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%
%       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}%
    \else\xdef\GlobalInputList{\GlobalInputList, \lastreadfilename}%
      \immediate\write\psbj@inaux{\lastreadfilename,}%
    \fi%
    \ps@ldinput\lastreadfilename\relax%
    \let\readfilenamehook=\relax%
  }\readfilename%
}%
\expandafter\ifx\csname @@input\endcsname\relax    % then Plain
  \immediate\let\ps@ldinput=\input\def\input{\psnewinput}%
\else
  \immediate\let\ps@ldinput=\@@input
  \def\@@input{\psnewinput}%
\fi%
%
\def\nowarnopenout{%
 \def\warnopenout##1##2{%
   \readfilename##2\relax
   \message{\lastreadfilename}%
   \immediate\openout##1=\lastreadfilename\relax}}%
\def\warnopenout#1#2{%
 \readfilename#2\relax
 \def\t@mp{TrashMe,psbjoin.aux,psbjoint.tex,}\uncatcode\t@mp
 \if\matchexpin{\t@mp}{\lastreadfilename,}%
 \else
   \immediate\openin\pst@mpin=\lastreadfilename\relax
   \ifeof\pst@mpin
     \else
     \errhelp{If the content of this file is so precious to you, abort (ie
press x or e) and rename it before retrying.}%
     \errmessage{I'm just about to replace your file named \lastreadfilename}%
   \fi
   \immediate\closein\pst@mpin
 \fi
 \message{\lastreadfilename}%
 \immediate\openout#1=\lastreadfilename\relax}%
% % will have an unusual catcode below; use * instead
%\vbox
{\catcode`\%=12\catcode`\*=14
\gdef\splitfile#1{*
 \readfilename#1\relax
 \immediate\openin\j@insplitin=\lastreadfilename\relax
 \ifeof\j@insplitin
   \message{! I couldn't find and split \lastreadfilename!}*
 \else
   \immediate\openout\j@insplitout=TrashMe
   \message{< Splitting \lastreadfilename\space into}*
   \loop
     \ifeof\j@insplitin
       \immediate\closein\j@insplitin\n@teoffalse
     \else
       \n@teoftrue
       \executeinspecs{\global\read\j@insplitin to\spl@tinline\expandafter
         \ch@ckbeginnewfile\spl@tinline%Beginning-Of-File-Named:%\endcheck}*
       \ifc@ntrolline
       \else
         \toks0=\expandafter{\spl@tinline}*
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \fi
   \ifn@teof\repeat
   \immediate\closeout\j@insplitout
 \fi\message{>}*
}*
\gdef\ch@ckbeginnewfile#1%Beginning-Of-File-Named:#2%#3\endcheck{*
 \def\t@mp{#1}*
 \ifx\@mpty\t@mp
   \def\t@mp{#3}*
   \ifx\@mpty\t@mp
     \global\c@ntrollinefalse
   \else
     \immediate\closeout\j@insplitout
     \warnopenout\j@insplitout{#2}*
     \global\c@ntrollinetrue
   \fi
 \else
   \global\c@ntrollinefalse
 \fi}*
\gdef\joinfiles#1\into#2{*
 \message{< Joining following files into}*
 \warnopenout\j@insplitout{#2}*
 \message{:}*
 {*
 \edef\w@##1{\immediate\write\j@insplitout{##1}}*
\w@{% This collection of files was produced with CERN psbox package}*
\w@{% To decompose and tex it:}*
\w@{%-save this with a filename CONTAINING ONLY LETTERS and a .TEX}*
\w@{% extension (say, JOINTFIL.TEX), in some uncrowded directory;}*
\w@{%-make sure you can \string\input\space psbox.tex (version>=1.3);}*
\w@{%  (else ftp cs.nyu.edu(=128.122.140.24):pub/TeX/psbox/, then get}*
\w@{%  and tex the file psboxall.tex; more info in psbREAD.ME)}*
\w@{%-tex JOINTFIL.TEX using Plain, or LaTeX, or whatever is needed by}*
\w@{%  the first file in the joining (after splitting JOINTFIL.TEX into}*
\w@{%  it's constituents, TeX will try to process it as it stands).}*
\w@{\string\input\space psbox.tex}*
\w@{\string\splitfile{\string\jobname}}*
\w@{\string\let\string\autojoin=\string\relax}*
}*
 \expandafter\tre@tfilelist#1, \endtre@t
 \immediate\closeout\j@insplitout
 \message{>}*
}*
\gdef\tre@tfilelist#1, #2\endtre@t{*
 \readfilename#1\relax
 \ifx\@mpty\lastreadfilename
 \else
   \immediate\openin\j@insplitin=\lastreadfilename\relax
   \ifeof\j@insplitin
     \errmessage{I couldn't find file \lastreadfilename}*
   \else
     \message{\lastreadfilename}*
     \immediate\write\j@insplitout{%Beginning-Of-File-Named:\lastreadfilename}*
     \executeinspecs{\global\read\j@insplitin to\oldj@ininline}*
     \loop
       \ifeof\j@insplitin\immediate\closein\j@insplitin\n@teoffalse
       \else\n@teoftrue
         \executeinspecs{\global\read\j@insplitin to\j@ininline}*
         \toks0=\expandafter{\oldj@ininline}*
         \let\oldj@ininline=\j@ininline
         \immediate\write\j@insplitout{\the\toks0}*
       \fi
     \ifn@teof
     \repeat
   \immediate\closein\j@insplitin
   \fi
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\begin{document}
{\flushright
 RUB-TPII-10/00\\
 JINR-E2-2000-176\\}


\begin{center}
{\large \bf New shapes of light-cone distributions
 of the transversely polarized $\rho$-mesons}\\[0.5cm]

A.~P.~Bakulev\footnote{E-mail: bakulev@thsun1.jinr.ru},\ \
S.~V.~Mikhailov\footnote{E-mail: mikhs@thsun1.jinr.ru}\\[0.5cm]

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

\end{center}
\begin{abstract}
The leading twist light-cone distributions for
transversely polarized $\rho$-, $\rho'$- and $b_1$-mesons
are re-analyzed in the framework of QCD sum rules
with nonlocal condensates.
Using different kinds of sum rules to obtain reliable predictions,
we estimate the 2-, 4-, 6- and 8-th moments for transversely
polarized $\rho$- and $\rho'$-meson distributions and re-estimate tensor
couplings $f^T_{\rho,\rho',b_1}$.
We stress that the results of standard sum rules also support our
estimation of the second moment of the transversely polarized
$\rho$-meson distribution.
New models for light-cone distributions of these mesons are constructed.
Phenomenological consequences from these distributions
are briefly discussed.
Our results are compared with those found by Ball and Braun (1996),
and the latter is shown to be incomplete.
\end{abstract}
\vspace {1cm}

PACS: 11.15.Tk, 12.38.Lg, 14.40.Cs \\
Keywords: QCD sum rules, nonlocal condensates,
 meson distribution amplitudes, tensor coupling constants,
 duality
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Introduction}
  \la{sect-1}
   \renewcommand\thefootnote{\arabic{footnote}}
    \setcounter{footnote}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this paper, we complete our investigation of the leading twist
light-cone distribution amplitudes (DAs) for lightest
transversely polarized mesons with quantum numbers $J^{PC} = 1^{-
-}$ ($\rho_{\perp}$, $\rho'_{\perp}$), $1^{+ -}$ ($b_{1 \perp}$)
in the framework of QCD sum rules (SRs) with nonlocal condensates
(NLC). These DAs are important ingredients of the
``factorization" formalism \cite{ERBL} for any hard exclusive
reactions involving $\rho$-mesons. For this reason, the DAs have
been attractive for theorists for a long time: the main points
are presented in \cite{CZ82,GRDW87}, a detailed revised version
of the standard approach is in \cite{BB96}, and a generalization
to the next twists is in \cite{BBKT98}. The leading twist DA
$\varphi^T_{\rho,\rho',b_1}(x,\mu^2)$ parameterizes the matrix
elements of the tensor current with transversely polarized
$\rho(770)$- and $\rho'(1465)$-mesons ($J^{PC} =1^{--}$) \be
 \va{0\mid\bar u(z)\sigma_{\mu\nu}d(0)\mid \rho_{\perp}(p,\lambda)}\Big|_{z^2=0}
 = if_{\rho_{\perp}}^{T}
  \left(\varepsilon_{\mu}(p,\lambda)p_{\nu}
       -\varepsilon_{\nu}(p,\lambda)p_{\mu}\right)
   \int^1_0 dx\ e^{ix(zp)}\ \varphi^T_{\rho_{\perp}}(x,\mu^2) + \ldots,
 \label{eq-Rho_DA}
\ee
and the $b_1 (1235)$-meson ($J^{PC} =1^{+-}$)
\ba
 \va{0\mid\bar u(z)\sigma_{\mu\nu}d(0)\mid b_{1}^+(p,\lambda)}\Big|_{z^2=0}
 = f^T_{b_1} \epsilon_{\mu\nu\alpha\beta}
    \varepsilon^\alpha(p,\lambda)p^{\beta}
        \int^1_0 dx\ e^{ix(zp)}\ \varphi^T_{b_1}(x,\mu^2)+ \ldots
 \label{eq-B1_DA}
\ea
(here dots represent higher-twist contributions,
 explicitly defined in Appendix A,
 see Eqs.(\ref{eq-AWF_rho})-(\ref{eq-AWF_b1}) and ref.\cite{BBKT98}).
In the above definitions, $p_{\nu}$ and $\varepsilon_{\mu}(p,\lambda)$
are the momentum and the polarization vector of a meson, respectively,
and $\mu^2$ is normalization point.

In the framework of the standard approach, one should restrict oneself
to an estimate of the second moment $\langle \xi^2 \rangle$
of the DA to restore its shape\footnote{%
We should note in this respect that the standard approach could
not provide a reliable estimate even for the second moment of DA,
see \cite{MR86,MR92,BM98,Rad97}}. In other words, the variety of
different DA shapes is reduced to the 1-parameter family of
``admissible" DAs: $\varphi(x;a_2) = 6 x (1-x) \left[ 1 + a_2
C^{3/2}_2(2x-1)\right]$. This family includes both the asymptotic
DA ($a_2=0$) and Chernyak--Zhitnitsky model \cite{CZ82} for the
pion DA ($a_2^{\pi|\itxt{CZ}}=-2/3$). For the pion case, one can
think it is rather enough: most of debates (see
\cite{CZ82,MR86,BrFil89,SSK99,SchmYa99} and refs. therein) about
the shape of this DA are concerned just with the value of
coefficient $a_2$
 -- is it close to $0$ or to $a_2^{\pi|\itxt{CZ}}$?
In our opinion, advocated since 1986 \cite{MR86}, the shape of
the pion DA is not far from the asymptotic one
\cite{MR86,MR92,Rad94,BM95,BM98}. Only recently, researchers have
tried to extract the next Gegenbauer coefficient \cite{SchmYa99}
and other parameters of the pion DA \cite{RR96} from experimental
data. But, in general, there is no principle to exclude a more
rich structure for a hadron DA. In this case, the standard
approach is definitely out of its applicability range, and one
should use more refined techniques, e.g., the QCD SRs with NLC.

