\documentstyle[12pt]{article}
\topmargin 0mm
\textwidth 16cm
\textheight 22.5cm
\evensidemargin 0mm
\oddsidemargin 0mm
%\parskip=\medskipamount
%\def\baselinestretch{1.2}
\newcommand{\ch}{\cosh}
\newcommand{\sh}{\sinh}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\renewcommand{\a}{\alpha}
\newcommand{\ad}{\dot{\alpha}}
\renewcommand{\b}{\beta}
\newcommand{\bd}{\dot{\beta}}
\newcommand{\q}{\theta}
\newcommand{\pa}{\partial}
\renewcommand{\l}{\lambda}
\newcommand{\g}{\gamma}
\newcommand{\f}{\phi}
\newcommand{\F}{\Phi}
\newcommand{\bD}{\bar{D}}
\renewcommand{\L}{\Lambda}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\begin{document}

\immediate\write16{<<WARNING: LINEDRAW macros work with emTeX-dvivers
                    and other drivers supporting emTeX \special's
                    (dviscr, dvihplj, dvidot, dvips, dviwin, etc.) >>}

%% Macros for drawing Feynman graphs and other complex diagrams
%% Designed by A.V.Voronin (1993); modified in 1995
%% Steklov Math. Inst., e-mail: av@voronin.mian.su
%%
\newdimen\Lengthunit       \Lengthunit  = 1.5cm
\newcount\Nhalfperiods     \Nhalfperiods= 9
\newcount\magnitude        \magnitude = 1000


\catcode`\*=11
\newdimen\L*   \newdimen\d*   \newdimen\d**
\newdimen\dm*  \newdimen\dd*  \newdimen\dt*
\newdimen\a*   \newdimen\b*   \newdimen\c*
\newdimen\a**  \newdimen\b**
\newdimen\xL*  \newdimen\yL*
\newdimen\rx*  \newdimen\ry*
\newdimen\tmp* \newdimen\linwid*



\newcount\k*   \newcount\l*   \newcount\m*
\newcount\k**  \newcount\l**  \newcount\m**
\newcount\n*   \newcount\dn*  \newcount\r*
\newcount\N*   \newcount\*one \newcount\*two  \*one=1 \*two=2
\newcount\*ths \*ths=1000
\newcount\angle*  \newcount\q*  \newcount\q**
\newcount\angle** \angle**=0
\newcount\sc*     \sc*=0



\newtoks\cos*  \cos*={1}
\newtoks\sin*  \sin*={0}







\catcode`\[=13



\def\rotate(#1){\advance\angle**#1\angle*=\angle**
\q**=\angle*\ifnum\q**<0\q**=-\q**\fi
\ifnum\q**>360\q*=\angle*\divide\q*360\multiply\q*360\advance\angle*-\q*\fi
\ifnum\angle*<0\advance\angle*360\fi\q**=\angle*\divide\q**90\q**=\q**
\def\sgcos*{+}\def\sgsin*{+}\relax
\ifcase\q**\or
 \def\sgcos*{-}\def\sgsin*{+}\or
 \def\sgcos*{-}\def\sgsin*{-}\or
 \def\sgcos*{+}\def\sgsin*{-}\else\fi
\q*=\q**
\multiply\q*90\advance\angle*-\q*
\ifnum\angle*>45\sc*=1\angle*=-\angle*\advance\angle*90\else\sc*=0\fi
\def[##1,##2]{\ifnum\sc*=0\relax
\edef\cs*{\sgcos*.##1}\edef\sn*{\sgsin*.##2}\ifcase\q**\or
 \edef\cs*{\sgcos*.##2}\edef\sn*{\sgsin*.##1}\or
 \edef\cs*{\sgcos*.##1}\edef\sn*{\sgsin*.##2}\or
 \edef\cs*{\sgcos*.##2}\edef\sn*{\sgsin*.##1}\else\fi\else
\edef\cs*{\sgcos*.##2}\edef\sn*{\sgsin*.##1}\ifcase\q**\or
 \edef\cs*{\sgcos*.##1}\edef\sn*{\sgsin*.##2}\or
 \edef\cs*{\sgcos*.##2}\edef\sn*{\sgsin*.##1}\or
 \edef\cs*{\sgcos*.##1}\edef\sn*{\sgsin*.##2}\else\fi\fi
\cos*={\cs*}\sin*={\sn*}\global\edef\gcos*{\cs*}\global\edef\gsin*{\sn*}}\relax
\ifcase\angle*[9999,0]\or
[999,017]\or[999,034]\or[998,052]\or[997,069]\or[996,087]\or
[994,104]\or[992,121]\or[990,139]\or[987,156]\or[984,173]\or
[981,190]\or[978,207]\or[974,224]\or[970,241]\or[965,258]\or
[961,275]\or[956,292]\or[951,309]\or[945,325]\or[939,342]\or
[933,358]\or[927,374]\or[920,390]\or[913,406]\or[906,422]\or
[898,438]\or[891,453]\or[882,469]\or[874,484]\or[866,499]\or
[857,515]\or[848,529]\or[838,544]\or[829,559]\or[819,573]\or
[809,587]\or[798,601]\or[788,615]\or[777,629]\or[766,642]\or
[754,656]\or[743,669]\or[731,681]\or[719,694]\or[707,707]\or
\else[9999,0]\fi}



\catcode`\[=12



\def\GRAPH(hsize=#1)#2{\hbox to #1\Lengthunit{#2\hss}}



\def\Linewidth#1{\global\linwid*=#1\relax
\global\divide\linwid*10\global\multiply\linwid*\mag
\global\divide\linwid*100\special{em:linewidth \the\linwid*}}



\Linewidth{.4pt}
\def\sm*{\special{em:moveto}}
\def\sl*{\special{em:lineto}}
\let\moveto=\sm*
\let\lineto=\sl*
\newbox\spm*   \newbox\spl*
\setbox\spm*\hbox{\sm*}
\setbox\spl*\hbox{\sl*}



\def\mov#1(#2,#3)#4{\rlap{\L*=#1\Lengthunit
\xL*=#2\L* \yL*=#3\L*
\xL*=\xscale\xL* \yL*=\yscale\yL*
\rx* \the\cos*\xL* \tmp* \the\sin*\yL* \advance\rx*-\tmp*
\ry* \the\cos*\yL* \tmp* \the\sin*\xL* \advance\ry*\tmp*
\kern\rx*\raise\ry*\hbox{#4}}}



\def\rmov*(#1,#2)#3{\rlap{\xL*=#1\yL*=#2\relax
\rx* \the\cos*\xL* \tmp* \the\sin*\yL* \advance\rx*-\tmp*
\ry* \the\cos*\yL* \tmp* \the\sin*\xL* \advance\ry*\tmp*
\kern\rx*\raise\ry*\hbox{#3}}}



\def\lin#1(#2,#3){\rlap{\sm*\mov#1(#2,#3){\sl*}}}



\def\arr*(#1,#2,#3){\rmov*(#1\dd*,#1\dt*){\sm*
\rmov*(#2\dd*,#2\dt*){\rmov*(#3\dt*,-#3\dd*){\sl*}}\sm*
\rmov*(#2\dd*,#2\dt*){\rmov*(-#3\dt*,#3\dd*){\sl*}}}}



\def\arrow#1(#2,#3){\rlap{\lin#1(#2,#3)\mov#1(#2,#3){\relax
\d**=-.012\Lengthunit\dd*=#2\d**\dt*=#3\d**
\arr*(1,10,4)\arr*(3,8,4)\arr*(4.8,4.2,3)}}}



