{\ifx#1 \else#1\fi#2}%
{\catcode`\%=12
\xdef\B@undingBox{%%BoundingBox}}%
%% is not a true comment in PostScript, even if % is!
\def\ReadPSize#1{
 \readfilename#1\relax
 \let\PSfilename=\lastreadfilename
 \openin\pst@mpin=#1\relax
 \ifeof\pst@mpin \errhelp=\PShelp
   \errmessage{I haven't found your postscript file (\PSfilename)}%
   \psloc@lerr{was not found}%
   \s@tsize 0 0 142 142\@ndsize
   \closein\pst@mpin
 \else
% each entry in \GlobalInputList should be unique
   \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
   \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
     \immediate\write\psbj@inaux{\lastreadfilename,}%
   \fi%
   \loop
     \executeinspecs{\catcode`\ =10\global\read\pst@mpin to\n@xtline}%
     \ifeof\pst@mpin
       \errhelp=\PShelp
       \errmessage{(\PSfilename) is not an Encapsulated PostScript File:
           I could not find any \B@undingBox: line.}%
       \edef\v@lue{0 0 142 142:}%
       \psloc@lerr{is not an EPSFile}%
       \NotB@undingBoxfalse
     \else
       \expandafter\sc@nBBline\n@xtline:\@ndBBline
       \ifx\p@rameter\B@undingBox\NotB@undingBoxfalse
         \edef\t@mp{%
           \expandafter\g@bblefirstblank\v@lue\space\space\space}%
         \expandafter\s@tsize\t@mp\@ndsize
       \else\NotB@undingBoxtrue
       \fi
     \fi
   \ifNotB@undingBox\repeat
   \closein\pst@mpin
 \fi
\message{#1}%
}%
%
% \psboxto(xdim;ydim){psfilename}: you specify the dimensions and
%    TeX uniformly scales to fit the largest one. If xdim=0pt, the
%    scale is fully determined by ydim and vice versa.
%    Notice: psboxes are a real vboxes; couldn't take hbox otherwise all
%    indentation and all cr's would be interpreted as spaces (hugh!).
%
\def\psboxto(#1;#2)#3{\vbox{%
   \ReadPSize{#3}%
   \advance\pswdincr by \drawingwd
   \advance\pshtincr by \drawinght
   \divide\pswdincr by 1000
   \divide\pshtincr by 1000
   \d@mx=#1
   \ifdim\d@mx=0pt\xscale=1000
         \else \xscale=\d@mx \divide \xscale by \pswdincr\fi
   \d@my=#2
   \ifdim\d@my=0pt\yscale=1000
         \else \yscale=\d@my \divide \yscale by \pshtincr\fi
   \ifnum\yscale=1000
         \else\ifnum\xscale=1000\xscale=\yscale
                    \else\ifnum\yscale<\xscale\xscale=\yscale\fi
              \fi
   \fi
   \divide\drawingwd by1000 \multiply\drawingwd by\xscale
   \divide\drawinght by1000 \multiply\drawinght by\xscale
   \divide\psxoffset by1000 \multiply\psxoffset by\xscale
   \divide\psyoffset by1000 \multiply\psyoffset by\xscale
   \global\divide\pscm by 1000
   \global\multiply\pscm by\xscale
   \multiply\pswdincr by\xscale \multiply\pshtincr by\xscale
   \ifdim\d@mx=0pt\d@mx=\pswdincr\fi
   \ifdim\d@my=0pt\d@my=\pshtincr\fi
   \message{scaled \the\xscale}%
 \hbox to\d@mx{\hss\vbox to\d@my{\vss
   \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
      \kern-\psyoffset
      \PSspeci@l{\PSfilename}{\the\xscale}%
      \vss}\hss\ps@nnotation}%
   \global\wd\drawingBox=\the\pswdincr
   \global\ht\drawingBox=\the\pshtincr
   \global\drawingwd=\pswdincr
   \global\drawinght=\pshtincr
   \baselineskip=0pt
   \copy\drawingBox
 \vss}\hss}%
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm %should not be necessary
}}%
%
% \psboxscaled{scalefactor*1000}{PSfilename} allows to bypass the
%   rounding errors of TeX integer divisions for situations where the
%   TeX box should fit the original BoundingBox with a precision
%   better
%   than 1/1000.
%