This work was started in \cite{BM98}
where the ``mixed parity" NLC SR for DAs
of $\rho$- and $b_1$-mesons,
the particles possessing different P-parity, was analyzed.
We concluded that, to obtain a reliable result,
one should reduce model uncertainties due to the nonlocal gluon
contribution.
Separate SRs for each P-parity channel should be preferable
for this purpose,
and here we construct these ``pure parity" SRs for corresponding DAs.
The SR of this type possesses a low sensitivity to the gluon model
but involves contributions from higher twists\footnote{%
as was noted in \cite{BB96}.}.
To construct a refined ``pure parity" SR for twist 2 DA,
one must resolve the corresponding system of equations (see Appendix A).
We realize this solution using the duality transformation,
introduced in our previous work~\cite{BM99}.
The negative parity NLC SR for the transversely polarized
$\rho$-, $\rho'$-mesons works rather well and allows us to estimate
the 2-nd, 4-th, 6-th, and 8-th moments of the leading twist DAs.
The positive parity SR for the transversely polarized $b_1$-meson
can provide only the value of the $b_1$-meson tensor coupling,
$f_{b_1}^T$.
We suggest the models for these DAs and check their self-consistency,
based upon both ``pure" and ``mixed" NLC SR.
The DA shape $\varphi^T_{\rho_{\perp}}(x)$ differs noticeably from the known one.
Finally, we inspect
how these models can influence the $B\to\rho e \nu$ decay form factors.

The approach has been grounded in \cite{MR86,MR92,MS93}, the
calculation technique is the same as in \cite{MR92,BM98};
therefore, the corresponding details are omitted below. Some
important features of the NLC SRs approach would be briefly
recalled.
%\newpage
%The approach introduced in \cite{MR86} was successfully
%applied for determining light meson dynamic characteristics, DAs,
%form factors, see, \eg, \cite{MR92,BM98,Rad94,BM95}.
The original tools of NLC SR are nonlocal objects
like $M_S(z^2)=\langle \bar q(0)E(0,z)q(z) \rangle$ \footnote{Here
$E(0,z)=P\exp(i \int_0^z dt_{\mu} A^a_{\mu}(t)\tau_a)$ is the
Schwinger phase factor required for gauge invariance.} or
$M_V^{\mu}(z)=\langle \bar q(0)\gamma^{\mu}E(0,z)q(z) \rangle$,
rather than constant quantities of
$\langle \bar q(0)q(0)\rangle$-type.
Note that, in deriving  sum rules,
one can always make a Wick rotation
and treat all the coordinates as Euclidean, $z^2 = -\tilde{z}_{Eucl}^2 <0$.
NLC $M_S(z^2)$ can be expanded in the Taylor series over the standard
(local) condensates, $\langle \bar q(0)q(0)\rangle$,
$\langle \bar q(0)\nabla^2 q(0)\rangle$, and over ``higher dimensions"
(see details of the expansion of different NLCs in \cite{Gr95}),
\be \la{Taylor}
M_S(z^2)=\langle \bar q(0)q(0)\rangle
 -\frac{\tilde{z}^2}{8} \langle \bar q(0)\nabla^2 q(0)\rangle
 + \ldots .
\ee
% (here
%$\nabla_\mu = \partial_\mu - i g \hat{A}_{\mu}$ is the covariant
%derivative).
So, one can return to the standard SR by truncating this series.
But, in virtue of the cut off, one loses an important physical
property of nonperturbative vacuum -- the possibility of vacuum
quarks (gluons) to flow through vacuum with a nonzero momentum $
k_{q(g)} \neq 0$. The parameter $\langle k^2_{q} \rangle$, fixing
the average virtuality of vacuum quarks, can be interpreted as a
measure of condensate ``nonlocality" $\lambda_q^2$,
$$\langle
k^2_{q} \rangle = \lambda_q^2= \frac{\langle \bar q(0)\nabla^2
q(0)\rangle}{\langle \bar q(0)q(0)\rangle}= \frac{\langle \bar
q(0)\left(ig\sigma_{\mu \nu} G^{\mu \nu}
\right)q(0)\rangle}{2\langle \bar q(0)q(0) \rangle}
%\approx 0.4-0.5~\mbox{GeV$^2$~\cite{Piv91,DEM97} }.
~~[\mbox{chiral limit}]
$$
The $\lambda_q^2$ was estimated from the mixed condensate of
dimension 5, $\lambda_q^2 \approx
0.4-0.5$~GeV$^2$~\cite{Piv91,DEM97}. It is important that its
value is of an order of the characteristic hadronic scale,
$\lambda_q^2 \sim m^2_\rho \approx 0.6~\gev{2}$, therefore the
nonlocality effect can be large, and it should be taken into
account in QCD SR. Really, the second term in the expansion
(\ref{Taylor}) of $M_S(z^2)$ that is inverse of the first one in
sign becomes of an order of the first term at $|z^2| \sim
1/m_{\rho}^2$ due to the estimate $|\lambda_q^2 z^2| \sim 1$.
Moreover, we should take into account the whole set of
$\left(\lambda_q^2 z^2 \right)^n$-type corrections, appearing in
the Taylor expansion. These corrections just form the decay rate
of the NLC ($M_S(z^2)$) in the main. The sensitivity to this rate
is crucial for the DA moment SR: it leads to much softer behavior
of DA near the end points $x=0,1$ and allows one to extend QCD SR
to higher moments $\langle\xi^N\rangle \equiv \int_{0}^{1}
\varphi(x) (2x-1)^N dx$, as it was shown in  \cite{MR86}.
%The sum of this set corresponds
%to the main part of the $M_S(z^2)$.

Since neither QCD vacuum theory exists yet,  nor higher dimension
condensates are estimated, it is clear that merely the models of
NLC can be suggested (Appendix B). Here we apply the simplest
ansatz to NLC \cite{MR92,BM98} that takes into account only the
main effect $\langle k^2_{q} \rangle = \lambda_q^2 \neq 0$ and
fixes a length of the quark-gluon correlations in QCD vacuum
$\Lambda = 1/ \lambda_q \approx 0.8$(Fm) \cite{MR86,MR92}. This
suggestion leads to the simple Gaussian decay for $M_S(z^2)$,
while the coordinate behavior of other NLCs looks more
complicated. Certainly, the single scale of decay for all types
of NLC (see Appendix B) looks as a crude model. But, the model
can be rather crude if one deals with  SRs only for the first few
moments $\langle\xi^N\rangle$, because for these integral
characteristics the details of NLC behavior appears to be not
very important (see discussion in sect.5). An alternative case is
provided by a special SR \cite{Rad94,BM95} constructed directly
for the shape of DAs.

%It is important to note that the
%nonlocal character of the quark condensate was recently confirmed in
%direct lattice calculations
In nowadays, the lattice calculations of NLC provide an inspiring knowledge
~\cite{DDM99,DEJM2000} for QCD SR. The latter
measurement in \cite{DEJM2000} confirms the validity of the Gaussian
ansatz for $M_S(z^2)$ (at a small distance) as well as the value of the parameter
$\lambda_q^2$.

%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{``Duality'' transformation  }
 \la{sect-2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To obtain sum rule, we start with a 2-point correlator
$\Pi^{\mu \nu ;\alpha \beta }(q)$ of tensor currents
$J^{\mu \nu}_{(N)}(x)=\bar u(x)\sigma^{\mu\nu}\left(z\nabla\right)^{N}d(x)$
($z$ is a light-like vector, $z^2=0$),

\be
 \Pi^{\mu \nu ;\alpha \beta }_{(N)}(q) =
  i \int d^{4}x\ e^{iq\cdot x}
    \va{0|T\left[J^{\mu\nu+}_{(0)}(x)J^{\alpha\beta}_{(N)}(0)
          \right]|0}\,
 \label{eq-corr_mnab}
\ee
whose properties were  partially analyzed
in~\cite{GRDW87,BB96,BM99}.
It is well known that the correlator at $N=0$
can be decomposed in invariant form factors $\Pi_\pm$,
\cite{GRDW87,BB96}
\be
 \Pi^{\mu \nu ;\alpha \beta }_{(0)}(q) =
   \Pi_-(q^2)P^{\mu \nu ;\alpha \beta}_1
  +\Pi_+(q^2)P^{\mu \nu ;\alpha \beta}_2 \,
\label{eq-deco_0}
\ee
where the projectors $P_{1,2}$, obeying the projector-type relations
\ba
\label{eq-project}
 \left( P_i\cdot P_j \right)^{\mu \nu ;\alpha \beta}
\equiv
 P_i^{\mu \nu ;\sigma \tau}P_j^{\sigma \tau ;\alpha \beta} =
 \delta_{ij}P_i^{\mu \nu ;\alpha \beta}
 ~\mbox{(no sum over $i$)},~P_i^{\mu \nu ;\mu \nu}=3,
\ea
are presented in Appendix A.
For the general case $N\neq 0$, a similar decomposition
involves 4 new independent tensors $Q_i$;
they appear due to a new vector $z^{\alpha}$ introduced
into the composite tensor current operator,
\ba
 \Pi^{\mu \nu ;\alpha \beta }_{(N)}(q)
 &=& \Pi_-(q^2,qz)P^{\mu \nu ;\alpha \beta}_1
  + \Pi_+(q^2,qz)P^{\mu \nu ;\alpha \beta}_2
  + K_1(q^2,qz)Q^{\mu \nu ;\alpha \beta}_1
\nn \\
 &+& K_3(q^2,qz)Q^{\mu \nu ;\alpha \beta}_3
  + K_z(q^2,qz)Q^{\mu \nu ;\alpha \beta}_z
  + K_q(q^2,qz)Q^{\mu \nu ;\alpha \beta}_q\ .
 \label{eq-deco_N}
\ea

Contributions of DAs,
defined in Eqs.(\ref{eq-AWF_rho})-(\ref{eq-AWF_b1}),
to different tensor structures in decomposition (\ref{eq-deco_N})
are mixed, see Eqs.(\ref{eq-Pi-})-(\ref{eq-Pi+}).
The most effective way to disentangle them
in practical OPE calculations
is to use explicit properties of different OPE terms
under the duality transformation $\hat D$
(introduced in our previous work~\cite{BM99})
mapping any rank-4 tensor $T^{\mu \nu;\alpha \beta}$
to another rank-4 tensor
$T_D^{\mu \nu;\alpha\beta} = (\hat D T)^{\mu \nu;\alpha \beta}$
with
\be
 D^{\mu \nu;\alpha \beta}_{\mu' \nu';\alpha' \beta'} =
 \frac{-1}4 \epsilon^{\mu\nu}_{\phantom{\mu\nu}\mu'\nu'}
             \epsilon_{\alpha'\beta'}^{\phantom{\alpha'\beta'}\alpha\beta}\
             ~\mbox{and} ~\hat D^2 = 1 .
 \la{eq-13}
\ee
Our projectors $P_1^{\mu \nu;\alpha \beta}$, $P_2^{\mu \nu;\alpha \beta}$
$Q_1^{\mu \nu;\alpha \beta}$, $Q_3^{\mu \nu;\alpha \beta}$,
$Q_z^{\mu \nu;\alpha \beta}$, and $Q_q^{\mu \nu;\alpha \beta}$
transform into each other under the action of $\hat D$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%   D[P1] = P2 ;                            D[P2] = P1 ;            %%%%%
%%%%%   D[Q1] = P1 + P2 - Q3 ;                  D[Q3] = P1 + P2 - Q1 ;  %%%%%
%%%%%   D[QQ] = QQ - QZ + Q1 + Q3 - P1 - P2 ;   D[QZ] = - QZ ;          %%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\be
 \left(\hat D P_1\right)^{\mu \nu;\alpha \beta}
   = P_2^{\mu \nu;\alpha \beta} \ ;\ \ \ \
 \left(\hat D Q_1\right)^{\mu \nu;\alpha \beta}
   = \left[P_1 + P_2 - Q_3\right]^{\mu \nu;\alpha \beta}\ ;
 \la{eq-DualP1Q1}
\ee
\be
 \left(\hat D P_2\right)^{\mu \nu;\alpha \beta}
   = P_1^{\mu \nu;\alpha \beta}\ ; \ \ \ \
 \left(\hat D Q_3\right)^{\mu \nu;\alpha \beta}
   = \left[P_1 + P_2 - Q_1\right]^{\mu \nu;\alpha \beta}\ ;
 \la{eq-DualProP2Q3}
\ee
\be
 \left(\hat D Q_z\right)^{\mu \nu;\alpha \beta}
   = - Q_z^{\mu \nu;\alpha \beta}; \ \ \ \
 \left(\hat D Q_q\right)^{\mu \nu;\alpha \beta}
   = \left[Q_q - Q_z + Q_1 + Q_3 - P_1 - P_2\right]^{\mu \nu;\alpha \beta}
 \ . \la{eq-DualProQzQq}
\ee
We have shown in~\cite{BM99}
that all terms in OPE could be divided into two classes,
self-dual
($\hat{D} X_{\itxt{SD}} = X_{\itxt{SD}}$)
and anti-self-dual
($\hat{D} X_{\itxt{ASD}} = - X_{\itxt{ASD}}$).
For example, the perturbative term is of ASD type,
whereas the 4-quark scalar condensate contribution to OPE
is of SD type.