\def\arrlin#1(#2,#3){\rlap{\L*=#1\Lengthunit\L*=.5\L*
\lin#1(#2,#3)\rmov*(#2\L*,#3\L*){\arrow.1(#2,#3)}}}



\def\dasharrow#1(#2,#3){\rlap{{\Lengthunit=0.9\Lengthunit
\dashlin#1(#2,#3)\mov#1(#2,#3){\sm*}}\mov#1(#2,#3){\sl*
\d**=-.012\Lengthunit\dd*=#2\d**\dt*=#3\d**
\arr*(1,10,4)\arr*(3,8,4)\arr*(4.8,4.2,3)}}}



\def\clap#1{\hbox to 0pt{\hss #1\hss}}



\def\ind(#1,#2)#3{\rlap{\L*=.1\Lengthunit
\xL*=#1\L* \yL*=#2\L*
\rx* \the\cos*\xL* \tmp* \the\sin*\yL* \advance\rx*-\tmp*
\ry* \the\cos*\yL* \tmp* \the\sin*\xL* \advance\ry*\tmp*
\kern\rx*\raise\ry*\hbox{\lower2pt\clap{$#3$}}}}



\def\sh*(#1,#2)#3{\rlap{\dm*=\the\n*\d**
\xL*=\xscale\dm* \yL*=\yscale\dm* \xL*=#1\xL* \yL*=#2\yL*
\rx* \the\cos*\xL* \tmp* \the\sin*\yL* \advance\rx*-\tmp*
\ry* \the\cos*\yL* \tmp* \the\sin*\xL* \advance\ry*\tmp*
\kern\rx*\raise\ry*\hbox{#3}}}



\def\calcnum*#1(#2,#3){\a*=1000sp\b*=1000sp\a*=#2\a*\b*=#3\b*
\ifdim\a*<0pt\a*-\a*\fi\ifdim\b*<0pt\b*-\b*\fi
\ifdim\a*>\b*\c*=.96\a*\advance\c*.4\b*
\else\c*=.96\b*\advance\c*.4\a*\fi
\k*\a*\multiply\k*\k*\l*\b*\multiply\l*\l*
\m*\k*\advance\m*\l*\n*\c*\r*\n*\multiply\n*\n*
\dn*\m*\advance\dn*-\n*\divide\dn*2\divide\dn*\r*
\advance\r*\dn*
\c*=\the\Nhalfperiods5sp\c*=#1\c*\ifdim\c*<0pt\c*-\c*\fi
\multiply\c*\r*\N*\c*\divide\N*10000}



\def\dashlin#1(#2,#3){\rlap{\calcnum*#1(#2,#3)\relax
\d**=#1\Lengthunit\ifdim\d**<0pt\d**-\d**\fi
\divide\N*2\multiply\N*2\advance\N*\*one
\divide\d**\N*\sm*\n*\*one\sh*(#2,#3){\sl*}\loop
\advance\n*\*one\sh*(#2,#3){\sm*}\advance\n*\*one
\sh*(#2,#3){\sl*}\ifnum\n*<\N*\repeat}}



\def\dashdotlin#1(#2,#3){\rlap{\calcnum*#1(#2,#3)\relax
\d**=#1\Lengthunit\ifdim\d**<0pt\d**-\d**\fi
\divide\N*2\multiply\N*2\advance\N*1\multiply\N*2\relax
\divide\d**\N*\sm*\n*\*two\sh*(#2,#3){\sl*}\loop
\advance\n*\*one\sh*(#2,#3){\kern-1.48pt\lower.5pt\hbox{\rm.}}\relax
\advance\n*\*one\sh*(#2,#3){\sm*}\advance\n*\*two
\sh*(#2,#3){\sl*}\ifnum\n*<\N*\repeat}}



\def\shl*(#1,#2)#3{\kern#1#3\lower#2#3\hbox{\unhcopy\spl*}}



\def\trianglin#1(#2,#3){\rlap{\toks0={#2}\toks1={#3}\calcnum*#1(#2,#3)\relax
\dd*=.57\Lengthunit\dd*=#1\dd*\divide\dd*\N*
\divide\dd*\*ths \multiply\dd*\magnitude
\d**=#1\Lengthunit\ifdim\d**<0pt\d**-\d**\fi
\multiply\N*2\divide\d**\N*\sm*\n*\*one\loop
\shl**{\dd*}\dd*-\dd*\advance\n*2\relax
\ifnum\n*<\N*\repeat\n*\N*\shl**{0pt}}}



\def\wavelin#1(#2,#3){\rlap{\toks0={#2}\toks1={#3}\calcnum*#1(#2,#3)\relax
\dd*=.23\Lengthunit\dd*=#1\dd*\divide\dd*\N*
\divide\dd*\*ths \multiply\dd*\magnitude
\d**=#1\Lengthunit\ifdim\d**<0pt\d**-\d**\fi
\multiply\N*4\divide\d**\N*\sm*\n*\*one\loop
\shl**{\dd*}\dt*=1.3\dd*\advance\n*\*one
\shl**{\dt*}\advance\n*\*one
\shl**{\dd*}\advance\n*\*two
\dd*-\dd*\ifnum\n*<\N*\repeat\n*\N*\shl**{0pt}}}



\def\w*lin(#1,#2){\rlap{\toks0={#1}\toks1={#2}\d**=\Lengthunit\dd*=-.12\d**
\divide\dd*\*ths \multiply\dd*\magnitude
\N*8\divide\d**\N*\sm*\n*\*one\loop
\shl**{\dd*}\dt*=1.3\dd*\advance\n*\*one
\shl**{\dt*}\advance\n*\*one
\shl**{\dd*}\advance\n*\*one
\shl**{0pt}\dd*-\dd*\advance\n*1\ifnum\n*<\N*\repeat}}



\def\l*arc(#1,#2)[#3][#4]{\rlap{\toks0={#1}\toks1={#2}\d**=\Lengthunit
\dd*=#3.037\d**\dd*=#4\dd*\dt*=#3.049\d**\dt*=#4\dt*\ifdim\d**>10mm\relax
\d**=.25\d**\n*\*one\shl**{-\dd*}\n*\*two\shl**{-\dt*}\n*3\relax
\shl**{-\dd*}\n*4\relax\shl**{0pt}\else
\ifdim\d**>5mm\d**=.5\d**\n*\*one\shl**{-\dt*}\n*\*two
\shl**{0pt}\else\n*\*one\shl**{0pt}\fi\fi}}



\def\d*arc(#1,#2)[#3][#4]{\rlap{\toks0={#1}\toks1={#2}\d**=\Lengthunit
\dd*=#3.037\d**\dd*=#4\dd*\d**=.25\d**\sm*\n*\*one\shl**{-\dd*}\relax
\n*3\relax\sh*(#1,#2){\xL*=\xscale\dd*\yL*=\yscale\dd*
\kern#2\xL*\lower#1\yL*\hbox{\sm*}}\n*4\relax\shl**{0pt}}}



\def\shl**#1{\c*=\the\n*\d**\d*=#1\relax
\a*=\the\toks0\c*\b*=\the\toks1\d*\advance\a*-\b*
\b*=\the\toks1\c*\d*=\the\toks0\d*\advance\b*\d*
\a*=\xscale\a*\b*=\yscale\b*
\rx* \the\cos*\a* \tmp* \the\sin*\b* \advance\rx*-\tmp*
\ry* \the\cos*\b* \tmp* \the\sin*\a* \advance\ry*\tmp*
\raise\ry*\rlap{\kern\rx*\unhcopy\spl*}}