\def\psboxscaled#1#2{\vbox{%
  \ReadPSize{#2}%
  \xscale=#1
  \message{scaled \the\xscale}%
  \divide\pswdincr by 1000 \multiply\pswdincr by \xscale
  \divide\pshtincr by 1000 \multiply\pshtincr by \xscale
  \divide\psxoffset by1000 \multiply\psxoffset by\xscale
  \divide\psyoffset by1000 \multiply\psyoffset by\xscale
  \divide\drawingwd by1000 \multiply\drawingwd by\xscale
  \divide\drawinght by1000 \multiply\drawinght by\xscale
  \global\divide\pscm by 1000
  \global\multiply\pscm by\xscale
  \global\setbox\drawingBox=\hbox to 0pt{\kern\psxoffset\vbox to 0pt{%
     \kern-\psyoffset
     \PSspeci@l{\PSfilename}{\the\xscale}%
     \vss}\hss\ps@nnotation}%
  \advance\pswdincr by \drawingwd
  \advance\pshtincr by \drawinght
  \global\wd\drawingBox=\the\pswdincr
  \global\ht\drawingBox=\the\pshtincr
  \global\drawingwd=\pswdincr
  \global\drawinght=\pshtincr
  \baselineskip=0pt
  \copy\drawingBox
  \global\psxoffset=0pt
  \global\psyoffset=0pt
  \global\pswdincr=0pt
  \global\pshtincr=0pt % These are local to one figure
  \global\pscm=1cm
}}%
%
%  \psbox{PSfilename} makes a TeX box having the minimal size to
%      enclose the picture
\def\psbox#1{\psboxscaled{1000}{#1}}%
%------------------------------------------------------
%  \joinfiles file1, file2, ...n \into joinedfilename .
%     makes one file out of many
%  \splitfile joinedfilename
%     the opposite
\newif\ifn@teof\n@teoftrue
\newif\ifc@ntrolline
\newif\ifmatch
\newread\j@insplitin
\newwrite\j@insplitout
\newwrite\psbj@inaux
\immediate\openout\psbj@inaux=psbjoin.aux
\immediate\write\psbj@inaux{\string\joinfiles}%
\immediate\write\psbj@inaux{\jobname,}%
%
% INPUT REDEFINITION
%
% works if #1 is a single character
\def\toother#1{\ifcat\relax#1\else\expandafter%
  \toother@ux\meaning#1\endtoother@ux\fi}%
\def\toother@ux#1 #2#3\endtoother@ux{\def\tmp{#3}%
  \ifx\tmp\@mpty\def\tmp{#2}\let\next=\relax%
  \else\def\next{\toother@ux#2#3\endtoother@ux}\fi%
\next}%
%
% \readfilename defs:
%
\let\readfilenamehook=\relax
\def\re@d{\expandafter\re@daux}% spares typing 10 \expandafter's...
\def\re@daux{\futurelet\nextchar\stopre@dtest}%
\def\re@dnext{\xdef\lastreadfilename{\lastreadfilename\nextchar}%
  \afterassignment\re@d\let\nextchar}%
\def\stopre@d{\egroup\readfilenamehook}%
\def\stopre@dtest{%
  \ifcat\nextchar\relax\let\nextread\stopre@d
  \else
    \ifcat\nextchar\space\def\nextread{%
      \afterassignment\stopre@d\chardef\nextchar=`}%
    \else\let\nextread=\re@dnext
      \toother\nextchar
      \edef\nextchar{\tmp}%
    \fi
  \fi\nextread}%
\def\readfilename{\bgroup%
  \let\\=\backslashother \let\%=\percentother \let\~=\tildeother
  \let\#=\sharpother \xdef\lastreadfilename{}%
  \re@d}%
%
% redefines \input using \readfilename
%
\xdef\GlobalInputList{\jobname}%
\def\psnewinput{%
  \def\readfilenamehook{% each entry in \GlobalInputList should be unique
    \if\matchexpin{\GlobalInputList}{, \lastreadfilename}%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\unitlength 1mm
\begin{picture}(10,10)
 \put(25,-100){\psboxscaled{500}{beta.ps}}
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\vspace{65mm}
 \label{fig:beta}
  \caption{The curve of the factor $\gamma_g(0)/\gamma_g(A)$,
           the arrow on the picture corresponds to the point
   $A=1/\pi ~(\ie ~\alpha_s=6/N_f)$, the first singularity appears
           at $A=A_0=5/2$.}
    \end{figure}
%***********************************************************************