Below we introduce the shorthand notation
for contributions of DAs to decomposition (\ref{eq-deco_N}):
$v_0$, $v_1$, and $v_2$ stand
for $1^{--}$ ($\rho_{\perp}$, $\rho_{\perp}'$);
and $u_0$, $u_1$, and $u_2$, for $1^{+-}$ ($b_1$),
see Appendix~A for details.
For SD parts of OPE
%the $b_1$-meson contributions are opposite in sign to the $\rho$-meson ones,
$u_i = - v_i$,
% $i = 0 \div 2$,
and the system of equations simplifies to:
\be
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% File "LHS_RHO": page 5 %%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%                     Self-Duality: D[P] = P                      %%%%%%
%%%%%%          for P = c1*P1+c2*P2+x1*Q1+x3*Q3+xz*QZ+xq*QQ            %%%%%%
%%%%%%                                                                 %%%%%%
%%%%%%          c2 = c1;     x1 + x3 = xq;    xq = -2*xz;              %%%%%%
%%%%%%                                                                 %%%%%%
%%%%%%          U0 = - V0;     U1 = - V1;      U2 = - V2;              %%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%   c1 = + V0 - U1 - U2 ;
 \frac{\Pi_{\mp}(q^2,qz)}{2(qz)^{N} q^{2}}
 = \mp v_0 - v_1 - v_2\ ;\
%%%%%%   x1 = + V1 + U2 ;  x3 = + U1 + U2 ;
 \frac{K_{1,3}(q^2,qz)}{2(qz)^{N} q^{2}}
 = \mp v_1 + v_2\ ;\
%%%%%%  xz = - U2 ;  xq = - V2 + U2 ;
 \frac{K_q(q^2,qz) (= -2 K_z(q^2,qz))}{4(qz)^{N} q^{2}}
 = v_2\ , \label{eq-ProSD}
\ee
whereas for ASD parts $u_i = v_i$,
and we have:
\be
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%                 Anti-Self-Duality: D[P] = - P                   %%%%%%
%%%%%%          for P = c1*P1+c2*P2+x1*Q1+x3*Q3+xz*QZ+xq*QQ            %%%%%%
%%%%%%                                                                 %%%%%%
%%%%%%          c2 = - c1 - 2*x1;     x3 = x1;    xq = 0;              %%%%%%
%%%%%%                                                                 %%%%%%
%%%%%%           U0 = V0;       U1 = V1;       U2 = V2;                %%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%   c1 = + V0 - U1 - U2 ;  c2 = - U0 - U1 - U2 ;
 \frac{\Pi_{\mp}(q^2,qz)}{2(qz)^{N} q^{2}}
 = \mp v_0 + v_1 + v_2\ ;\
%%%%%%   x1 = + V1 + U2 ;  x3 = + U1 + U2 ;
 \frac{K_{1,3}(q^2,qz)}{2(qz)^{N} q^{2}}
 = - v_1 - v_2\ ;\
%%%%%%  xz = - U2 ;
 \frac{K_z(q^2,qz)}{2(qz)^{N} q^{2}}
 = + v_2\ ; \
%%%%%%  xq = - V2 + U2 ;
 K_q(q^2,qz) = 0\ .\label{eq-ProASD}
\ee
By these formulas, it is possible to determine
$\rho$- and $b_1$-meson DA contributions
of leading and higher twists.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{The ``mixed parity" sum rule}
  \la{sect-3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The usual way ~\cite{CZ82,BB96} to extract the moments of the
function $\varphi^T(x)$ appeals to a correlator $J_{(N,0)}(q^2)$
of currents $J^{\mu \alpha}_{(N)}(0)z^{\alpha}$ and $J^{\mu
\beta}_{(0)}(x)z^{\beta}$ defined as \be
 -2i^n\left(zq\right)^{N+2} J_{(N,0)}(q^2)
 \equiv
 \Pi_{(N)}^{\mu \nu ;\alpha \beta }(q)
 \left( z^{\nu} z^{\beta} g^{\mu\alpha}\right)
 = \frac{\Pi_-(q^2)-\Pi_+(q^2)}{q^2}(qz)^2,
\la{cor-mixed}
\ee
the latter equality in (\ref{cor-mixed}) follows
from (\ref{eq-deco_N}) and Eqs.(\ref{norm-muz})
in Appendix A.
This correlator contains the contributions
from states with different parity, $\Pi_-(q^2)$ and $\Pi_+(q^2)$
(see the analysis in \cite{BB96}),
therefore, the contamination from $b_1$-meson
$\left(J^{PC}=1^{+-}\right)$ in the phenomenological part
of the corresponding SR is mandatory.
The contamination makes it difficult to reliably extract
the meson characteristics from this ``mixed" SR.

The main feature of the theoretical part of $J_{(N,0)}(q^2)$ is
the cancellation of the self-dual part,
represented by the four-quark condensate,
in the anti-self-dual expression (\ref{cor-mixed}).
The remaining ``condensate'' parts of Eq.
(\ref{cor-mixed}) contain, after the Borel transformation, the
same 5 universal elements $\Delta \Phi_{\Gamma}(x; M^2)$ as for
the $\rho^L$-, $\pi$-cases and, besides, an additional gluon
contribution $\Delta\Phi_G'(x; M^2)$ (see Appendix B).
This term affects the values of moments rather strong,
as was shown in \cite{BM98}.
The contributions from the different kinds of NLC,
 $\Delta \Phi_{\Gamma}(x; M^2)$,
are symbolically noted in the r.h.s. of SR (\ref{eq:srf_rb}). So,
here we get rid of the four-quark condensate that is not known
very well due to a possible vacuum dominance violation. But, the
price we pay for it is a high sensitivity to an ill-known gluon
contribution $\Delta\Phi_G'(x;M^2)$.

The method of calculation of the NLC contributions
$\Delta\Phi_\Gamma\left(x;M^2\right)$ to the theoretical part of
SR is described in \cite{MR86,MR92,BM98}. The corrected final
results of the calculation are presented in Appendix B that
contains all the needed explicit expressions of $\Delta
\Phi_{\Gamma}(x; M^2)$ for the simplest physically motivated
Gaussian ansatz. The final SR including DAs of $\rho$-meson and
next resonances $\rho'$ and $b_1$ into the phenomenological
(left) part is as follows: \ba
 \left(f_\rho^T\right)^2\varphi_\rho^T(x) e^{-m^2_\rho/M^2}
 + (\rho\to\rho')
 + \left(f_{b_1}^T\right)^2\varphi_{b_1}^T(x)e^{-m^2_{b_1}/M^2}
 = \int_{0}^{s_{b}^T}\rho^{\itxt{mixed}}_T\left(x,s;s_\rho^T,s_b^T\right)
      e^{-s/M^2}ds \nonumber &&\\
 +\ \Delta\Phi_G(x;M^2) + \Delta\Phi_G'(x;M^2)
 + \Delta\Phi_V(x;M^2) + \Delta\Phi_{T}(x;M^2)\ , &&\label{eq:srf_rb}
\ea
where $s_{\rho}^T$ and $s_{b}^T$ are the effective continuum thresholds
in $\rho$- and $b_1$-channels.
Recall again that the variation of the ill-known part of gluon contribution
$\Delta\Phi_G'\left(x;M^2\right)$
can reduce the second moment significantly \cite{BM98}.
In that paper, we suggest the following naive model:
instead of the constant contribution
$\Ds \Delta\varphi'_{G}\left(x;M^2\right)
\equiv \langle \alpha_s GG \rangle /(6\pi M^2)$
(as in the standard approach), we put
$$ \Delta\Phi'_{G}\left(x;M^2\right) =
    \Delta\varphi'_{G}\left(x;M^2\right)
     \frac{\theta\left(\Delta<x\right)\theta\left(x<1-\Delta\right)}
           {1-2\Delta}.$$
This simulation eliminates end-point ($x=0, 1$) effects
due to the influence of the vacuum gluon nonlocality
inspired by the analysis in \cite{MS93}
and our experience in the nonlocal quark case.
The corresponding SR leads to estimate
$\langle \xi^2 \rangle_\rho^T = 0.329(11)$ (see Fig.2(a) ).
However, this value  drastically changes, $\langle \xi^2 \rangle_{\rho}^T \to 0.231(8)$,
if we take the local expression
$\Ds \Delta\varphi'_{G}\left(x,M^2\right)$ unchanged.
Therefore, the estimate $\langle \xi^2 \rangle_\rho^T = 0.329$
contains a significant model uncertainty,
and the real value seems to be smaller.


Which prediction for this quantity
can be obtained
within the standard QCD SR approach?
As one can see from Fig.2(b) (long-dashed line),
the value of $\langle \xi^2 \rangle_{\rho}^T$
cannot be estimated with a reasonable accuracy,
because the standard SR {\bf does not have real stability}.
Nevertheless, the authors of~\cite{BB96}
bravely deduce an estimate
$\langle \xi^2 \rangle_{\rho~\itxt{[B\&B]}}^T = 0.27(4)$.
We discuss this attempt in comparison
with processing other SRs in greater detail
in section~5.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{The ``pure parity" sum rules}\la{sect-4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Using the approach of Section 2, we calculate OPE terms for
$\Pi_{\mp}$, $K_{1,3}$, and $K_{z,q}$ correlators and extract the
contributions to DAs of the $\rho$- and $b_1$-mesons. This allows
us to write down the SRs for DAs of the $\rho$- and $b_1$-mesons
separately:
\ba
 &\Ds
   \left(m_{\rho} f_\rho^T\right)^2
    \varphi_\rho^T(x) e^{-m^2_\rho/M^2}
 + \left(m_{\rho'} f_{\rho'}^T\right)^2
    \varphi_{\rho'}^T(x) e^{-m^2_{\rho'}/M^2}
 = \frac12\int_{0}^{s_{\rho}^T}
    \rho^{{\itxt{pert}}}_T(x;s)\ s\ e^{-s/M^2} ds
&\nonumber \\&\Ds\phantom{.}
 + \Delta\tilde\Phi_G(x;M^2)
 + \Delta\tilde\Phi_S(x;M^2)
 + \Delta\tilde\Phi_V(x;M^2)
 + \Delta\tilde\Phi_{T}(x;M^2)\ ;
&\label{eq:srf_ro}
\ea
\vspace*{-13.4mm}

\ba&\Ds
   \left(m_{b_1} f_{b_1}^T\right)^2
    \varphi_{b_1}^T(x) e^{-m^2_{b_1}/M^2}
 = \frac12\int_{0}^{s_{b}^T}
    \rho^{{\itxt{pert}}}_T(x;s)\ s\ e^{-s/M^2} ds
&\nonumber \\&\Ds\phantom{.}
 + \Delta\tilde\Phi_G(x;M^2)
 - \Delta\tilde\Phi_S(x;M^2)
 + \Delta\tilde\Phi_V(x;M^2)
 + \Delta\tilde\Phi_{T}(x;M^2)\ .
& \label{eq:srf_b1}
\ea
where $s_{\rho; b}^T$ are the effective continuum thresholds
in the $\rho$- and the $b_1$-meson cases, respectively.
The perturbative spectral density $\rho^{pert}_{T}(x;s)$
is presented in an order of $O(\alpha_s)$ in \cite{BB96,BM98} (Appendix B).
Here we also define ``tilded" functions
\be  \label{eq:connection}
 \Delta\tilde\Phi_{\Gamma}(x;M^2)
  \equiv \frac12 M^4 \dpa_{M^2} \Delta\Phi_{\Gamma}(x;M^2)\ ,
\ee
and the whole tensor NLC contribution
\be
 \Delta\tilde\Phi_{T}(x;M^2)
 \equiv \Delta\tilde\Phi_{T_1}(x;M^2)
      + \Delta\tilde\Phi_{T_2}(x;M^2)
      - \Delta\tilde\Phi_{T_3}(x;M^2).
\ee
The later noticeably differs from the case of longitudinally polarized
$\rho$-meson due to the opposite sign of $T_3$-term, cf. \cite{BM98}.
The theoretical ``condensate'' part in
(\ref{eq:srf_ro})-(\ref{eq:srf_b1})
contains 5 elements obtained from (\ref{eq:connection}) with the same
$\Delta\Phi_{\Gamma}(x;M^2)$
as for the $\rho^L$-meson case,
whereas the self-dual four-quark contribution
$\Delta\tilde\Phi_S(x;M^2)$ is a new element of the analysis.
Note, just this self-dual part $\Delta\tilde\Phi_S(x;M^2)$,
entering in the SRs (\ref{eq:srf_ro}) and (\ref{eq:srf_b1}) with different sign,
provides the different properties of the $\rho$- and $b_1$-mesons \cite{BM99}.