\def\wlin*#1(#2,#3)[#4]{\rlap{\toks0={#2}\toks1={#3}\relax
\c*=#1\l*\c*\c*=.01\Lengthunit\m*\c*\divide\l*\m*
\c*=\the\Nhalfperiods5sp\multiply\c*\l*\N*\c*\divide\N*\*ths
\divide\N*2\multiply\N*2\advance\N*\*one
\dd*=.002\Lengthunit\dd*=#4\dd*\multiply\dd*\l*\divide\dd*\N*
\divide\dd*\*ths \multiply\dd*\magnitude
\d**=#1\multiply\N*4\divide\d**\N*\sm*\n*\*one\loop
\shl**{\dd*}\dt*=1.3\dd*\advance\n*\*one
\shl**{\dt*}\advance\n*\*one
\shl**{\dd*}\advance\n*\*two
\dd*-\dd*\ifnum\n*<\N*\repeat\n*\N*\shl**{0pt}}}



\def\wavebox#1{\setbox0\hbox{#1}\relax
\a*=\wd0\advance\a*14pt\b*=\ht0\advance\b*\dp0\advance\b*14pt\relax
\hbox{\kern9pt\relax
\rmov*(0pt,\ht0){\rmov*(-7pt,7pt){\wlin*\a*(1,0)[+]\wlin*\b*(0,-1)[-]}}\relax
\rmov*(\wd0,-\dp0){\rmov*(7pt,-7pt){\wlin*\a*(-1,0)[+]\wlin*\b*(0,1)[-]}}\relax
\box0\kern9pt}}



\def\rectangle#1(#2,#3){\relax
\lin#1(#2,0)\lin#1(0,#3)\mov#1(0,#3){\lin#1(#2,0)}\mov#1(#2,0){\lin#1(0,#3)}}



\def\dashrectangle#1(#2,#3){\dashlin#1(#2,0)\dashlin#1(0,#3)\relax
\mov#1(0,#3){\dashlin#1(#2,0)}\mov#1(#2,0){\dashlin#1(0,#3)}}



\def\waverectangle#1(#2,#3){\L*=#1\Lengthunit\a*=#2\L*\b*=#3\L*
\ifdim\a*<0pt\a*-\a*\def\x*{-1}\else\def\x*{1}\fi
\ifdim\b*<0pt\b*-\b*\def\y*{-1}\else\def\y*{1}\fi
\wlin*\a*(\x*,0)[-]\wlin*\b*(0,\y*)[+]\relax
\mov#1(0,#3){\wlin*\a*(\x*,0)[+]}\mov#1(#2,0){\wlin*\b*(0,\y*)[-]}}



\def\calcparab*{\ifnum\n*>\m*\k*\N*\advance\k*-\n*\else\k*\n*\fi
\a*=\the\k* sp\a*=10\a*\b*\dm*\advance\b*-\a*\k*\b*
\a*=\the\*ths\b*\divide\a*\l*\multiply\a*\k*
\divide\a*\l*\k*\*ths\r*\a*\advance\k*-\r*\dt*=\the\k*\L*}



\def\arcto#1(#2,#3)[#4]{\rlap{\toks0={#2}\toks1={#3}\calcnum*#1(#2,#3)\relax
\dm*=135sp\dm*=#1\dm*\d**=#1\Lengthunit\ifdim\dm*<0pt\dm*-\dm*\fi
\multiply\dm*\r*\a*=.3\dm*\a*=#4\a*\ifdim\a*<0pt\a*-\a*\fi
\advance\dm*\a*\N*\dm*\divide\N*10000\relax
\divide\N*2\multiply\N*2\advance\N*\*one
\L*=-.25\d**\L*=#4\L*\divide\d**\N*\divide\L*\*ths
\m*\N*\divide\m*2\dm*=\the\m*5sp\l*\dm*\sm*\n*\*one\loop
\calcparab*\shl**{-\dt*}\advance\n*1\ifnum\n*<\N*\repeat}}



\def\arrarcto#1(#2,#3)[#4]{\L*=#1\Lengthunit\L*=.54\L*
\arcto#1(#2,#3)[#4]\rmov*(#2\L*,#3\L*){\d*=.457\L*\d*=#4\d*\d**-\d*
\rmov*(#3\d**,#2\d*){\arrow.02(#2,#3)}}}



\def\dasharcto#1(#2,#3)[#4]{\rlap{\toks0={#2}\toks1={#3}\relax
\calcnum*#1(#2,#3)\dm*=\the\N*5sp\a*=.3\dm*\a*=#4\a*\ifdim\a*<0pt\a*-\a*\fi
\advance\dm*\a*\N*\dm*
\divide\N*20\multiply\N*2\advance\N*1\d**=#1\Lengthunit
\L*=-.25\d**\L*=#4\L*\divide\d**\N*\divide\L*\*ths
\m*\N*\divide\m*2\dm*=\the\m*5sp\l*\dm*
\sm*\n*\*one\loop\calcparab*
\shl**{-\dt*}\advance\n*1\ifnum\n*>\N*\else\calcparab*
\sh*(#2,#3){\xL*=#3\dt* \yL*=#2\dt*
\rx* \the\cos*\xL* \tmp* \the\sin*\yL* \advance\rx*\tmp*
\ry* \the\cos*\yL* \tmp* \the\sin*\xL* \advance\ry*-\tmp*
\kern\rx*\lower\ry*\hbox{\sm*}}\fi
\advance\n*1\ifnum\n*<\N*\repeat}}



\def\*shl*#1{\c*=\the\n*\d**\advance\c*#1\a**\d*\dt*\advance\d*#1\b**
\a*=\the\toks0\c*\b*=\the\toks1\d*\advance\a*-\b*
\b*=\the\toks1\c*\d*=\the\toks0\d*\advance\b*\d*
\rx* \the\cos*\a* \tmp* \the\sin*\b* \advance\rx*-\tmp*
\ry* \the\cos*\b* \tmp* \the\sin*\a* \advance\ry*\tmp*
\raise\ry*\rlap{\kern\rx*\unhcopy\spl*}}



\def\calcnormal*#1{\b**=10000sp\a**\b**\k*\n*\advance\k*-\m*
\multiply\a**\k*\divide\a**\m*\a**=#1\a**\ifdim\a**<0pt\a**-\a**\fi
\ifdim\a**>\b**\d*=.96\a**\advance\d*.4\b**
\else\d*=.96\b**\advance\d*.4\a**\fi
\d*=.01\d*\r*\d*\divide\a**\r*\divide\b**\r*
\ifnum\k*<0\a**-\a**\fi\d*=#1\d*\ifdim\d*<0pt\b**-\b**\fi
\k*\a**\a**=\the\k*\dd*\k*\b**\b**=\the\k*\dd*}



\def\wavearcto#1(#2,#3)[#4]{\rlap{\toks0={#2}\toks1={#3}\relax
\calcnum*#1(#2,#3)\c*=\the\N*5sp\a*=.4\c*\a*=#4\a*\ifdim\a*<0pt\a*-\a*\fi
\advance\c*\a*\N*\c*\divide\N*20\multiply\N*2\advance\N*-1\multiply\N*4\relax
\d**=#1\Lengthunit\dd*=.012\d**
\divide\dd*\*ths \multiply\dd*\magnitude
\ifdim\d**<0pt\d**-\d**\fi\L*=.25\d**
\divide\d**\N*\divide\dd*\N*\L*=#4\L*\divide\L*\*ths
\m*\N*\divide\m*2\dm*=\the\m*0sp\l*\dm*
\sm*\n*\*one\loop\calcnormal*{#4}\calcparab*
\*shl*{1}\advance\n*\*one\calcparab*
\*shl*{1.3}\advance\n*\*one\calcparab*
\*shl*{1}\advance\n*2\dd*-\dd*\ifnum\n*<\N*\repeat\n*\N*\shl**{0pt}}}