(i)~the range of convergence of PT series corresponds to the left zero
 of the function $\gamma^{(q)}_g(A)$ and is equal to
 $ A_0 = 5/2,$ that corresponds to $\alpha_{s}^0 = 15\pi /N_f$, so,
 this range looks very broad
 \footnote{Here we consider the evolution kernel $P(z, A)$ by itself.
We take out of the scope that the factorization scale $\mu^2$ of
hard processes would be chosen large enough, $\mu^2 \geq m_{\rho}^2$,
where the $\rho$--meson mass $m_{\rho}$ represents the characteristic
hadronic scale. Following this reason, the used coupling
$\alpha_s(\mu^2)$ could not be too large.},
  $\alpha_s < 5\pi$ at $N_f=3$;

(ii) in spite of a wide range of PT fidelity, the resummation into
 $P^{(1)}_q(z; A)$ is substantial -- two zeros of the function
 $P^{(1)}_q(z; A)$ in $A$ appear within the range of convergence
(it depends on a certain \MSbar scheme);

(iii) the factor $\gamma_g^{(q)}(0) /\gamma_g^{(q)}(A)$ decays quickly
with the growth of the argument $A$.
Really, if we take the naive boundary of
the standard PT applicability, $\alpha_s =1$
at ~$N_f=3$, $~A^{(q)}=1/(2\pi)$, then this factor falls
approximately to
~$ 0.7$, at $N_f=6$, $~A^{(q)}=1/\pi$ it falls to 0.5, see arrow in
Fig. 2; thus, the resummation is numerically important in this range.

Note at the end that Eq.(\ref{Psum}) could not provide valid
asymptotic behavior of the kernels for $z \to 0$.
A similar $z$-behavior is determined by the double-logarithmic
corrections which are most singular at
zero, like \as $\left(\as \ln^2[z] \right)^n$~\cite{bv96}.
These contributions appear due to
renormalization of the composite operator in the diagrams
by ladder graphs, \etc
rather than by the triangular ones.

\section{Analysis of the NNA assumption for kernel calculations }
The expansion of $P^{(1)}_q(z; A)$ in $A$ provides
the leading $\as \left(\as N_f \ln[1/z]\right)^n$ dependence of the
kernels with a large number $N_f$ in any order $n$ of PT \cite{M97}.
But these contributions do not numerically dominate for real numbers
of flavours $N_f=4, ~5, ~6$. That may be verified by comparing
the total numerical results for the 2-- and 3--loop AD's of composite
operators (ADCO) presented in
\cite{LRV94} with their $N_f$-leading terms (see ADCO in Table 1).
Therefore, to obtain a satisfactory agreement at least with the second
order results, one should take into account the contribution
from {\bf subleading} $N_f$-terms.
As a first step, let us consider the contribution
from the completed renormalization of the gluon line -- it should
generate {\bf a part of subleading} terms.
Below we shall examine an exceptional  choice of the gauge parameter
$\xi = -3$.
~For this gauge the coefficient of one-loop gluon AD
$\gamma_g(0, -3)$ coincides with $b_0$, the one-loop coefficient of the
$\beta$-function \footnote{ Here, for the $\beta(\as)$-function
we adapt $\beta(\as)= -b_0 \as^2 + \ldots$,
$\Ds ~b_0 = \frac{11}{3}C_A - \frac4{3}N_fT_R$}.
Therefore this gauge may be used for a reformulation of the
so-called  \cite{BrGr95} NNA proposition  to kernel calculations.
Note, just this value of $\xi$ has been used in \cite{Ch96} to
estimate the total gluon contribution only from the gluon bubble
in order of $\as^2$ to the process of $e^+ ~e^-$ annihilation.