 For better understanding of the SR properties it is
instructive to reduce them to standard version for $\va{\xi^N}$-moments.
To this end, let us take the limits $\lambda_q^2 \to 0$,
~$\Delta \Phi_{\Gamma}(x, M^2)\to\Delta \varphi_{\Gamma}(x, M^2)$
%and $\rho^{pert}_{L}(x,s) \to \rho^{Born}_{L}(x,s)$
in eqs.(\ref{eq:srf_ro})-(\ref{eq:srf_b1}) and integrate in $x$
with weights $(1-2x)^N$
to obtain the local limit version of moment SR:
\ba \label{eq:srf_ro_loc}
&\Ds
 \left(m_{\rho} f_\rho^T\right)^2\va{\xi^N}_\rho^T e^{-m^2_\rho/M^2}
% + \left(m_{\rho'} f_{\rho'}^T\right)^2\va{\xi^N}_{\rho'}^T e^{-m^2_{\rho'}/M^2}
 \ = & \\ \nn
&\Ds\phantom{.}
\frac12\int_{0}^{s_{\rho}^T}
    \rho^{{\itxt{pert}}}_T(x;s)\ s\ e^{-s/M^2} ds
%   \frac{3M^4\left[1-\exp\left(-\frac{s_{\rho}^T}{M^2}\right)
%             \left(1+\frac{s_{\rho}^T}{M^2}\right)\right]}
%        {8(N+1)(N+3)\pi^2}
 - \frac{\langle \alpha_s GG \rangle}{24\pi}
       \left(\frac{N-1}{N+1}\right)
 - \frac{16\pi}{81}\frac{\langle\sqrt{\as}\bar q(0)q(0)\rangle^2}{M^2}(4N-13)\ .&
\ea
%\ba
%\Delta\va{\xi^N}_{\tilde\Phi_T}
%  = -\frac{8A_0}{M^2}(2N-5)\ ;
%\ea
This SR demonstrates a considerably lower sensitivity
to the gluon condensate contribution: the gluon part does not
depend on the Borel parameter $M^2$ at all, and its relative value
is 6 times as low as that in the ``mixed" SR.
The r.h.s. of Eq. (\ref{eq:srf_ro_loc}) is reduced
at $N=0$ to the known expression, see \cite{BB96},
that is not sensitive to the $\rho'$ contribution,
while its nonlocal version analyzed in \cite{BM99}
makes it possible to analyse the $\rho'$ meson.
For $N > 0$, the SR is unstable
due to the effect of radiative corrections,
and to obtain the moment estimates,
we should return to the nonlocal version, Eq.(\ref{eq:srf_ro}).

But the price one pays for this is high, the fidelity windows of
the SRs are significantly reduced.
  For the $\rho$-meson case, fidelity windows
of the Borel parameters $M^2$ shrink to $M^2 = 0.7 - 1.15~\gev{2}$
(to be compared with $M^2 = 0.75 - 2.25~\gev{2}$ in ``mixed" SR)
and demand one to take into account the $\rho'$-meson explicitly.
Here we cannot obtain the $\rho'$-meson mass from SR
(\ref{eq:srf_ro}) because of the enhanced perturbative spectral
density ($\sim s$; this means that the differentiated SR has a
spectral density $\sim s^2$ and presumably, is not stable at all);
instead, we use the $\rho'$-meson mass extracted in our previous
paper on the longitudinally polarized $\rho$-meson DA
\cite{BM98}, $m_{\rho'} = 1496\pm 37$ MeV, rather close to the
Particle Data Group value $m_{\rho'} = 1465\pm 22$ MeV
\cite{PDG98}.

  In the case of $b_1$-meson, one can analyze only the SR
for the zeroth moment (decay constant $f_{b_1}^T$) of the DA
(see Fig.3),
the SRs for higher moments appearing to be invalid.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Processing different SRs and comparison of the results}
  \la{sect-5}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% FIGURE: MIXED (f_{\rho}) and PURE (f_{\rho}) PARITY SRs %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\input{psbox}\vspace*{-2mm}
 \begin{figure}[hbt]\noindent
 $$\psboxto(0cm;0.34\textwidth){fig1.eps}$$
    \myfig{\label{fig:stab_frors}}\vspace*{-10mm}
     \caption{\footnotesize $f^T_{\rho}$ as a function of the Borel
       parameter $M^2$ obtained from: (a) the ``mixed parity'' NLC SR,
       Eq.~(\ref{eq:srf_rb}), with $s_0=2.9~\gev{2}$;
       (b) the ``pure parity'' NLC SR, Eq.~(\ref{eq:srf_ro}),
       with $s_0=2.8~\gev{2}$.
       The fidelity windows for both figures coincide with
       the whole depicted range of $M^2$.}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace*{2mm}
 We start with considering the results of processing
both the types of SRs for $f^{T}_{\rho}$.
Its dependence on the Borel parameter $M^2$ obtained from
the ``mixed parity'' NLC SR, Eq.~(\ref{eq:srf_rb}),
with $s_0=2.9~\gev{2}$ is shown in Fig.~\ref{fig:stab_frors}(a).
Figure ~\ref{fig:stab_frors}(b) shows $f^{T}_{\rho}$ as
a function of the Borel parameter $M^2$ obtained from
the ``pure parity'' NLC SR, Eq.~(\ref{eq:srf_ro}),
with $s_0=2.8~\gev{2}$.
Both kinds of SRs are rather sensitive to the $\rho'$-meson
contribution and, for this reason, they
were processed with taking it into account
(see numerical results in Table~1).
Solid lines correspond to the optimal thresholds $s_0$;
the dashed lines --- to the curves with the
10-fold variation of $\chi^2_{\mbox{\tiny min}}$
(this corresponds approximately to the $5\%$-variation of $s_0$;
 definition of $\chi^2$, see in Appendix C, Eq.(\ref{eq-xi2})).
So, one can conclude that both types of NLC SRs agree rather well
about the value of $f^{T}_{\rho}$. Note that the presented
$f^{T}_{\rho}$ is rather close to the standard estimation
$f^{T}_{\rho}=0.160(10)~\gev{}$~\cite{BB96} and to the lattice
one $f^T_{\rho Latt}(4 \gev{2}) =0.165(11)~\gev{}$~\cite{Beci98},
and differs significantly from the result
$f^{T}_{\rho}=0.140~\gev{}$ in~\cite{Pol98}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                 T A B L E   1                                       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{5mm}

\noindent\hspace*{0.02\textwidth}
\begin{minipage}{0.96\textwidth}
\begin{tabular}{|cl||c|c|c|c|c|}\hline
%_______________________________________________________________
&&\multicolumn{5}{|c|}{\strut\vphantom{\vbox to 6mm{}}
 {\bf Table 1}:
 The moments $\langle \xi^N \rangle_M(\mu^2)$
  at $\mu^2 \sim 1$ GeV$^2$}\\
&&\multicolumn{5}{|c|}{
(errors are depicted in brackets in a standard manner)
 $_{\vphantom{\vbox to 4mm{}}}$}\\ \cline{3-7}
%_______________________________________________________________
\multicolumn{2}{|c||}{\strut\vphantom{\vbox to 6mm{}}
 Type of SR$_{\vphantom{\vbox to 4mm{}}}$}
   & $f_{M}\left(1 \mbox{GeV}^2\right)$
     &$\hspace{4mm}N=2\hspace{4mm}$
       &$\hspace{4mm}N=4\hspace{4mm}$
         &$\hspace{4mm}N=6\hspace{4mm}$
           &$\hspace{4mm}N=8\hspace{4mm}$ \\ \hline \hline
%________________________________________________________________
\multicolumn{2}{|c||}{\strut\vphantom{\vbox to 6mm{}}
  Asympt. WF$_{\vphantom{\vbox to 4mm{}}}$}
    & $1$ & $0.2$ & $0.086$ & $0.047$ & $0.030$ \\ \hline \hline
%________________________________________________________________
 {\strut\vphantom{\vbox to 6mm{}} \itxt{NLC SR~Eq.(\ref{eq:srf_ro})}
  $_{\vphantom{\vbox to 4mm{}}}$:}
 & $\rho^T$
   & $0.157(5)$ & $0.296(20)$  & $0.196(6)$
     & $0.132(5)$ & $0.089(4)$  \\ \cline{3-7}
 {\strut\vphantom{\vbox to 6mm{}} \itxt{NLC SR~Eq.(\ref{eq:srf_rb})}
  $_{\vphantom{\vbox to 4mm{}}}$:}
 & $\rho^T$
   & $0.162(5)$ & $0.329(11)$ & -- & -- & -- \\ \cline{3-7}
 {\strut\vphantom{\vbox to 6mm{}} \itxt{B\&B ~SR}
  $_{\vphantom{\vbox to 4mm{}}}$:}
 & $\rho^T$
    & $0.160(10)$ & $0.304(40)$\protect\footnotemark{}
%       & \itxt{does not work}
%& \itxt{does not work}
        & \multicolumn{3}{|c|}{\itxt{does not work}}  \\ \hline \hline
%_______________________________________________________________
 {\strut\vphantom{\vbox to 6mm{}} \itxt{NLC SR~Eq.(\ref{eq:srf_ro})}
  $_{\vphantom{\vbox to 4mm{}}}$:}
 & $\rho'^T$
    & $0.140(10)$ & $0.086(6)$  & $0.010(1)$
       & $0.013(1)$ & $0.022(2)$ \\ \hline \hline
%_______________________________________________________________
 {\strut\vphantom{\vbox to 6mm{}} \itxt{NLC SR~Eq.(\ref{eq:srf_b1})}
  $_{\vphantom{\vbox to 4mm{}}}$:}
 & $b_1^T$
   & $0.184(5)$ & \multicolumn{4}{|c|}{\itxt{does not work}}
% \itxt{does not work} & \itxt{does not work}
%     & \itxt{does not work} & \itxt{does not work}
\\ \cline{3-7}
 {\strut\vphantom{\vbox to 6mm{}} \itxt{NLC SR~Eq.(\ref{eq:srf_rb})}
  $_{\vphantom{\vbox to 4mm{}}}$:}
 & $b_1^T$
   & $0.181(5)$ & $0.144(15)$ & -- & -- & --  \\ \cline{3-7}
 {\strut\vphantom{\vbox to 6mm{}} \itxt{B\&B SR}
  $_{\vphantom{\vbox to 4mm{}}}$:}
 & $b_1^T$
    & 0.175(5) &\multicolumn{4}{|c|}{\itxt{does not work}}
%\itxt{does not work} & \itxt{does not work}
%       & \itxt{does not work} & \itxt{does not work}
\\ \hline \hline
%_______________________________________________________________
\end{tabular}
\end{minipage}
\footnotetext{%
The estimate presented in this cell has been obtained by
processing the ``mixed parity" SR established in~\cite{BB96},
whereas in the original paper~\cite{BB96} this value
amounts to $0.27(4)$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