\def\triangarcto#1(#2,#3)[#4]{\rlap{\toks0={#2}\toks1={#3}\relax
\calcnum*#1(#2,#3)\c*=\the\N*5sp\a*=.4\c*\a*=#4\a*\ifdim\a*<0pt\a*-\a*\fi
\advance\c*\a*\N*\c*\divide\N*20\multiply\N*2\advance\N*-1\multiply\N*2\relax
\d**=#1\Lengthunit\dd*=.012\d**
\divide\dd*\*ths \multiply\dd*\magnitude
\ifdim\d**<0pt\d**-\d**\fi\L*=.25\d**
\divide\d**\N*\divide\dd*\N*\L*=#4\L*\divide\L*\*ths
\m*\N*\divide\m*2\dm*=\the\m*0sp\l*\dm*
\sm*\n*\*one\loop\calcnormal*{#4}\calcparab*
\*shl*{1}\advance\n*2\dd*-\dd*\ifnum\n*<\N*\repeat\n*\N*\shl**{0pt}}}



\def\hr*#1{\L*=\xscale\Lengthunit\ifnum
\angle**=0\clap{\vrule width#1\L* height.1pt}\else
\L*=#1\L*\L*=.5\L*\rmov*(-\L*,0pt){\sm*}\rmov*(\L*,0pt){\sl*}\fi}



\def\shade#1[#2]{\rlap{\Lengthunit=#1\Lengthunit
\special{em:linewidth .001pt}\relax
\mov(0,#2.05){\hr*{.994}}\mov(0,#2.1){\hr*{.980}}\relax
\mov(0,#2.15){\hr*{.953}}\mov(0,#2.2){\hr*{.916}}\relax
\mov(0,#2.25){\hr*{.867}}\mov(0,#2.3){\hr*{.798}}\relax
\mov(0,#2.35){\hr*{.715}}\mov(0,#2.4){\hr*{.603}}\relax
\mov(0,#2.45){\hr*{.435}}\special{em:linewidth \the\linwid*}}}



\def\dshade#1[#2]{\rlap{\special{em:linewidth .001pt}\relax
\Lengthunit=#1\Lengthunit\if#2-\def\t*{+}\else\def\t*{-}\fi
\mov(0,\t*.025){\relax
\mov(0,#2.05){\hr*{.995}}\mov(0,#2.1){\hr*{.988}}\relax
\mov(0,#2.15){\hr*{.969}}\mov(0,#2.2){\hr*{.937}}\relax
\mov(0,#2.25){\hr*{.893}}\mov(0,#2.3){\hr*{.836}}\relax
\mov(0,#2.35){\hr*{.760}}\mov(0,#2.4){\hr*{.662}}\relax
\mov(0,#2.45){\hr*{.531}}\mov(0,#2.5){\hr*{.320}}\relax
\special{em:linewidth \the\linwid*}}}}



\def\vdot{\rlap{\kern-1.9pt\lower1.8pt\hbox{$\scriptstyle\bullet$}}}
\def\vtimes{\rlap{\kern-3pt\lower1.8pt\hbox{$\scriptstyle\times$}}}
\def\vDot{\rlap{\kern-2.3pt\lower2.7pt\hbox{$\bullet$}}}
\def\vTimes{\rlap{\kern-3.6pt\lower2.4pt\hbox{$\times$}}}



\def\arc(#1)[#2,#3]{{\k*=#2\l*=#3\m*=\l*
\advance\m*-6\ifnum\k*>\l*\relax\else
{\rotate(#2)\mov(#1,0){\sm*}}\loop
\ifnum\k*<\m*\advance\k*5{\rotate(\k*)\mov(#1,0){\sl*}}\repeat
{\rotate(#3)\mov(#1,0){\sl*}}\fi}}



\def\dasharc(#1)[#2,#3]{{\k**=#2\n*=#3\advance\n*-1\advance\n*-\k**
\L*=1000sp\L*#1\L* \multiply\L*\n* \multiply\L*\Nhalfperiods
\divide\L*57\N*\L* \divide\N*2000\ifnum\N*=0\N*1\fi
\r*\n*  \divide\r*\N* \ifnum\r*<2\r*2\fi
\m**\r* \divide\m**2 \l**\r* \advance\l**-\m** \N*\n* \divide\N*\r*
\k**\r* \multiply\k**\N* \dn*\n* 
\advance\dn*-\k** \divide\dn*2\advance\dn*\*one
\r*\l** \divide\r*2\advance\dn*\r* \advance\N*-2\k**#2\relax
\ifnum\l**<6{\rotate(#2)\mov(#1,0){\sm*}}\advance\k**\dn*
{\rotate(\k**)\mov(#1,0){\sl*}}\advance\k**\m**
{\rotate(\k**)\mov(#1,0){\sm*}}\loop
\advance\k**\l**{\rotate(\k**)\mov(#1,0){\sl*}}\advance\k**\m**
{\rotate(\k**)\mov(#1,0){\sm*}}\advance\N*-1\ifnum\N*>0\repeat
{\rotate(#3)\mov(#1,0){\sl*}}\else\advance\k**\dn*
\arc(#1)[#2,\k**]\loop\advance\k**\m** \r*\k**
\advance\k**\l** {\arc(#1)[\r*,\k**]}\relax
\advance\N*-1\ifnum\N*>0\repeat
\advance\k**\m**\arc(#1)[\k**,#3]\fi}}



\def\triangarc#1(#2)[#3,#4]{{\k**=#3\n*=#4\advance\n*-\k**
\L*=1000sp\L*#2\L* \multiply\L*\n* \multiply\L*\Nhalfperiods
\divide\L*57\N*\L* \divide\N*1000\ifnum\N*=0\N*1\fi
\d**=#2\Lengthunit \d*\d** \divide\d*57\multiply\d*\n*
\r*\n*  \divide\r*\N* \ifnum\r*<2\r*2\fi
\m**\r* \divide\m**2 \l**\r* \advance\l**-\m** \N*\n* \divide\N*\r*
\dt*\d* \divide\dt*\N* \dt*.5\dt* \dt*#1\dt*
\divide\dt*1000\multiply\dt*\magnitude
\k**\r* \multiply\k**\N* \dn*\n* \advance\dn*-\k** \divide\dn*2\relax
\r*\l** \divide\r*2\advance\dn*\r* \advance\N*-1\k**#3\relax
{\rotate(#3)\mov(#2,0){\sm*}}\advance\k**\dn*
{\rotate(\k**)\mov(#2,0){\sl*}}\advance\k**-\m**\advance\l**\m**\loop\dt*-\dt*
\d*\d** \advance\d*\dt*
\advance\k**\l**{\rotate(\k**)\rmov*(\d*,0pt){\sl*}}%
\advance\N*-1\ifnum\N*>0\repeat\advance\k**\m**
{\rotate(\k**)\mov(#2,0){\sl*}}{\rotate(#4)\mov(#2,0){\sl*}}}}