 \begin{center}
  {\bf\Large Table $1$.}
  \end{center}
 The results of $\Gamma_{(1,2)}(n)$ calculations
 ( $\Gamma(n)=\int^1_0 dx x^n P(z)$)
performed in different ways, exact numerical results from [15]
and approximation obtained from $P(z, A, \xi)$ with  $\xi=-3$;
both numerical and analytical {\bf exact results}
are emphasized by the bold print.

\begin{tabular}{|c||c|c||c|c|c|}\hline
%&\multicolumn{2}{|c||}{} &\multicolumn{3}{|c|}{} \\
&\multicolumn{2}{|c||}{{\strut\vphantom{\vbox to 6mm{}}
$\Gamma_{(1)}(n)$ $_{\vphantom{\vbox to 4mm{}}}$}}&
\multicolumn{3}{|c|}{$\Gamma_{(2)}(n)$}\\
%&\multicolumn{2}{|c||}{} &\multicolumn{3}{|c|}{} \\
\cline{2-6}
%___________________________________________________________________
 &{\strut\vphantom{\vbox to 6mm{}}
 \hspace{0.1mm}$C_F C_A$\hspace{0.1mm}
   $_{\vphantom{\vbox to 4mm{}}}$}
   &\hspace{1mm}$N_f \cdot C_F$\hspace{1mm}
     &\hspace{1mm}$C_A^2 C_F$\hspace{1mm}
       &\hspace{1mm}$N_f \cdot C_FC_A$\hspace{1mm}
         &\hspace{1mm}$N_f^2 \cdot C_F$\hspace*{1mm}\\
\hline
%___________________________________________________________________
{\strut\vphantom{\vbox to 6mm{}} n=2
$_{\vphantom{\vbox to 4mm{}}}$}& & & & & \\
{\bf Exact }
 &{\bf 13.9}
   &
     &{\boldmath $\Ds 86.1 + 21.3 \ \zeta(3)$}
       &{\boldmath $\Ds -12.9-21.3 \ \zeta(3)$}
         &  \\
& &{\boldmath $\Ds -2.3704$} & &
&{\boldmath $\Ds -0.9218$} \\ %\hline
{\strut\vphantom{\vbox to 6mm{}}
 $\hspace{1mm} \xi=-3 \hspace{1mm}$
  $_{\vphantom{\vbox to 4mm{}}}$}
 &$11.3$
   &
     &$-42.0$
       &$12.9$
         & \\
\hline
%___________________________________________________________________
{\strut\vphantom{\vbox to 6mm{}}n=4$_{\vphantom{\vbox to 4mm{}}}$:}
& & & & & \\
{\bf Exact }
 &{\bf 23.9}
   &
     & {\boldmath $140.0 + 19.2 \ \zeta(3)$}
       &{\boldmath $-18.1 -41.9 \ \zeta(3)$}
         &  \\
& &{\boldmath $\Ds -4.9152$} & &
&{\boldmath $\Ds -1.5814$} \\ %\hline
{\strut\vphantom{\vbox to 6mm{}}
$\hspace{1mm} \xi=-3 \hspace{1mm}$ $_{\vphantom{\vbox to 4mm{}}}$}
 &$23.5$
   &
     &$-76.0$
       &$23.$
         &  \\
\hline
%___________________________________________________________________
{\strut\vphantom{\vbox to 6mm{}}n=6 $_{\vphantom{\vbox to 4mm{}}}$}
& & & & & \\
{\bf Exact }
 &{\bf 29.7}
   &
     &{\boldmath $173+  19.01 \ \zeta(3)$}
       &{\boldmath $-20.4 -54.0 \ \zeta(3) $}
         &  \\
& &{\boldmath $\Ds -6.4719$} & &
&{\boldmath $\Ds -1.9279$}  \\ %\hline
{\strut\vphantom{\vbox to 6mm{}}
$\hspace{1mm} \xi=-3 \hspace{1mm}$ $_{\vphantom{\vbox to 4mm{}}}$}
 &$31.1$
   &
     &$-95.6$
       &$28.5$
         &\\
\hline
%___________________________________________________________________
{\strut\vphantom{\vbox to 6mm{}}n=8
$_{\vphantom{\vbox to 4mm{}}}$} & & & & & \\
{\bf Exact }
 &{\bf 33.9}
   &
     &{\boldmath $196.9 + 18.98 \ \zeta(3)$}
       &{\boldmath $-21.9 - 62.7 \ \zeta(3)$}
         &  \\
& &{\boldmath $\Ds -7.6094$} & &
&{\boldmath $\Ds - 2.1619$} \\ %\hline
$\hspace{1mm} \xi=-3 \hspace{1mm}$
 &$36.3$
   &
     &$-109.0$
       &$32.3$
         &  \\
\hline
%___________________________________________________________________
{\strut\vphantom{\vbox to 6mm{}}
n=10 $_{\vphantom{\vbox to 4mm{}}}$}& & & & & \\
 {\bf Exact }
 &{\bf 37.27}
   &
     &{\boldmath $216.0 + 18.96 \ \zeta(3)$}
       &{\boldmath $-23.2 - 69.6 \ \zeta(3)$}
         & \\
& &{\boldmath $-8.5095$} & & &{\boldmath $ -2.3366$} \\ %\hline
$\hspace{1mm} \xi=-3 \hspace{1mm}$
 &$41.00$
   &
     &$-119.28$
       &$35.24$
         &  \\
\hline
%___________________________________________________________________
{\strut\vphantom{\vbox to 6mm{}}
n=12 $_{\vphantom{\vbox to 4mm{}}}$}& & & & & \\
 {\bf Exact }
 &{\bf 40.02}
   &
     &{\bf ?}
       &{\bf ?}
         & \\
& &{\boldmath $-9.2555$} & & &{\boldmath $ -2.4753$} \\ %\hline
$\hspace{1mm} \xi=-3 \hspace{1mm}$
 &$44.64$
   &
     &$-127.61$
       &$37.58$
         &  \\
\hline
%___________________________________________________________________
\end{tabular}
\vspace*{2mm}