 Now we consider the results of processing SRs for the second moment
$\langle \xi^2\rangle^{T}_{\rho}$.
First, we demonstrate the results of the ``standard" approach:
$\langle \xi^2\rangle^{T}_{\rho}$ from Eq.(3.21) in  \cite{BB96}
as a function of $M^2$ is shown in Fig.2(b) by a long-dashed line.
This curve is not stable in $M^2$ at all,
therefore the SR can provide merely a range of admissible values,
$0.27 \leq\langle \xi^2\rangle^{T}_{\rho} \leq 0.4$.
As it is evident from Fig.\ref{fig:stab_x2},
this wide window agrees reasonably with both the estimates
from the ``mixed" (a) and ``pure" (b) NLC SRs.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% FIGURE: MIXED and PURE PARITY Srs for <\xi^2>_\rho %%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[hbt]\noindent
 $$\psboxto(0.98\textwidth;0cm){fig2.eps}$$
    \myfig{\label{fig:stab_x2}}
     \vspace*{-10mm}
      \caption{\footnotesize $\langle \xi^2\rangle^T_{\rho}$
       as a function of the Borel parameter $M^2$
       obtained from: (a) the ``mixed parity'' NLC SR,
       Eq.~(\ref{eq:srf_rb}), with $s_0=2.9~\gev{2}$;
       (b) the ``pure parity'' NLC SR, Eq.~(\ref{eq:srf_ro}),
       with $s_0=2.8~\gev{2}$.
      Both kinds of SRs were processed with taking the $\rho'$-meson
      into account, the arrows show the fidelity window
      (for figure (a) the window coincides with
       the whole depicted range of $M^2$).
      Solid lines correspond to the optimal thresholds $s_0$,
      the short-dashed lines on both the figures
      correspond to the curves with the $10\%$-variation of $s_0$ (a)
      or of $\chi^2_{\mbox{\tiny min}}$ (b).
      The long-dashed line in figure (b) represents
      the SR of Ball--Braun~\protect{\cite{BB96}}.}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Note, the authors of \cite{BB96} dealt with the quantity $a_2$,
the Gegenbauer coefficient in the expansion of DA.
The second moment of DA is trivially connected with this coefficient,
$\Ds \langle \xi^2\rangle
= 0.2 + (12/35) a_2$.
Using the SR of~\cite{BB96} for $a_2$, we obtain
the corresponding window,
$0.2 \leq a_2 \leq 0.4$,
that leads to the mean value
$\langle \xi^2\rangle^{T}_{\rho~\itxt{[Stand]}}=0.30$
being surprisingly close to our estimate from NLC SRs,
(see Table~1).
However Ball and Braun have obtained
the erroneous estimate $a_2=0.2 \pm 0.1$
producing, instead, the mean value
$\langle \xi^2\rangle^{T}_{\rho~\itxt{[B\&B]}}=0.27$.

The curves for the next higher moments, whose estimates are presented in Table 1,
have the fidelity windows and the stability behavior similar to
$\langle \xi^2\rangle^T_{\rho}(M^2)$ in Fig.2(b).
Finally, in Fig.3, we demonstrate a very good correspondence
between the values of $f^T_{b_1}$ obtained in different NLC SRs.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% FIGURE: MIXED and PURE PARITY Srs for f_{B_1} %%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[hbt]\noindent
   $$\psboxto(0.98\textwidth;0cm){fig3.eps}$$
    \myfig{\label{fig:stab_frob1}}
     \vspace{-10mm}
      \caption{\footnotesize The curves $f^T_{b_1}$ in $M^2$
       obtained from: (a) the ``mixed parity'' NLC SR
      (with taking the $\rho'$-meson into account with $f_{\rho'}$
       defined from ``pure parity'' SR~(\protect\ref{eq:srf_ro}));
      (b) the ``pure parity'' NLC SR~(\protect\ref{eq:srf_b1}).
      The arrows show the fidelity window
      (for the right figure, the window coincides with
       the whole depicted range of $M^2$).
      Solid lines correspond to the optimal thresholds;
      the short-dashed lines on both the figures,  -- to the curves
      with the 10-fold variation of $\chi^2_{\mbox{\tiny min}}$;
      the long-dashed line on the right figure corresponds
      to the real B\&B curve.}
 \end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{DA models and their check}
  \la{sect-6}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Possible models of DAs corresponding to the moments in Table 1
are of the form
\ba
 \varphi_\rho^{T,mod}(x, \mu^2) &=&
 1.382\left[\varphi^{as}(x)\right]^2
 \left(1+0.927 C^{3/2}_2(\xi)
        +0.729 C^{3/2}_4(\xi) \right) \nn \\
& =& \varphi^{as}(x)
 \left(1+0.29 C^{3/2}_2(\xi)
        +0.41 C^{3/2}_4(\xi) -0.32C^{3/2}_6(\xi) \right)\ ,
% (\mbox{SR (\ref{eq:srf_ro})}) ,
\label{eq:mod_tro} \\
 \varphi_{\rho'}^{T,mod}(x, \mu^2) &=& \varphi^{as}(x)
 \left(1-0.339 C^{3/2}_2(\xi)
        +0.003 C^{3/2}_4(\xi)
        +0.192 C^{3/2}_6(\xi)\right)\ ,
% (\mbox{SR (\ref{eq:srf_ro})})
\label{eq:mod_trs} \\
 \varphi_{b_1}^{mod}(x, \mu^2) &=&\varphi^{as}(x)
 \left(1-(0.175 \pm 0.05) C^{3/2}_2(\xi)
       \right),%  (\mbox{SR (\ref{eq:srf_rb})}),
\label{eq:mod_b1}
\ea
where $\xi\equiv 1-2x$, $C^{\nu}_{n}(\xi)$ are the
Gegenbauer polynomials (GP), and the norm $\mu^2\simeq 1$ GeV$^2$
corresponds to a mean value of $M^2$. Recall again that the value
of the important coefficient  $a_2=0.29$ in (\ref{eq:mod_tro}) is confirmed
by 3 sources: ``pure'' NLC SR (\ref{eq:srf_ro}),
``mixed'' NLC SR (\ref{eq:srf_rb}),
and a mean value from the ``mixed" standard SR.
Figures~\ref{fig-wf_rot},~\ref{fig-wf_rst}(a) contain curves of DA
corresponding to $\rho_{\perp}$, eqs.~(\ref{eq:mod_tro}),
and  $\rho_{\perp}'$ (\ref{eq:mod_trs}).
The arising 3-hump shape of DA for $\rho_{\perp}$
drastically differs from that obtained in \cite{BB96}
and from the one obtained in chiral effective theory \cite{Pol98}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%  FIGURE: WFs Models (\rho) %%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[hbt]\noindent
 $$\psboxto(0.98\textwidth;0cm){fig4.eps}$$
  \myfig{\label{fig-wf_rot}}
   \vspace*{-10mm}
    \caption{\footnotesize
    (a) The curves of $\varphi_{\rho}^{T,mod}(x,1~\gev{2})$:
        Solid lines correspond to the best fits for determined
        moments (see Table 1);
        the dashed line on the left figure corresponds
        to the B\&B curve (which fits only
        $\langle \xi^2 \rangle_{\rho}^{T} \approx 0.27$).
    (b) The rhs of Eq.(\protect\ref{eq:srf_ro})
        SR$_{\rho}^{T}(x,M^2)$ in $x$.
        Different lines here correspond
        to different values of Borel $M^2=0.7 - 0.9~\gev{2}$.}
\vspace*{-5mm}        
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%  FIGURE: WF Model (\rho')  %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 $$\psboxto(0.98\textwidth;0cm){fig5.eps}$$
    \myfig{\label{fig-wf_rst}}
     \vspace*{-10mm}
      \caption{\footnotesize
      (a) The curve of $\varphi_{\rho'}^{T,mod}(x,1~\gev{2})$ in $x$.
      (b) The rhs of Eq.(\protect\ref{eq:srf_ro})
          SR$_{\rho}^{T}(x,M_0^2)$ in $x$.
          Solid and dashed lines here correspond
          to different values of nonlocality parameter
          $\lambda_q^2=0.4 - 0.5~\gev{2}$
          with fixed value of Borel parameter $M_0^2 = 0.8~\gev{2}$.}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This difference mainly appears due to the higher moments, $N=4, 6, 8$,
involved into consideration.
Nevertheless, the hump shape is not an artifact
of the GP expansion series truncation.
These models really contain only 3 first GPs,
meanwhile, it is enough to reproduce all 4 moments up to $N=8$.
Moreover, an additionally smoothed\footnote{%
A certain smoothing of some $\delta$-functions in the r.h.s. of the SR
(see Appendix B) is not important.}
rhs of the NLC SR (\ref{eq:srf_ro})
demonstrates qualitatively the same behaviour in $x$
(at admissible $M^2$) as the model DA,
compare Figs.~\ref{fig-wf_rot}(a) and (b).
The stability of the DA shape
with respect to the variation of ansatz is also checked.
To this end, we show in Fig.~\ref{fig-wf_rst}(b)
the same r.h.s. of (\ref{eq:srf_ro}) as in Fig.~\ref{fig-wf_rot}(b),
but with different values of the single ansatz parameter
$\lambda_q^2=0.4 - 0.5~\gev{2}$ at fixed value $M_0^2 = 0.8~\gev{2}$.

Inverse moments of DAs often appear in perturbative QCD predictions
for exclusive reactions. The estimates for important
$\langle x^{-1}\rangle_{M}$ moments obtained from the model DAs
are presented here\footnote{The upper error $+0.4$ in (\ref{rho-1}) corresponds
  to an overestimate $\langle \xi^2\rangle=0.329$ from the ``mixed" SR }
\ba \la{rho-1}
\langle x^{-1}\rangle_{\rho}\equiv \int_{0}^{1}
\frac{\varphi_\rho^{T}(x,1~\gev{2})}{x}\ dx
 &=& \left\{
   \begin{array}{lll}
    4.15^{+0.4}_{-0.1} & (\mbox{here})\\
    3.6    & (\mbox{B\&B model})
   \end{array} \right. \\
\la{rhoprim-1}
 \langle x^{-1}\rangle_{\rho'}\equiv \int_{0}^{1}
  \frac{\varphi_{\rho'}^{T}(x,1~\gev{2})}{x}\ dx
 &=& 2.57 \pm 0.20 \ (\mbox{here})  \\
\la{b_1-1}
\langle x^{-1}\rangle_{b_1}\equiv \int_{0}^{1}
 \frac{\varphi_{b_1}^{T}(x,1~\gev{2})}{x}\ dx
 &=& 2.48 \pm 0.20 \ (\mbox{here})
\ea
It is useful to construct {\bf an independent} SR for these
inverse moments to verify the DA models
(\ref{eq:mod_tro}, \ref{eq:mod_trs}, \ref{eq:mod_b1}).
Namely, the weighted sum $C(M^2)$ of these moments
\be
 C(M^2)
 \equiv \langle x^{-1}\rangle_{\rho}
 + \langle x^{-1}\rangle_{\rho'}
   \left(\frac{f_{\rho'}^T}{f_{\rho}^T}\right)^2
    e^{-(m_{\rho'}^2 -m_{\rho}^2)/M^2}
 + \langle x^{-1}\rangle_{b_1}
   \left(\frac{f_{b_1}^T}{f_{\rho}^T}\right)^2
    e^{-(m_{b_1}^2-m_{\rho}^2)/M^2}
 \la{eq:C(M)}
\ee
can be obtained by integrating the rhs of the ``mixed"
NLC SR (\ref{eq:srf_rb}) with the weight $1/x$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%  FIGURE: X^{-1}-Moments  %%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht ]\noindent
 $$\psboxto(0.5\textwidth;0cm){fig6.eps}$$
    \myfig{\label{fig-xm1_mom}}
     \vspace*{-8mm}
      \caption{\footnotesize
      $C(M^2)$ as a function of $M^2$ (solid line)
      determined by Eq.(\ref{eq:C(M)})
      and integrating in $x$ Eq.(\ref{eq:srf_rb})
      in comparison with the lhs of Eq.(\ref{eq:SR_C(M)}) (long-dashed line).
      The dotted line corresponds to $\langle x^{-1}\rangle_{\rho}=4.15$;
      whereas the dashed line -- to the lhs of Eq.(\ref{eq:SR_C(M)})
      with upper values of corresponding moments.}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A comparison of the function $C(M^2)$
with the corresponding combination of model estimates
(\ref{rho-1}, \ref{rhoprim-1}, \ref{b_1-1}) obtained in
different kinds of NLC SRs 
(mainly from the ``pure" ones) leads to an approximate equation
\ba
  4.15
+ 2.57 \left(\frac{f_{\rho'}^T}{f_{\rho}^T}\right)^2
       e^{-(m_{\rho'}^2-m_{\rho}^2)/M^2}
+ 2.48 \left(\frac{f_{b_1}^T}{f_{\rho}^T}\right)^2
      e^{-(m_{b_1}^2-m_{\rho}^2)/M^2}
\approx C(M^2)\ ,
 \la{eq:SR_C(M)}
\ea
illustrated in Fig.\ \ref{fig-xm1_mom}.