\def\wavearc#1(#2)[#3,#4]{{\k**=#3\n*=#4\advance\n*-\k**
\L*=4000sp\L*#2\L* \multiply\L*\n* \multiply\L*\Nhalfperiods
\divide\L*57\N*\L* \divide\N*1000\ifnum\N*=0\N*1\fi
\d**=#2\Lengthunit \d*\d** \divide\d*57\multiply\d*\n*
\r*\n*  \divide\r*\N* \ifnum\r*=0\r*1\fi
\m**\r* \divide\m**2 \l**\r* \advance\l**-\m** \N*\n* \divide\N*\r*
\dt*\d* \divide\dt*\N* \dt*.7\dt* \dt*#1\dt*
\divide\dt*1000\multiply\dt*\magnitude
\k**\r* \multiply\k**\N* \dn*\n* \advance\dn*-\k** \divide\dn*2\relax
\divide\N*4\advance\N*-1\k**#3\relax
{\rotate(#3)\mov(#2,0){\sm*}}\advance\k**\dn*
{\rotate(\k**)\mov(#2,0){\sl*}}\advance\k**-\m**\advance\l**\m**\loop\dt*-\dt*
\d*\d** \advance\d*\dt* \dd*\d** \advance\dd*1.3\dt*
\advance\k**\r*{\rotate(\k**)\rmov*(\d*,0pt){\sl*}}\relax
\advance\k**\r*{\rotate(\k**)\rmov*(\dd*,0pt){\sl*}}\relax
\advance\k**\r*{\rotate(\k**)\rmov*(\d*,0pt){\sl*}}\relax
\advance\k**\r*
\advance\N*-1\ifnum\N*>0\repeat\advance\k**\m**
{\rotate(\k**)\mov(#2,0){\sl*}}{\rotate(#4)\mov(#2,0){\sl*}}}}



\def\gmov*#1(#2,#3)#4{\rlap{\L*=#1\Lengthunit
\xL*=#2\L* \yL*=#3\L*
\rx* \gcos*\xL* \tmp* \gsin*\yL* \advance\rx*-\tmp*
\ry* \gcos*\yL* \tmp* \gsin*\xL* \advance\ry*\tmp*
\rx*=\xscale\rx* \ry*=\yscale\ry*
\xL* \the\cos*\rx* \tmp* \the\sin*\ry* \advance\xL*-\tmp*
\yL* \the\cos*\ry* \tmp* \the\sin*\rx* \advance\yL*\tmp*
\kern\xL*\raise\yL*\hbox{#4}}}



\def\rgmov*(#1,#2)#3{\rlap{\xL*#1\yL*#2\relax
\rx* \gcos*\xL* \tmp* \gsin*\yL* \advance\rx*-\tmp*
\ry* \gcos*\yL* \tmp* \gsin*\xL* \advance\ry*\tmp*
\rx*=\xscale\rx* \ry*=\yscale\ry*
\xL* \the\cos*\rx* \tmp* \the\sin*\ry* \advance\xL*-\tmp*
\yL* \the\cos*\ry* \tmp* \the\sin*\rx* \advance\yL*\tmp*
\kern\xL*\raise\yL*\hbox{#3}}}



\def\Earc(#1)[#2,#3][#4,#5]{{\k*=#2\l*=#3\m*=\l*
\advance\m*-6\ifnum\k*>\l*\relax\else\def\xscale{#4}\def\yscale{#5}\relax
{\angle**0\rotate(#2)}\gmov*(#1,0){\sm*}\loop
\ifnum\k*<\m*\advance\k*5\relax
{\angle**0\rotate(\k*)}\gmov*(#1,0){\sl*}\repeat
{\angle**0\rotate(#3)}\gmov*(#1,0){\sl*}\relax
\def\xscale{1}\def\yscale{1}\fi}}



\def\dashEarc(#1)[#2,#3][#4,#5]{{\k**=#2\n*=#3\advance\n*-1\advance\n*-\k**
\L*=1000sp\L*#1\L* \multiply\L*\n* \multiply\L*\Nhalfperiods
\divide\L*57\N*\L* \divide\N*2000\ifnum\N*=0\N*1\fi
\r*\n*  \divide\r*\N* \ifnum\r*<2\r*2\fi
\m**\r* \divide\m**2 \l**\r* \advance\l**-\m** \N*\n* \divide\N*\r*
\k**\r*\multiply\k**\N* \dn*\n* \advance\dn*-\k** \divide\dn*2\advance\dn*\*one
\r*\l** \divide\r*2\advance\dn*\r* \advance\N*-2\k**#2\relax
\ifnum\l**<6\def\xscale{#4}\def\yscale{#5}\relax
{\angle**0\rotate(#2)}\gmov*(#1,0){\sm*}\advance\k**\dn*
{\angle**0\rotate(\k**)}\gmov*(#1,0){\sl*}\advance\k**\m**
{\angle**0\rotate(\k**)}\gmov*(#1,0){\sm*}\loop
\advance\k**\l**{\angle**0\rotate(\k**)}\gmov*(#1,0){\sl*}\advance\k**\m**
{\angle**0\rotate(\k**)}\gmov*(#1,0){\sm*}\advance\N*-1\ifnum\N*>0\repeat
{\angle**0\rotate(#3)}\gmov*(#1,0){\sl*}\def\xscale{1}\def\yscale{1}\else
\advance\k**\dn* \Earc(#1)[#2,\k**][#4,#5]\loop\advance\k**\m** \r*\k**
\advance\k**\l** {\Earc(#1)[\r*,\k**][#4,#5]}\relax
\advance\N*-1\ifnum\N*>0\repeat
\advance\k**\m**\Earc(#1)[\k**,#3][#4,#5]\fi}}



\def\triangEarc#1(#2)[#3,#4][#5,#6]{{\k**=#3\n*=#4\advance\n*-\k**
\L*=1000sp\L*#2\L* \multiply\L*\n* \multiply\L*\Nhalfperiods
\divide\L*57\N*\L* \divide\N*1000\ifnum\N*=0\N*1\fi
\d**=#2\Lengthunit \d*\d** \divide\d*57\multiply\d*\n*
\r*\n*  \divide\r*\N* \ifnum\r*<2\r*2\fi
\m**\r* \divide\m**2 \l**\r* \advance\l**-\m** \N*\n* \divide\N*\r*
\dt*\d* \divide\dt*\N* \dt*.5\dt* \dt*#1\dt*
\divide\dt*1000\multiply\dt*\magnitude
\k**\r* \multiply\k**\N* \dn*\n* \advance\dn*-\k** \divide\dn*2\relax
\r*\l** \divide\r*2\advance\dn*\r* \advance\N*-1\k**#3\relax
\def\xscale{#5}\def\yscale{#6}\relax
{\angle**0\rotate(#3)}\gmov*(#2,0){\sm*}\advance\k**\dn*
{\angle**0\rotate(\k**)}\gmov*(#2,0){\sl*}\advance\k**-\m**
\advance\l**\m**\loop\dt*-\dt* \d*\d** \advance\d*\dt*
\advance\k**\l**{\angle**0\rotate(\k**)}\rgmov*(\d*,0pt){\sl*}\relax
\advance\N*-1\ifnum\N*>0\repeat\advance\k**\m**
{\angle**0\rotate(\k**)}\gmov*(#2,0){\sl*}\relax
{\angle**0\rotate(#4)}\gmov*(#2,0){\sl*}\def\xscale{1}\def\yscale{1}}}