To obtain the NNA result in a usual way, one should substitute the
coefficient
$b_0$ for $\gamma^{(q)}_g(0)$ into the expression for $A^{(q)}$ by
hand (see, \eg, \cite{GK97}).
Note, the use of such an NNA procedure to improve $P^{(1)}_q(z; A)$
leads to poor results even for $\as^2~P_1(z)$ term of the expansion; a
similar observation was also done in ~\cite{MMS97}.
The NNA trick expresses common hope that the main logarithmic
contribution
may follow from the renormalization of the coupling constant.
This renormalization appears as a sum of contributions from all
the sources of renormalization of \as, corresponding diagrammatic
analysis for two-loop kernels is presented in \cite{MR85,MR86}.
In the case of the $\xi=-3$ gauge the one-loop gluon renormalization
imitates the contributions from these other sources and the coefficient
$b_0$ appears naturally.

The expansion of kernel $P^{(1)}(z; A, \xi)$
generates partial kernels
$\as^2 P_{(1)}(z), \ \as^3P_{(2)}(z), \ldots$
which in their turn produce ADCO
$\as^2~\Gamma_{(1)}(n), \ \as^3~\Gamma_{(2)}(n), \ldots$
according to the relation $\Gamma(n) = \int^1_0 dz z^n P(z)$.
These elements of  ADCO and a few numerical exact results from
\cite{LRV94} are collected in Table 1, let us compare them:

(i) we consider there the contribution to the coefficient
$\Gamma_{(1)}(n)$ which is generated by the gluon loops
and associated with Casimirs $C_F C_A/2$, the ~$C_F^2$--term is missed, but
its contribution is  numerically insignificant.
It is seen that in this order the $C_F C_A$--terms are rather close
to exact values (the accuracy is about $10\%$ for $n > 2$)
and our approximation works rather well;

(ii) in the next order the contributions to $\Gamma_{(2)}(n)$
associated  with
the coefficients $N_f \cdot C_F C_A$ and $C_A^2 C_F$ are generated, while
the terms with the coefficients $C_F^3, ~N_f \cdot C_F^2 ,~C_F^2 C_A$ are
missed. In the third order, contrary to the previous item, all the
generated terms are opposite in sign to the exact values,
and the ``$\xi=-3$ approximation" doesn't work at all.

So, we need the next step to improve the agreement -- to obtain the
subleading $N_f$-terms by the {\bf exact calculation}.

\section{The non-forward ER-BL evolution kernel}
Here we present the results of the bubble resummation for the
ER-BL kernel $V(x,y)$. It can be derived in the same manner as it was
done for the DGLAP kernel $P(z)$, see Appendix A in \cite{M97}. On the
other hand $V(x,y)$ can be obtained as a ``byproduct" of the previous
results for  $P(z)$, \ie,  we shall use again \cite{M97} the exact
relations between the $V$ and $P$ kernels established in any order of PT
\cite{MR85} for triangular diagrams.
These relations were obtained by comparing counterterms
for the same triangular diagrams considered in ``forward", Fig.1a, and
``nonforward",  Fig.1d, kinematics.