As a result, one can conclude:
\ben
 \item
 The ``mixed" NLC SR is highly sensitive
 to $b_1$- and $\rho'$-meson contributions,
 the difference in the behavior of $C(M^2)$ (solid line)
 and in the $\rho$-contribution alone (dotted line) illustrates this point.
 \item The curve $C(M^2)$ lies between  mean and upper estimates
 for the lhs of (\ref{eq:SR_C(M)}),
 so it is in reasonable agreement with the estimates
 (\ref{rho-1}, \ref{rhoprim-1}, \ref{b_1-1}).
It also demonstrates
an overestimation of DA moments
in the ``mixed" SR as compared to that obtained from the ``pure" one.
\een

\section{DA models and the $B \to \rho e \nu$ decay form factors}
The new DA shapes result in different pQCD predictions for exclusive
reactions with the $\rho$-meson.
As an example, we re-estimate form factors  $V(t)$, $A_{1,2}(t)$
corresponding to
the transition matrix element $\va{\rho,\lambda|(V-A)_{\mu}|B}$
of the process $B \to \rho ~e \nu$,
in the framework of the light-cone SR approach, ~\cite{BMR00}.
That was done earlier by Ball and Braun in
\cite{BB97},~\cite{BB98} on the base of DAs from \cite{BB96}.
Thus, to estimate the influence of the new nonperturbative
input presented in the previous sections, we have used the LC SR in the
leading twist approximation (cf. \cite{BB97}).
Just as in the case of the LC expansion
for the transition amplitude $\gamma^*\gamma \to \pi^0$,
one might expect high sensitivity to the end-point behavior of the DAs,
as they enter into convolution integrals like
$\langle x^{-1}\rangle_{M}$ estimated in (\ref{rho-1}).

However, there are some essential differences which effectively
soften our expectations. First, the DAs also enter into the
``phenomenological'' side of the SR in the ``continuum''
contribution of higher excited states in the channel with
$B$-meson quantum numbers. This, actually, is a specific feature
of any LC SR. By subtracting the ``continuum'', one actually
obtains ``infrared safe quantities'' like $\int_{\epsilon}^1 dx
\varphi(x)/x$ where $\epsilon\simeq (m_b^2-t)/(s_0^B-t)$,
$m_b\simeq 4.8\,\Gev$, and $s_0^B\simeq 34~\gev{2}$ is the
continuum threshold in the $B$-channel\footnote{As we shall see
below, the LC SRs ``prefer'' a higher value.} as defined from the
2-point QCD SRs for the $B$-meson decay constant $f_B$
(see~\cite{Eletsky83}). For $t\approx 0$, $\epsilon\simeq
0.5-0.6$ and the LC SR should not be so sensitive to the
end-point region $x \sim 0$. Obviously, the end-point region
becomes to be important for higher momentum transfers $t$.
However, for $t\ge 20~\gev{2}$ the LC expansion would hardly make
sense. The second factor which eventually decreases the
importance of the end-point region is connected with the standard
Borel transformation of the SR with respect to the virtuality of
the $B$-meson current: $-p_B^2 \to M_B^2$. The corresponding
contribution from the coefficient function produces a standard
suppression exponent: $\exp(\bar{x}(t-m_b^2)/x\,M_B^2)$.
Numerically, it occurred to be less important.


We have treated the LC SRs using the same input parameters and the same
procedure  of extracting the physical form factors as in
Ref.\cite{BB97}. However, if one tries to fix the onset of the "continuum"
by hand to the value $s_0^B \simeq 34~\gev{2}$ dictated by the 2-point
SRs for $f_B$, one encounters {\bf inadmissible} uncertainties
in the determination of the form factors when using our new
nonperturbative input DAs. To get a stable SR, one is
forced to allow a higher value for the $s_0^B$ parameter.


%The source of the difference can be traced to the difference
%of the estimates like (\ref{rho-1}) for the simplest integrals.
Below, the form factor values are written at a zero momentum
transfer ($t=0$)
as compared with B\&B results:
\ba  %
 V(0)
 &=& \left\{
   \begin{array}{l c}
    0.37(1) & (\mbox{here}~~[s_0=50~\gev{2}], ~\chi^2\approx 0.4) \nn \\
    0.35(2) & (\mbox{\cite{BB97}}~~[s_0=34~\gev{2}], ~\chi^2\approx 3.4)
\nn\end{array} \right.\\
 A_1(0)
 &=& \left\{
   \begin{array}{l c}
    0.283(4) & (\mbox{here}~~[s_0=45~\gev{2}], ~\chi^2\approx 0.1) \la{form factors}\\
     0.27(1) & (\mbox{\cite{BB97}}~~[s_0=34~\gev{2}], ~\chi^2\approx 1.1)
   \end{array}\right.\\
 A_2(0)
 &=& \left\{
   \begin{array}{l c}
    0.30(1) & (\mbox{here}~~[s_0=50~\gev{2}],  ~\chi^2\approx 0.2 ) \\
    0.28(1) & (\mbox{\cite{BB97}}~~[s_0=34~\gev{2}], ~\chi^2\approx 1.1)
  \la{formfactors}
 \nn\end{array}\right.
\ea
%All evaluated form factors  are a little bit larger
%than the corresponding estimations with the B\&B leading twist DAs.
Our form factors are slightly higher than those in \cite{BB97}
and possess a  better accuracy
(compare $\chi^2$ in (\ref{formfactors})).
The difference becomes more pronounced
for a large value of the momentum transfer $t$,
($m_b^2-t \sim {\cal O}(m_b)$).
The last is not surprising due to higher sensitivity
to the end-point behavior of the input DA in this region.
The form factors presented are determined
with new ``optimal'' thresholds $s_0^B$
providing few times better processing accuracy.
Note that the parameters of the usual ``pole'' parameterization
of the form factors change significantly as compared to that in
\cite{BB97}, {\it e.g.},
\begin{eqnarray}
% \label{A-1}
 A_1(t)= \frac{0.283}{1-0.157(t/m_B^2)-0.837(t/m_B^2)^2} \nonumber
\end{eqnarray}
The important form factor $A_1(t)$ (solid line)
increases about $5 - 10 \% $ in comparison with the B\&B result
(the bars in the figure show the errors of the  B\&B calculations),
with an optimal threshold $s_0^B \simeq 45$ GeV$^2$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
 $$\psboxto(0.5\textwidth;0cm){fig7.eps}$$
    \myfig{\label{fig-A1(t)}}
     \vspace*{-4mm}
      \caption{\footnotesize Form factor $A_1(t)$; solid line
       corresponds to our processing of LC QCD SR, dashed line --
       to processing following the B\&B formulas (\cite{BB97})
       (the bars in the figure show the errors of the B\&B calculations).}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Discussion}
%From a physical point of view one should
%consider the duality interval
%$s_0^B$ as a characteristic of the spectra in the $B$-channel.
%Thus, in a self-consistent approach it is desirable to obtain the same
%(physical) value for $s_0^B$ from different SRs.

%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Conclusion}
  \la{sect-7}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Let us summarize the main results of this paper:
\begin{enumerate}

\item We construct NLC SRs for DA for each P-parity channels,
based on the properties of the duality transformation.
The negative parity NLC SR for transversely polarized
$\rho$-, $\rho'$-mesons works rather well and allows us to estimate
the 2-nd, 4-th, 6-th, and 8-th moments of the leading twist DAs.
The positive parity SR for the transversely polarized $b_1$-meson
can provide only the value of the $b_1$-meson lepton decay constant,
$f_{b_1}^T$.
It should be emphasized that an analogous evaluation of the moments within
the standard QCD SR approach is impossible.

\item Results of processing different NLC SRs of the ``pure"
(see Figs. 1b,~2b,~3b)
and ``mixed" (see Figs. 1a,~2a,~3a) parity are compared,
and a reasonable agreement between them is found.
The ``mixed" SR in the standard version admits merely a window
of possible values
of the second moment $\langle \xi^2 \rangle$ (see, \eg,~\cite{BB96});
the position of the window is corrected here and, as a result,
agrees with the NLC SR results presented in Table 1.

\item  The models for the leading twist DAs of the $\rho_{\perp}$- and
$\rho_{\perp}'$-mesons, (\ref{eq:mod_tro},\ref{eq:mod_trs}),
and of the $b_1^{\perp}$-meson, (\ref{eq:mod_b1}), are suggested.
The shape of a new $\rho_{\perp}$-meson distribution (see Fig. 4a)
drastically differs from that obtained by Ball and Braun \cite{BB96}
only on the basis of the value $a_2=0.2$. The latter estimate is discussed
in sect. 5.
%It should be emphasized that the Ball--Braun SR
%for the coefficient $a_2$, Eq.(3.20) of \cite{BB96},
%produces the estimate $a_2 = 0.3 \pm 0.1$.

\item We estimate important integrals appearing
in perturbative QCD predictions for different exclusive reactions,
$\Ds \langle x^{-1}\rangle_{M}\equiv
 \int_{0}^{1} \frac{\varphi_M^{T}(x)}{x}\ dx $
in (\ref{rho-1})-(\ref{b_1-1}),
based on our results
for the DA shapes.
We check the self-consistency of these results
by comparing them with those obtained
from an independent ``mixed" QCD SR for the inverse moment
$\langle x^{-1}\rangle_{M}$ and find an agreement.

\item Form factors of the process $B \to \rho ~e \nu$, $V(t)$, $A_{1,2}(t)$
where $t$ is momentum transfer are also re-estimated
in the framework of the light-cone SR approach \cite{BB97}
on the basis of the new model for the $\rho$-meson DAs;
the results are slightly higher and have uncertainties a few times
as small as those obtained by Ball and Braun.

\end{enumerate}

Finally, we can conclude that the nonlocal condensate QCD SR approach
to distribution amplitudes is self-consistent and gives reliable results.
An open problem of this approach is to determine
well-established models of distribution functions
$f_\Gamma(\nu)$ from the theory of nonperturbative QCD vacuum.
First direct attempts to calculate quark NLC have been done
in lattice simulations in ~\cite{DDM99}.
The ``short distance" correlation length of NLC
has also been extracted later in \cite{DEJM2000};
it turns out to be reasonably close to the value of $1/\lambda_q$
and confirms the validity of our Gaussian NLC model.