\def\waveEarc#1(#2)[#3,#4][#5,#6]{{\k**=#3\n*=#4\advance\n*-\k**
\L*=4000sp\L*#2\L* \multiply\L*\n* \multiply\L*\Nhalfperiods
\divide\L*57\N*\L* \divide\N*1000\ifnum\N*=0\N*1\fi
\d**=#2\Lengthunit \d*\d** \divide\d*57\multiply\d*\n*
\r*\n*  \divide\r*\N* \ifnum\r*=0\r*1\fi
\m**\r* \divide\m**2 \l**\r* \advance\l**-\m** \N*\n* \divide\N*\r*
\dt*\d* \divide\dt*\N* \dt*.7\dt* \dt*#1\dt*
\divide\dt*1000\multiply\dt*\magnitude
\k**\r* \multiply\k**\N* \dn*\n* \advance\dn*-\k** \divide\dn*2\relax
\divide\N*4\advance\N*-1\k**#3\def\xscale{#5}\def\yscale{#6}\relax
{\angle**0\rotate(#3)}\gmov*(#2,0){\sm*}\advance\k**\dn*
{\angle**0\rotate(\k**)}\gmov*(#2,0){\sl*}\advance\k**-\m**
\advance\l**\m**\loop\dt*-\dt*
\d*\d** \advance\d*\dt* \dd*\d** \advance\dd*1.3\dt*
\advance\k**\r*{\angle**0\rotate(\k**)}\rgmov*(\d*,0pt){\sl*}\relax
\advance\k**\r*{\angle**0\rotate(\k**)}\rgmov*(\dd*,0pt){\sl*}\relax
\advance\k**\r*{\angle**0\rotate(\k**)}\rgmov*(\d*,0pt){\sl*}\relax
\advance\k**\r*
\advance\N*-1\ifnum\N*>0\repeat\advance\k**\m**
{\angle**0\rotate(\k**)}\gmov*(#2,0){\sl*}\relax
{\angle**0\rotate(#4)}\gmov*(#2,0){\sl*}\def\xscale{1}\def\yscale{1}}}





\newcount\CatcodeOfAtSign
\CatcodeOfAtSign=\the\catcode`\@
\catcode`\@=11
\def\@arc#1[#2][#3]{\rlap{\Lengthunit=#1\Lengthunit
\sm*\l*arc(#2.1914,#3.0381)[#2][#3]\relax
\mov(#2.1914,#3.0381){\l*arc(#2.1622,#3.1084)[#2][#3]}\relax
\mov(#2.3536,#3.1465){\l*arc(#2.1084,#3.1622)[#2][#3]}\relax
\mov(#2.4619,#3.3086){\l*arc(#2.0381,#3.1914)[#2][#3]}}}



\def\dash@arc#1[#2][#3]{\rlap{\Lengthunit=#1\Lengthunit
\d*arc(#2.1914,#3.0381)[#2][#3]\relax
\mov(#2.1914,#3.0381){\d*arc(#2.1622,#3.1084)[#2][#3]}\relax
\mov(#2.3536,#3.1465){\d*arc(#2.1084,#3.1622)[#2][#3]}\relax
\mov(#2.4619,#3.3086){\d*arc(#2.0381,#3.1914)[#2][#3]}}}



\def\wave@arc#1[#2][#3]{\rlap{\Lengthunit=#1\Lengthunit
\w*lin(#2.1914,#3.0381)\relax
\mov(#2.1914,#3.0381){\w*lin(#2.1622,#3.1084)}\relax
\mov(#2.3536,#3.1465){\w*lin(#2.1084,#3.1622)}\relax
\mov(#2.4619,#3.3086){\w*lin(#2.0381,#3.1914)}}}



\def\bezier#1(#2,#3)(#4,#5)(#6,#7){\N*#1\l*\N* \advance\l*\*one
\d* #4\Lengthunit \advance\d* -#2\Lengthunit \multiply\d* \*two
\b* #6\Lengthunit \advance\b* -#2\Lengthunit
\advance\b*-\d* \divide\b*\N*
\d** #5\Lengthunit \advance\d** -#3\Lengthunit \multiply\d** \*two
\b** #7\Lengthunit \advance\b** -#3\Lengthunit
\advance\b** -\d** \divide\b**\N*
\mov(#2,#3){\sm*{\loop\ifnum\m*<\l*
\a*\m*\b* \advance\a*\d* \divide\a*\N* \multiply\a*\m*
\a**\m*\b** \advance\a**\d** \divide\a**\N* \multiply\a**\m*
\rmov*(\a*,\a**){\unhcopy\spl*}\advance\m*\*one\repeat}}}


\catcode`\*=12

\newcount\n@ast

\def\n@ast@#1{\n@ast0\relax\get@ast@#1\end}
\def\get@ast@#1{\ifx#1\end\let\next\relax\else
\ifx#1*\advance\n@ast1\fi\let\next\get@ast@\fi\next}



\newif\if@up \newif\if@dwn
\def\up@down@#1{\@upfalse\@dwnfalse
\if#1u\@uptrue\fi\if#1U\@uptrue\fi\if#1+\@uptrue\fi
\if#1d\@dwntrue\fi\if#1D\@dwntrue\fi\if#1-\@dwntrue\fi}



\def\halfcirc#1(#2)[#3]{{\Lengthunit=#2\Lengthunit\up@down@{#3}\relax
\if@up\mov(0,.5){\@arc[-][-]\@arc[+][-]}\fi
\if@dwn\mov(0,-.5){\@arc[-][+]\@arc[+][+]}\fi
\def\lft{\mov(0,.5){\@arc[-][-]}\mov(0,-.5){\@arc[-][+]}}\relax
\def\rght{\mov(0,.5){\@arc[+][-]}\mov(0,-.5){\@arc[+][+]}}\relax
\if#3l\lft\fi\if#3L\lft\fi\if#3r\rght\fi\if#3R\rght\fi
\n@ast@{#1}\relax
\ifnum\n@ast>0\if@up\shade[+]\fi\if@dwn\shade[-]\fi\fi
\ifnum\n@ast>1\if@up\dshade[+]\fi\if@dwn\dshade[-]\fi\fi}}



\def\halfdashcirc(#1)[#2]{{\Lengthunit=#1\Lengthunit\up@down@{#2}\relax
\if@up\mov(0,.5){\dash@arc[-][-]\dash@arc[+][-]}\fi
\if@dwn\mov(0,-.5){\dash@arc[-][+]\dash@arc[+][+]}\fi
\def\lft{\mov(0,.5){\dash@arc[-][-]}\mov(0,-.5){\dash@arc[-][+]}}\relax
\def\rght{\mov(0,.5){\dash@arc[+][-]}\mov(0,-.5){\dash@arc[+][+]}}\relax
\if#2l\lft\fi\if#2L\lft\fi\if#2r\rght\fi\if#2R\rght\fi}}



\def\halfwavecirc(#1)[#2]{{\Lengthunit=#1\Lengthunit\up@down@{#2}\relax
\if@up\mov(0,.5){\wave@arc[-][-]\wave@arc[+][-]}\fi
\if@dwn\mov(0,-.5){\wave@arc[-][+]\wave@arc[+][+]}\fi
\def\lft{\mov(0,.5){\wave@arc[-][-]}\mov(0,-.5){\wave@arc[-][+]}}\relax
\def\rght{\mov(0,.5){\wave@arc[+][-]}\mov(0,-.5){\wave@arc[+][+]}}\relax
\if#2l\lft\fi\if#2L\lft\fi\if#2r\rght\fi\if#2R\rght\fi}}



\catcode`\*=11

\def\Circle#1(#2){\halfcirc#1(#2)[u]\halfcirc#1(#2)[d]\n@ast@{#1}\relax
\ifnum\n@ast>0\L*=\xscale\Lengthunit
\ifnum\angle**=0\clap{\vrule width#2\L* height.1pt}\else
\L*=#2\L*\L*=.5\L*\special{em:linewidth .001pt}\relax
\rmov*(-\L*,0pt){\sm*}\rmov*(\L*,0pt){\sl*}\relax
\special{em:linewidth \the\linwid*}\fi\fi}