Let the diagram in Fig.1a have a contribution to the DGLAP kernel
in the form $P(z)= p(z) + \delta(1-z) \cdot C$;
then its contribution to the ER-BL kernel (Fig.1d) is
\ba \la{PV1}
\Ds V(x,y) = {\cal C} \left(\theta(y > x) \int^{ \frac{x}{y}}_0
\frac{p(z)}{\bar z} dz \right)+ \delta(y-x) \cdot C,
\ea
where $ {\cal C} \equiv 1 + \left(x \to \bar x, y \to \bar y \right)$
to take into account the mirror-conjugate diagram.
From relation (\ref{PV1}) and Eqs. (\ref{Pqcd_a}), (\ref{Pqcd_c}) for
$P^{(1a,c)}$ we immediately derive the expression for the sum of
contributions $V^{(1a+1c)}$,
\ba \la{V1a}
\Ds V^{(1a+1c)}(x,y; A, \xi) = a_s C_F 2 \cdot  {\cal C}
\left[ \theta(y > x) (1-A)\left(\frac{x}{y}\right)^{1-A}
 - \frac1{2} \delta(y-x)
\frac{(1-A)}{(2-A)} \right]
\frac{\gamma_g(0, \xi)}{\gamma_g(A, \xi)},
\ea
that may naturally be represented in  the ``plus form".
Expression (\ref{V1a}) can be independently verified by other relations
reducing any
$V$ to $P$ \cite{MR85,DMRGH88} (see formulae for the $V \to P$ reduction
there) and we came back to the same Eqs.(\ref{Pqcd_a}), (\ref{Pqcd_c}) for
$P^{(1a,c)}$. Moreover,
the first terms of the Taylor expansion of $V^{(1a,c)}(x,y; A)$ in $A$
coincide with the results of the two-loop calculation in \cite{MR85}.
The relation $P \to V$ similar to Eq.(\ref{PV1}) has also been derived
for the diagram in Fig. 1b
\ba \la{PV11}
\Ds V^{(1b)}(x,y) =  {\cal C} \left[\theta(y > x)
 \frac1{2y}P^{(1b)}\left(\frac{x}{y} \right) \right]_+;
\ea
therefore, substituting Eq.(\ref{Pqcd_b}) into (\ref{PV11}) we obtain
\ba \la{V1b}
\Ds V^{(1b)}(x,y; A, \xi) = a_s C_F 2 \cdot  {\cal C}
\left[ \theta(y > x)
\left(\frac{x}{y}\right)^{1-A} \frac{1}{y-x} \right]_{+}
\frac{\gamma_g(0, \xi)}{\gamma_g(A, \xi)}.
\ea
Collecting the results in (\ref{V1a}) and (\ref{V1b}) we arrive at
the final expression for $V^{(1)}$ in the ``main bubbles'' approximation
\ba \la{Vsum}
V^{(1)}(x,y; A, \xi) = \as C_F 2 \cdot {\cal C} \left[ \theta(y > x)
\left( \frac{x}{y} \right)^{1-A} \left(1-A + \frac{1}{y-x} \right)
 \right]_{+} \frac{\gamma_g(0, \xi)}{\gamma_g(A, \xi)},
\ea
which has a ``plus form'' again due to the vector current conservation.
The contribution $V^{(1)}$ in (\ref{Vsum}) should dominate for
$N_f \gg 1$ in the
kernel $V$. Besides, the function $V^{(1)}(x,y;A, \xi)$ possesses an
important symmetry of its arguments $x$ and $y$. Indeed, the function
${\cal V}(x,y; A, \xi)=V^{(1)}(x,y; A, \xi) \cdot (\y y)^{1-A}$
is symmetrical under the change $x \leftrightarrow y$,
${\cal V}(x,y)={\cal V}(y,x)$. This symmetry allows us to obtain the
eigenfunctions $\psi_n(x)$ of the ``reduced'' evolution equation
\cite{MR86}
\ba \la{ev}
&& \int\limits_{0}^{1} V^{(1)}(x,y; A)\psi_n(y;A) dy= \Gamma(n;A)
\psi_n(x;A), \\
&&\Ds \psi_n(y;A) \sim (\y y)^{d_{\psi}(A) - \frac1{2}}
~C_{n}^{d_{\psi}(A)} (y-\y),
~\mbox{here}
~~d_{\psi}(A) = (D_A-1)/2, ~~D_A=4-2A, \la{solv1}
\ea
and $d_{\psi}(A)$ is the
effective dimension of the quark field when the
AD $A$ is taken into account; $C_{n}^{(\alpha)}(z)$ are the
Gegenbauer polynomials of an order of $\alpha$.
The partial solutions $\Phi(x; \as, l)$ of the
original ER-BL--equation ( where $l \equiv \ln(\mu^2/\mu_0^2)$)
\be \la{BL}
\left(\mu^2 \partial_{\mu^2} + \beta(\as)\partial_{\as}\right)
\Phi(x; \as, l) =
\int^1_0 V^{(1)}(x,y; A)~\Phi(y; \as, l) dy
\ee
are proportional to these eigenfunctions $\psi_n(x; A)$
for the special case $\beta(\as)=0$, see, \eg ~\cite{M97}.

In the general case $\beta (\as) \not=0$ let us start with an ansatz
for the partial solution of Eq.(\ref{BL}), $\Phi_n(x; \as, l)$
$\sim \chi_n(\as, l) \cdot \psi_n(x; A)$,
and the boundary condition is $\chi_n(\as, 0)=1$;
$\Phi_n(x; \as, 0) \sim \psi_n(x; A)$. For this ansatz
Eq.(\ref{BL}) reduces to
\be \la{BLn}
\left(\mu^2 \partial_{\mu^2} + \beta(\as)\partial_{\as}\right)
\ln\left(\Phi_n(x; \as, l)\right) = \Gamma(n; A).
\ee
In the case $n=0$ the AD of the vector current ~$\Gamma(0; A)=0$,
and the solution of the homogeneous equation in (\ref{BLn}) provides
the ``asymptotic wave function"
\be \la{asympsol}
\Phi_0(x; \as, l) = \psi_0(x; \bar{A})
\sim ((1-x) x)^{(1-\bar{A})},
\ee
where $\bar{A} = - \bar{a}_s(\mu^2) \gamma(0,\xi)$
 and
$\bar{a}_s(\mu^2)$ is the running coupling corresponding to $\beta (\as)$.
A similar solution has been discussed in ~\cite{GK97} in the framework
of the standard NNA approximation.
Solving simultaneously Eq. (\ref{BLn}) and
the renormalization-group equation for the coupling constant
$\bar{a}_s$ we arrive at
the partial solution $\Phi_n(x;\bar{\as}, l)$ in the form
\be \la{BLsol}
\Phi_n(x,\bar{\as}) \sim \chi_n(\mu^2) \cdot \psi_n(x;\bar A);
~\mbox{where} ~\chi_n(\mu^2) =
\exp
\left\{- \int^{\as(\mu^2)}_{\as(\mu_0^2)}
\frac{\Gamma(n,A)}{\beta(a)}da\right\}
\ee
Recently, a form of the solution $ \sim \psi_n(x; A)$
with  $A = - \as b_0$ has been confirmed in \cite{BeMul97} by the
consideration of conformal constraints \cite{Mul94} on the meson wave
functions in the limit $N_f \gg 1$.