\vspace*{4mm}

\noindent\textbf{\large Acknowledgments} \vspace*{3mm}

This work was partially supported by
the Russian Foundation for Basic Research (grant N 00-02-16696)
and Heisenberg--Landau Program.
We are grateful to R.~Ruskov for help in a better understanding
of $B$ decay (sect. 8), to O.~V.~Teryaev,
 M.~Polyakov, and N.~G.~Stefanis for fruitful discussions.
We gratefully acknowledge the warm hospitality
of Prof. K.~Goeke and Dr.~Dr.~N.~G.~Stefanis at Bochum University,
where this work was completed.
%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{appendix}
\vspace*{12mm}
\appendix
\hspace*{2mm}{\Large \bf Appendix}
\vspace{-3mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Decomposition of rank-4 tensor $\Pi^{\mu \nu ;\alpha \beta }_{(N)}$}
 \renewcommand{\theequation}{\thesection.\arabic{equation}}
  \la{subs-A.1}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \ba
 P_1^{\mu \nu;\alpha \beta}
 &\equiv & \frac1{2q^2}
   \left[g^{\mu\alpha}q^\nu q^\beta
       - g^{\nu\alpha}q^\mu q^\beta
       - g^{\mu\beta}q^\nu q^\alpha
       + g^{\nu\beta}q^\mu q^\alpha
   \right] \ ;\label{eq-AP1} \\
 P_2^{\mu \nu ;\alpha \beta}
 &\equiv & \frac12
  \left[g^{\mu\alpha}g^{\nu\beta}
      - g^{\mu\beta}g^{\nu\alpha}\right]
 - P_1^{\mu \nu;\alpha \beta}\ ;\label{eq-AP2} \\
 Q_1^{\mu \nu ;\alpha \beta}
 &\equiv & \frac1{2(qz)}
    \left[g^{\mu\alpha}q^{\nu}z^{\beta}
        + g^{\nu\beta}q^{\mu}z^{\alpha}
        - g^{\mu\beta}q^{\nu}z^{\alpha}
        - g^{\nu\alpha}q^{\mu}z^{\beta}
    \right]\ ;\label{eq-AQ1} \\
 Q_3^{\mu \nu ;\alpha \beta} &\equiv &
   \frac1{2(qz)}
    \left[g^{\mu\alpha}z^{\nu}q^{\beta}
        + g^{\nu\beta}z^{\mu}q^{\alpha}
        - g^{\mu\beta}z^{\nu}q^{\alpha}
        - g^{\nu\alpha}z^{\mu}q^{\beta}
    \right]\ ;\label{eq-AQ3} \\
 Q_z^{\mu \nu ;\alpha \beta} &\equiv &
   \frac{q^2}{2(qz)^2}
    \left[g^{\mu\alpha}z^{\nu}z^{\beta}
        + g^{\nu\beta}z^{\mu}z^{\alpha}
        - g^{\mu\beta}z^{\nu}z^{\alpha}
        - g^{\nu\alpha}z^{\mu}z^{\beta}
    \right]\ ;\label{eq-AQZ} \\
 Q_q^{\mu \nu ;\alpha \beta} &\equiv &
   \frac{1}{2(qz)^2}
    \left(q^{\alpha}z^{\beta} - q^{\beta}z^{\alpha}\right)
    \left(q^{\mu}z^{\nu} - q^{\nu}z^{\mu}
     \vphantom{q^{\alpha}z^{\beta}}\right)\ .\label{eq-AQQ}
\ea

\ba \la{norm-muz}
g^{\mu\alpha}z^{\nu}z^{\beta}
P_1^{\mu \nu;\alpha \beta}
\equiv P_1^{\mu z;\mu z}
=-P_2^{\mu z;\mu z}=\frac{(qz)^2}{q^2};
~Q_1^{\mu z;\mu z}=
~Q_3^{\mu z;\mu z}=
~Q_z^{\mu z;\mu z}=
~Q_q^{\mu z;\mu z}=0;\\
\la{norm-qz}
q^{\mu}q^{\alpha}z^{\nu}z^{\beta} P_1^{\mu \nu;\alpha \beta}\equiv P_1^{q z;q z}=Q_1^{q z;q z}=
Q_3^{q z;q z}=-Q_q^{q z;q z}=-\frac{(qz)^2}{2};~P_2^{q z;q z}=Q_z^{q z;q z}=0.
\ea
Let us write down the parameterization of matrix elements
of a composite tensor current operator, see, \eg, \cite{BB98}:
\ba
 \va{0\mid\bar{d}(z)\sigma_{\mu\nu}u(0)
      \mid \rho_{\perp}(p,\lambda)}\Big|_{z^2=0}
 &=&\ if_{\rho_{\perp}}^{T}\left[
      \left(\varepsilon_{\mu}(p,\lambda)p_{\nu}
           -\varepsilon_{\nu}(p,\lambda)p_{\mu}
      \right)
       \int^1_0 dx\ \varphi^T_{\rho}(x)\ e^{ix(zp)} \right.
\nonumber\\ &&\ \ \quad
    + \left(\varepsilon_{\mu}(p,\lambda)z_{\nu}
           -\varepsilon_{\nu}(p,\lambda)z_{\mu}
      \right)p^2
       \int^1_0 dx\ V_{1}(x)\ e^{ix(zp)}\nonumber\\
 &&\left.\ \ \quad
    + \left(p^{\mu}z_{\nu}-p^{\nu}z_{\mu}\right)
       {(\varepsilon(p,\lambda)z)}p^2
       \int^1_0 dx\ V_{2}(x)\ e^{ix(zp)}
    \right]\ . \label{eq-AWF_rho}\\
\va{0\mid\bar{d}(z)\sigma_{\mu\nu}u(0)
     \mid b_{1}(p,\lambda)}\Big|_{z^2=0}
 &=&\ f_{b_{1}}^{T}\left[
      \epsilon_{\mu\nu\alpha\beta}
       \varepsilon^{\alpha}(p,\lambda)p^{\beta}
        \int^1_0 dx\ \varphi_{b_{1}}(x)\ e^{ix(zp)} \right.
\nonumber\\ &&\ \
   \quad + \epsilon_{\mu\nu\alpha\beta}
            \varepsilon^{\alpha}(p,\lambda)z^{\beta}p^2
             \int^1_0 dx\ U_{1}(x)\ e^{ix(zp)}\nonumber\\
 &&\left.\ \
   \quad + \epsilon_{\mu\nu\alpha\beta}p^{\alpha}z^{\beta}
            {(\varepsilon(p,\lambda)z)}p^2
             \int^1_0 dx\ U_{2}(x)\ e^{ix(zp)}
 \right]\ .
 \label{eq-AWF_b1}
\ea
Here we decode our shorthand notation used in Section~2:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% File "LHS_RHO": page 5 %%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%   c1 = + V0 - U1 - U2 ;   x1 = + V1 + U2 ;   xz = - U2 ;        %%%%%%
%%%%%   c2 = - U0 - U1 - U2 ;   x3 = + U1 + U2 ;   xq = - V2 + U2 ;   %%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
 v_0 \equiv \left|f_{\rho_{\perp}}^{T}\right|^2\va{x^N}_{\rho_{\perp}}\ ;
 \qquad
 v_1 \equiv \left|f_{\rho_{\perp}}^{T}\right|^2\va{-iNx^{N-1}}_{V_1}\ ;
 \qquad
 v_2 \equiv \left|f_{\rho_{\perp}}^{T}\right|^2\va{-N(N-1)x^{N-2}}_{V_2}
\ ; $$ $$
 u_0 \equiv \left|f_{b_{\perp}}^{T}\right|^2\va{x^N}_{b_{\perp}}\ ;
 \qquad
 u_1 \equiv \left|f_{b_{\perp}}^{T}\right|^2\va{-iNx^{N-1}}_{U_1}\ ;
 \qquad
 u_2 \equiv \left|f_{b_{\perp}}^{T}\right|^2\va{-N(N-1)x^{N-2}}_{U_2}
\ , $$
(with $\va{f(x)}_U \equiv \int_0^1  dx\ f(x)U(x)$).
In the general case, the whole system of equations for
different twist DA contributions is of the following form
\ba
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%   c1 = + V0 - U1 - U2 ;
 \frac{\Pi_-(q^2,qz)}{2(qz)^{N} q^{2}}
 = - v_0 + u_1 + u_2\ ;\quad
%%%%%%   x1 = + V1 + U2 ;
 \frac{K_1(q^2,qz)}{2(qz)^{N} q^{2}}
 = - v_1 - u_2\ ;\quad
%%%%%%  xz = - U2 ;
 \frac{K_z(q^2,qz)}{2(qz)^{N} q^{2}}
 = + u_2\ ; ~~~~&& \label{eq-Pi-}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%   c2 = - U0 - U1 - U2 ;
 \frac{\Pi_+(q^2,qz)}{2(qz)^{N} q^{2}}
 = + u_0 + u_1 + u_2\ ;\quad
%%%%%%  x3 = + U1 + U2 ;
 \frac{K_3(q^2,qz)}{2(qz)^{N} q^{2}}
 = - u_1 - u_2\ ;\quad
%%%%%%  xq = - V2 + U2 ;
 \frac{K_q(q^2,qz)}{2(qz)^{N} q^{2}}
 = v_2 - u_2\ . && \label{eq-Pi+}
\ea