\catcode`\*=12

\def\wavecirc(#1){\halfwavecirc(#1)[u]\halfwavecirc(#1)[d]}
\def\dashcirc(#1){\halfdashcirc(#1)[u]\halfdashcirc(#1)[d]}





\def\xscale{1}

\def\yscale{1}



\def\Ellipse#1(#2)[#3,#4]{\def\xscale{#3}\def\yscale{#4}\relax
\Circle#1(#2)\def\xscale{1}\def\yscale{1}}



\def\dashEllipse(#1)[#2,#3]{\def\xscale{#2}\def\yscale{#3}\relax
\dashcirc(#1)\def\xscale{1}\def\yscale{1}}



\def\waveEllipse(#1)[#2,#3]{\def\xscale{#2}\def\yscale{#3}\relax
\wavecirc(#1)\def\xscale{1}\def\yscale{1}}



\def\halfEllipse#1(#2)[#3][#4,#5]{\def\xscale{#4}\def\yscale{#5}\relax
\halfcirc#1(#2)[#3]\def\xscale{1}\def\yscale{1}}



\def\halfdashEllipse(#1)[#2][#3,#4]{\def\xscale{#3}\def\yscale{#4}\relax
\halfdashcirc(#1)[#2]\def\xscale{1}\def\yscale{1}}



\def\halfwaveEllipse(#1)[#2][#3,#4]{\def\xscale{#3}\def\yscale{#4}\relax
\halfwavecirc(#1)[#2]\def\xscale{1}\def\yscale{1}}





\catcode`\@=\the\CatcodeOfAtSign

%\begin{titlepage}
%\thispagestyle{empty}

\begin{center}
{\Large\bf Quantum superfield supersymmetry}

{A.Yu. Petrov}

\footnotesize
{\it
 Departamento de F\'{i}sica e Matematica,\\
Instituto de F\'{i}sica,\\
Universidade de S\~{a}o Paulo,\\
S\~{a}o Paulo, Brazil}\\
and\\
{\it
 Department of Theoretical Physics,\\
Tomsk State Pedagogical University\\
Tomsk 634041, Russia}\\

\end{center}

\begin{abstract}
Superfield approach in supersymmetric quantum field theory is
described.
Many examples of its applications to different 
superfield models are considered.
\end{abstract}
\section{Introduction. General properties of superspace}

This paper presents itself as lecture notes in superfield
supersymmetry 
based on lectures given
at Instituto de F\'{i}sica, Universidade de S\~{a}o Paulo.

Idea of supersymmetry is now considered as one of the basic
concepts of theoretical high energy physics (see f.e. \cite{GSH}). 
Supersymmetry, being a
fundamental symmetry of bosons and fermions, provides possibilities to
construct theories with essentially better renormalization properties
since some bosonic and fermionic contributions cancel each other.
Moreover, there are essentially finite supersymmetry theories {\bf without}
higher derivatives, f.e. $N=4$ super-Yang-Mills theory.
Now most specialists in quantum field theory suggest that unified
theory of all interactions must be supersymmetric.

Concept of supersymmetry was introduced in known papers by Volkov and
Akulov \cite{VA} and Golfand and Lichtman \cite{GL} in early 70's and
received further development in \cite{WZ}. However, the essential break
through in supersymmetric field theory was achieved with introducing
the idea of superfield \cite{WZ1} (see also \cite{SS,FWZ}). 
The superfield approach in
supersymmetric quantum field theory is a main topic of these lectures. We use
notations introduced in \cite{WB,BK0}. 

A superfield is a function of bosonic coordinates $x^a$ and fermionic
(Grassmann) ones $\q^{i\alpha},\bar{\q}^j_{\dot{\alpha}}$. The
fermionic coordinates are transformed under spinor representation of
Lorentz group. The indices $i,j$ in general case take values from 1 to
$N$ in the case of $N$-extended supersymmetry. Here and further we are
generally interested in $N=1$ case. However, we note that all theories
with $N$-extended supersymmetry possess $N=1$ formulation.
The supersymmetry transformations for coordinates are
\bea
\delta\q^{\a}=\epsilon^{\a};\ \delta\bar{\q}_{\ad}=\epsilon_{\ad};\ 
\delta x^a=i(\epsilon\sigma^a\bar{\q}-\bar{\epsilon}\sigma^a \q)
\eea 
Here $\epsilon^{\a},\bar{\epsilon}^{\ad}$ are fermionic parameters.
The general form of superfield is (see f.e. \cite{WB,OM}):
\bea
F(x,\q,\bar{\q})&=&A(x)+\q^{\a}\psi_{\a}(x)+\bar{\q}_{\ad}\zeta^{\ad}(x)+
\q^2F(x)+\bar{\q}^2G(x)+i(\bar{\q}\sigma^a \q)A_a(x)+
\nonumber\\&+&\bar{\q}^2\q^{\a}
\chi_{\a}(x)+\q^2\bar{\q}_{\ad}\xi^{\ad}(x)+\q^2\bar{\q}^2H(x)
\eea
We note that this power series is finite due to anticommutation of
Grassmann numbers $\q,\bar{\q}$ which enforces $\q^n,\bar{\q}^n$ to vanish at
$n\geq 3$. Further we will see that there are some restrictions on
structure of superfields caused by the form of representation of
supersymmetry algebra. Here $f(x),\psi_{\a}(x),\ldots$ are bosonic and
fermionic fields forming component content of superfield $F$.
If a theory describing dynamics of these fields is supersymmetric its
action should be invariant under supersymmetry transformations,
i.e. symmmetry transformations with fermionic parameters.

{\bf Example.} In Wess-Zumino model \cite{WB} these transformations
have the form
\bea
\delta A(x)&=&\epsilon^{\a}\psi_{\a}(x)\nonumber\\
\delta \psi_{\a}(x)&=&\epsilon_{\a}F(x)-\bar{\epsilon}^{\ad}i
\pa_{\a\ad}A(x)\nonumber\\
\delta F(x)&=&\bar{\epsilon}_{\ad}i\pa^{\a\ad}\psi_{\a}.
\eea

Variation of arbitrary superfield $F(x,\q,\bar{\q})$ has the form
\bea
\label{svar}
\delta F(x,\q,\bar{\q})=
(\epsilon^{\a}Q_{\a}+\bar{\epsilon}_{\ad}\bar{Q}^{\ad})
F(x,\q,\bar{\q})
\eea
Here $Q_{\a},\bar{Q}_{\ad}$ are generators of supersymmetry possessing
anticommutation relations
\bea
\label{gencom}
\{Q_{\a},\bar{Q}_{\ad} \}&=&2i\sigma_{\a\ad}^m\partial_m ;\
\{Q_{\a},Q_{\b} \}=\{\bar{Q}_{\ad},\bar{Q}_{\bd}\}=0;\
[ Q_{\a},\partial_{m} ]=0
\eea
The variation (\ref{svar}) is a translation in some space.

As a result we need in introducing some extended space
parametrized by bosonic and fermionic coordinates 
$(x^a, \q^{\a},\bar{\q}^{\ad})$ which includes standard space-time as
subspace. This extended space is called superspace.
Translations on superspace are given by standard Poincare translations
and transformations (\ref{svar}). It is easy to see that (\ref{svar})
is a manifestly Lorentz covariant transformation. The superspace is
parametrized by 4 bosonic coordinates $x^a$ and 4 fermionic ones 
$\q^{\a},\bar{\q}^{\ad}$ so it is 8-dimensional and is denoted as 
$R^{4|4}$. It is natural to consider superfields as fields on
superspace. The $z^A=(x,\q^{\a},\bar{\q}^{\ad})$ are coordinates on 
superspace.
Our task is to develop quantum theory for superfields based on
principles of standard quantum field theory.
Here we suppose that reader knows general properties of classical
superfield theory.