\section{Conclusion}
In this paper, I present closed expressions in the ``all order"
approximation
for the DGLAP kernel $P(z)$ and ER-BL kernel
$V(x,y)$ appearing as a result of the resummation of a certain class
of QCD diagrams with the renormalon chain insertions. The contributions
from these diagrams, $P^{(1)}(z; A)$ and $V^{(1)}(z; A)$,
give the leading $N_f$ dependence of the kernels for a large
number of flavours $N_f \gg 1$.
These ``improved" kernels are generating functions to obtain
contributions to partial kernels like $\as^{(n+1)}P_{(n)}(z)$ in any
order $n$ of perturbation expansion.
Here $A \sim \as$ is a new expansion parameter that coincides
(in magnitude) with the anomalous dimension of the gluon field.
On the other hand, the method of calculation suggested in
~\cite{M97} does not depend
on the nature of self-energy insertions and does not appeal to
the value of the parameters $N_f T_R, ~C_A/2$ or ~$C_F$
associated with different loops. This allows us to obtain contributions
from chains with different kinds of self-energy insertions, both quark
and gluon (ghost) loops, see \cite{MS98}.
The prize for this generalization is
gauge dependence of the final results for $P^{(1)}(z; A)$ and
$V^{(1)}(z; A)$ on the gauge parameter $\xi$.

The result for the DGLAP non-singlet kernel  $P^{(1)}(z; A(\xi), \xi)$
is presented in (\ref{Psum}) in the covariant $\xi$-gauge.
The analytic properties of this kernel in the variable \as
are discussed for quark bubble chains only, and
in the general case for an exceptional gauge parameter $\xi= -3$.
For the latter case, $P^{(1)}(z; A(-3), -3)$ reproduces two-loop
anomalous dimensions $ \as^2 \Gamma_{(1)}(n)$ with good accuracy,
while the standard ``naive nonabelianization"  proposition
fails at this level.  But in the  third order in \as the ``$ \xi=-3$
approximation" is insufficient, see quantities $\Gamma_{(2)}(n)$ in
Table 1.

The contribution $V^{(1)}(x,y; A(\xi),\xi)$ to the non-forward ER-BL kernel
(\ref{Vsum}) is obtained for the same classes of diagrams as a
``byproduct" of the previous technique \cite{MS98,MR85}.
The partial solutions (\ref{asympsol}, \ref{BLsol}) to the ER-BL
equation are derived.

The obtained results are certainly useful for an independent check of
complicated computer calculations in higher orders of perturbation
theory (PT), similar to \cite{LRV94}; they may be a starting point for
further approximation procedures.
\vspace{2mm}

\centerline{\bf Acknowledgements}
\vspace{2mm}
The author is grateful to
Dr. M. Beneke, Dr. K. Chetyrkin, Dr. A. Grozin, Dr. D. Mueller,
Dr. A. Kataev, Dr. M. Polyakov, R. Ruskov and Dr.Dr. N. Stefanis
for fruitful discussions of the results, author would like to thank the
hosts of QUARKS'98 their extremely kind hospitality.
This investigation has been supported
by the Russian Foundation for Basic Research (RFBR) 98-02-16923.

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\end{document}
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