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Expressions for nonlocal contributions to SR}
 \la{subs-B.1}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To construct SR for distribution amplitudes, it is useful to
parameterize NLC behaviors by the ``distribution functions''
\cite{MR92,BM98,Rad94,BM95} {\em a'la} $\alpha$-representation of
propagators, \eg, $f_S(\alpha)$ for the scalar condensate
$M_S(z^2)$ \footnote{ In deriving these sum rules we can always
make a Wick rotation, i.e., we assume that all coordinates are
Euclidean, $z^2 <0$.} \ba
 M_S\left(z^2\right)
  = \langle\bar q(0)q(0)\rangle
     \int_{0}^{\infty} e^{\alpha z^2/4}\, f_S(\alpha)\, d\alpha,~
 \mbox{where}~\int_{0}^{\infty}\, f_S(\alpha)\, d\alpha = 1,\
 \int_{0}^{\infty}\alpha f_S(\alpha) d\alpha =
 \frac{\lambda_q^2}2,
 \label{eq:qq}
%\nonumber
\ea
 and for the vector condensate $M^{\mu}_V(z^2)$,
\ba
 M_V^{\mu}(z) \equiv \langle \bar q(0)\gamma^{\mu}q(z) \rangle
  = -iz^{\mu}~\frac{A_S}{4}
     \int_{0}^{\infty} e^{\alpha z^2/4}\, f_V(\alpha)\, d\alpha,~
 \mbox{where}~\int_{0}^{\infty}\, f_V(\alpha)\, d\alpha = 0.
 \label{eq:qgammaq}
%\nonumber
\ea
Here and in the following we take quark and gluon fields in the
fixed point gauge $z^{\mu}A_{\mu}(z)=0$ where the path-ordered
exponential $E(0,z)=1$.
The appearing in the SR quark-gluon-quark NLC
$M_{\mu \nu} (\tilde M_{\mu \nu})$,
\ba
&& M_{\mu \nu}(y,z) \equiv  \va{\bar q(0)\gamma_{\nu}\hat A_{\mu}(z)q(y)}
 = \Bigl( y_{\mu}z_{\nu} - g_{\mu \nu}(zy) \Bigr) \cdot M_{T1} +
   \Bigl( z_{\mu}z_{\nu} - g_{\mu \nu} z^2 \Bigr) \cdot M_{T2} + \cdots
\label{eq:m12} \\
&&\tilde M_{\mu \nu}(y,z) \equiv \langle \bar q(0) \gamma_{\nu} (\gamma_{5})
 \hat A_{\mu}(z) q(y) \rangle
= \varepsilon_{\mu \nu \rho \sigma} z_{\rho} y_{\sigma} \cdot M_{T3} + \cdots,
\label{eq:m3}
\ea
can be decomposed in form factors $M_{T1-T3}$,
%the tensors  multiplied by the scalar form factors $M_{Ti}$,
where the tensors in front of them satisfy
the gauge condition
$z^{\mu}M_{\mu \nu} (\tilde M_{\mu \nu}) = 0 \
(\mbox{ since } z^{\mu} \hat A_{\mu}(z) = 0)$.
The NLC $M_{T1-T3}$ can be parameterized
by a triple integral representation
\be
M_{Ti}(z^2,y^2,(z-y)^2)
 =  A_{Ti} \int_{0}^{\infty}
     e^{(\alpha_1z^2/4+\alpha_2y^2/4+\alpha_3(z-y)^2/4)}
       f_i(\alpha_1,\alpha_2,\alpha_3)\,d\alpha_1d\alpha_2d\alpha_3,
 \label{eq:MMM}
\ee
where
$A_{Ti} = \bigl\{-\frac{3}{8}A_{S}, \frac{1}{2} A_{S}, \frac{3}{8}A_{S} \bigr\}$,
and $\Ds A_S=\frac{8\pi}{81}\langle\sqrt{\as}\bar q(0)q(0)\rangle^2$.
The function $f_S(\alpha)$
and other similar functions $f_\Gamma(\alpha)$
describe distributions of vacuum fields in virtuality $\alpha$
for every type ($\Gamma$) of NLC.
The convolutions $\Delta \Phi_{\Gamma}(x, M^2)$
of the distribution functions $f_\Gamma$
and coefficient functions completely determine the r.h.s. of SR's,
so $\Delta \Phi_{\Gamma}$ depends on the model of $f_{\Gamma}$.
For vacuum distribution functions $f_{\Gamma}(\alpha)$,
we use the set of the simplest ansatzes
\ba
 f_S(\alpha) &=& \delta\left(\alpha-\lambda_q^2/2\right)\ ;\quad \
 f_V(\alpha)\ =\ \delta^\prime\left(\alpha-\lambda_q^2/2\right)\ ;
\la{eq:ansv}\\
 f_{T_{1,2,3}}(\alpha_1,\alpha_2,\alpha_3)
          &=& \delta\left(\alpha_1-\lambda_q^2/2\right)
               \delta\left(\alpha_2-\lambda_q^2/2\right)
                \delta\left(\alpha_3-\lambda_q^2/2\right)\ .
\label{eq:anstril} \ea Their meaning and relation to initial NLCs
have been discussed in detail in \cite{MR86,MR92}. The
contributions $\Delta \Phi_{\Gamma}(x, M^2)$ to the r.h.s of SR,
 corresponding to these ansatzes,
are shown below.
The limit of these expressions to the standard (local) contributions
$\varphi_{\Gamma}(x, M^2)$ -- $\lambda_q^2\to 0$,
$\Delta \Phi_{\Gamma}(x, M^2)\to\Delta \varphi_{\Gamma}(x, M^2)$
are also written for comparison.
Hereafter $\Delta \equiv \lambda_q^2/(2M^2)$,
$\bar\Delta\equiv 1-\Delta$:
\ba
 \Delta\Phi_S\left(x,M^2\right)
  &=&\frac{A_S}{M^4}
      \frac{18}{\bar\Delta\Delta^2}
       \left\{
        \theta\left(\bar x>\Delta>x\right)
         \bar x\left[x+(\Delta-x)\ln\left(\bar x\right)\right]
       + \xx + \right. \nonumber \\
&&\qquad\qquad
\left. + \theta(1>\Delta)\theta\left(\Delta>x>\bar\Delta\right)
         \left[\bar\Delta
              +\left(\Delta-2\bar xx\right)\ln(\Delta)\right]
        \right\},
\la{eq:phi_s}\\
 \Delta\varphi_S\left(x, M^2\right)
  &=&\frac{A_S}{M^4}9
      \left(\delta(x)+\xx\right); \nn \\
      %~~\Delta\va{\xi^N}_{\varphi_S}  =  \frac{18A_S}{M^4}\ ; \nn \\
 \Delta\Phi_V\left(x, M^2\right) &=& \frac{A_S}{M^4}
     \left(x\delta'\left(\bar x-\Delta\right)+\xx\right), \\
 \Delta\varphi_V\left(x, M^2\right) &=& \frac{A_S}{M^4}
     \left(x\delta'\left(\bar x\right)+\xx\right);\\ %~ \Delta\va{\xi^N}_{\tilde\Phi_V}
  %= -\frac{8A_0}{M^2}(2N+1)\ \la{eq:phi_v}\\
 \Delta\Phi_{T_1}\left(x,M^2\right)
  &=& -\frac{3 A_S}{M^4}\theta(1>2\Delta)\left\{
      \left[\delta(x-2\Delta) - \delta(x-\Delta)\right]
       \left(\frac1{\Delta} - 2\right)
       + \theta(2\Delta>x) \cdot\nn \right.\\ &&\left.
    \theta(x>\Delta)
  \frac{\bar x}{\bar\Delta}
  \left[\frac{x-2\Delta}{\Delta \bar\Delta} \right]
  \right\} + \xx,
\la{eq:phi_t1} \\
 \Delta\varphi_{T_1}\left(x, M^2\right) &=& \frac{3A_S}{M^4}
     \left(\delta'\left(\bar x\right)+\xx\right); \nn \\
 \Delta\Phi_{T_2}\left(x,M^2\right)
  &=& \frac{4 A_S}{M^4}\bar x \theta(1>2\Delta)
     \left\{\frac{\delta(x-2\Delta)}{\Delta}
           -\theta(2\Delta>x)\theta(x>\Delta) \cdot
         \right. \nonumber \\ &&\left.
             \frac{1+2x-4\Delta}{\bar\Delta\Delta^2}\right\}
  + \xx,
\la{eq:phi_t2}\\
 \Delta\varphi_{T_2}\left(x, M^2\right) &=& -\frac{2 A_S}{M^4}
    \left(x\delta'\left(\bar x\right)+\xx\right); \nn \\
 \Delta\Phi_{T_3}\left(x, M^2\right)
  &=& \frac{3 A_S\bar x}{M^4\bar\Delta\Delta}
     \left\{\theta(2\Delta>x)\theta(x>\Delta)\theta(1>2\Delta)
      \left[2-\frac{\bar x}{\bar\Delta}-\frac{\Delta}{\bar\Delta}
      \right]\right\} \nn \\
      & & + \xx, \\
 \Delta\varphi_{T_3}\left(x, M^2\right) &=& \frac{3 A_S}{M^4}
     \left(\delta\left(\bar x\right) +\xx\right); \nn \\
 \Delta\Phi_G\left(x,M^2\right) &=&
      \frac{\langle \alpha_s GG \rangle}{24\pi M^2}
       \left(\delta\left(x-\Delta\right)+ \xx \right), \\
 \Delta\varphi_{G}\left(x, M^2 \right) &=&
      \frac{\langle \alpha_s GG \rangle}{24\pi M^2}
 \left(\delta\left(\bar x\right)+\xx\right); \nn  \\ %~ \Delta\va{\xi^N}_{\tilde\Phi_G}
%  = - \frac{\langle \alpha_s GG \rangle}{24\pi} \left(\frac{N-1}{N+1}\right)\ \nn \\
 \Delta\Phi_G'\left(x,M^2\right) &=&
    \frac{\langle \alpha_s GG \rangle}{6\pi M^2}
     \frac{\theta\left(\Delta<x\right)\theta\left(x<1-\Delta\right)}
          {1-2\Delta};
\la{eq:phi_gs}\\
 \Delta\varphi_G'\left(x,M^2\right) &=&
    \frac{\langle \alpha_s GG \rangle}{6\pi M^2}. \nn
\ea For quark and gluon condensates, we use the standard estimates
$\langle\sqrt{\as}\bar q(0)q(0)\rangle\approx (-0.238\
\mbox{GeV})^3$, $\Ds\frac{\langle\as GG\rangle}{12\pi}\approx
0.001$ GeV$^4$ \cite{SVZ} and $\Ds\lambda_q^2
 =\frac{\langle\bar q\left(ig\sigma_{\mu\nu}G^{\mu\nu}\right)q\rangle}
         {2\langle\bar qq\rangle}
 =  0.4 \pm 0.1$~GeV$^2$
normalized at $\mu^2 \approx 1$~GeV$^2$.

{\bf Expressions for perturbative spectral density}:
Radiative corrections reach 10 \% of the Born result at $s \sim 1~\gev{2}$.
\ba
% \rho^{\itxt{Born}}(x,s)
%  &=& \frac{3}{2\pi^2} x\bar x, \\
 \rho^{\itxt{pert}}_T(x,s)
  &=& \frac{3}{2\pi^2} x\bar x
     \left\{1+\frac{\alpha_s(\mu^2)C_F}{4\pi}
      \left(2\ln\left[\frac{s}{\mu^2}\right]
            + 6 - \frac{\pi^2}3 + \ln^2(\bar x/x) %\right.\right. \\
         +\ln(x\bar x)\right)\right\}.
\la{eq:rcrho}
\ea
Here $\mu^2 \sim 1~\gev{2}$ corresponds to the average value of
the Borel parameter $M^2$ in the stability window;
$\alpha_s\left(1\mbox{GeV}^2\right)\approx 0.52$.
We also use the `mixed' perturbative spectral density
suggested in \cite{BO97} in the ``mixed'' SR:
\be
 \rho^{\itxt{mixed}}_T(x,s;s_\rho^T,s_b^T)
  \equiv
   \rho^{\itxt{pert}}_T(x;s)
    \frac12\left[\theta\left(s_\rho^T-s\right)
               + \theta\left(s_{b}^T-s\right)
    \right]\ .
 \label{eq:RoMixed}
\ee

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{About $\chi^2$-definition in Sum Rules}
 \la{subs-C.1}\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  Let us discuss the definition of $\chi^2$ for the SR case.
We have here the function $F(M^2,s)$, and the problem is to find
the best value $s_0$, such that $F(M^2,s_0)$ is the most close to
a constant value for $M^2_{-} \le M^2 \le M^2_{+}$ (values of
$M^2_{\pm}$ are known and fixed from standard constraints of QCD
SR, see \cite{SVZ,BM98} ). We define the function $\chi^2(s)$ for
the curve $F(M^2,s)$ with $M^2 \in \left[M^2_{-},M^2_{+}\right]$
in the following manner: \be
 \chi^2(s)
 \equiv \frac{1}{(N-1)\epsilon^{2}}
         \sum_{k=0}^{N}
          \left[ F\left(M_{-}^2+k\delta, s\right)
               - \frac{1}{N+1}\sum_{k=0}^{N}
                 F\left(M_{-}^2+k\delta, s\right)
           \right]^2\ ,
 \label{eq-xi2}
\ee
where $\delta = (M_{+}^2-M_{-}^2)/N$, $N \simeq 10$,
and $\epsilon$ is of an order of the last decimal digit
in $F(M^2,s)$ we are interested in
(in the case of decay constant $f_{\rho} \approx 200~\Mev$,
$\epsilon \approx 1~\Mev$;
in the case of the second moment $\va{\xi^2}_{\rho} \approx 0.25$,
$\epsilon \approx 0.01$).
Then, if we obtain $\chi^2(s_0) \approx 1$, this tells us
that the mean deviation of $F(M^2,s_0)$ from a constant value
in the region $[M_{-}^2,M_{+}^2]$ is about $\epsilon$.
To find the minimum value of $\chi^2$ and the corresponding $s_0$,
we used the code Mathematica.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{appendix}

%\bibliographystyle{myplb}
%\bibliography{pion}
%\end{document}

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