To develop field theory on superspace we must introduce integration
and differentiation on superspace, i.e. with respect to Grassmann
coordinates. We can introduce left $\pa_L$ and right $\pa_R$ 
derivatives with respect to Grassmann coordinates as
\bea
\frac{\pa_L}{\pa\q^{\alpha_i}}
(\q^{\a_1}\q^{\a_{i-1}}\q^{\a_i}\q^{\a_{i+1}}\ldots \q^{\a_n})&=&
{(-1)}^{a_1+\ldots+a_{i-1}}
(\q^{\a_1}\q^{\a_{i-1}}\q^{\a_{i+1}}\ldots \q^{\a_n})
\nonumber\\
\frac{\pa_R}{\pa\q^{\alpha_i}}
(\q^{\a_1}\q^{\a_{i-1}}\q^{\a_i}\q^{\a_{i+1}}\ldots \q^{\a_n})&=&
{(-1)}^{a_{i+1}+\ldots+a_n}
(\q^{\a_1}\q^{\a_{i-1}}\q^{\a_{i+1}}\ldots \q^{\a_{n}})
\eea
Therefore these derivatives differ only by a sign factor. We can
choose f.e. left one and use it henceforth.

Integral is determined on the base of the following relation:
$\int d\theta \theta=1$, or, generally, 
$$\int d\q^{\a}\q_{\b}=\delta^{\a}_{\b}.$$ 
It is a convention. Note that $\q$ and $d\q$ have different
dimensions: the mass dimension of $\q$ is equal to $-\frac{1}{2}$, and of
$d\q$ -- to $\frac{1}{2}$, and variation $\delta\q$ cannot be mixed
with differential $d\q$. Then, integral from a constant is zero,
$$\int d\q 1=0 
$$
this identity is caused by translation invariance due to which
relation $\int d\q (\q+\lambda)= \int d\q\q$ for constant $\lambda$ 
must be satisfied, hence $\lambda\int d\q=0$. 
We introduce the following scalar measures for Grassmann integration:
\bea
d^2\q=-\frac{1}{4}d\q^{\a}d\q_{\a},\ d^2\bar{\q}=-\frac{1}{4}
\bar{d\q}_{\ad}\bar{d\q}^{\ad},\ d^4\q=d^2\q d^2\bar{\q} 
\eea
These measures satisfy the relations
\bea
\int d^2\q \q^2=\int d^2\bar{\q} \bar{\q}^2=\int d^4 \q \q^4 =1
\eea
(here and further we denote $\q^4\equiv \q^2\bar{\q}^2$).

Since $\frac{\pa \q^{\a}}{\pa \q^{\b}}=\delta^{\a}_{\b}$ as well as
$\int d\q^{\a}\q_{\b}=\delta^{\a}_{\b}$ we conclude that integration
and differentiation in Grassmann space are equivalent.
F.e. we see that
\bea
\int d^4\q F(x,\q,\bar{\q})&=&\frac{1}{16}\frac{\pa^2}{\pa\q^2}
\frac{\pa^2}{\pa\bar{\q}^2}F(x,\q,\bar{\q})=
\frac{1}{16}F(x,\q,\bar{\q})|_{\q^2\bar{\q}^2}\nonumber\\
\int d^2\q G(x,\q)&=&-\frac{1}{4}\frac{\pa^2}{\pa\q^2}G(x,\q)=
-\frac{1}{4}G(x,\q)|_{\q^2}
\eea
Here $|_{\q^2}, |_{\q^2\bar{\q}^2}$ denotes the corresponding component
of the superfield.
Of course, differentiations with respect to Grassmann coordinates
anticommute.

The supersymmetry generators possess several realizations in terms of
$\frac{\pa}{\pa x^m}$ and 
$\frac{\pa}{\pa \q_{\a}}, \frac{\pa}{\pa\bar{\q}_{\ad}}$, 
f.e.
\bea
\label{gen}
\bar{Q}_{\ad}=\frac{\pa}{\pa\bar{\q}^{\ad}}-
i\q^{\a}(\sigma^m)_{\a\ad}\pa_m, \ 
Q_{\a}=-\frac{\pa}{\pa\q^{\a}}+i\bar{\q}^{\bd}(\sigma^m)_{\bd\a}\pa_m
\eea
In general all realizations must satisfy relations 
(\ref{gencom}).

The spinor supercovariant derivatives $D_A$ also must be constructed from
$\frac{\pa}{\pa x^m}$ and 
$\frac{\pa}{\pa \q_{\a}}, \frac{\pa}{\pa\bar{\q}}_{\ad}$. 
They should anticommute with generators $Q_{\a},\bar{Q}_{\ad}$
which provides that $D_A\Phi$ is transformed covariantly, i.e. 
according to (\ref{svar}):
$$\delta (D_A\Phi)=(\epsilon Q+\bar{\epsilon}\bar{Q})D_A\Phi.
$$
F.e. if generators of supersymmetry are realized in terms of
(\ref{gen})
supercovariant derivatives are realized as
\bea
\label{gend}
\bar{D}_{\ad}&=&-i\bar{Q}_{\ad}+i\q^{\a}\pa_{\a\ad}
=-i\frac{\pa}{\pa\bar{\q}^{\ad}}, \\ 
D_{\a}&=&-iQ_{\a}+i\bar{\q}^{\ad}\pa_{\a\ad}
=i(-\frac{\pa}{\pa\q^{\a}}+2i\bar{\q}^{\bd}(\sigma^m)_{\bd\a}\pa_m)
\eea
Here and further $\pa_{\a\ad}=(\sigma^m)_{\a\ad}\pa_m$.
The spinor supercovariant derivatives satisfy the following
anticommutation relations
\bea
\label{salg}
\{D_{\a},\bar{D}_{\ad}\}=-2i\pa_{\a\ad};
\ \{D_{\a},D_{\b}\}=
\{\bar{D}_{\ad},\bar{D}_{\bd}\}=0
\eea
Hence we developed procedures of integration and differentiation 
in superspace.

The next step in developing field theory is in introducing of delta
function. It must satisfy the condition analogous to standard delta function
\bea
\int d^4\q' \delta^4(\q-\q')f(\q')=f(\q)
\eea 
This identity can be satisfied if we choose
\bea
\delta^4(\q-\bar{\q})=\frac{1}{16}(\q-\q')^2(\bar{\q}-\bar{\q}')^2
\eea
It is easy to see that this delta function satisfies the condition
\bea
\int d^4\q \delta^4(\q-\q')=1
\eea
We note the identity
\bea
\delta^4(\q_1-\q_2)D^2_1\bar{D}^2_2\delta^4(\q_1-\q_2)=16\delta^4(\q_1-\q_2)
\eea
Further we denote $\delta_{12}=\delta^4(\q_1-\q_2)$.
It is easy to see that $\delta^4_{12}\delta^4_{12}=
\delta^4_{12}D^{\a}\delta^4_{12}=\delta^4_{12}D^2\delta^4_{12}=
\delta^4_{12}\bar{D}_{\ad}\delta^4_{12}=0$.

A supermatrix is defined as a matrix $M=M^P_Q$ of the form
\bea
M=\left(\begin{array}{cc}
A&B\\
C&D
\end{array}\right) 
\eea
determining a quadratic form $z_P M^P_Q z^{'Q}$ with $z,z'$ are coordinates
on superspace. Here $A,B,C,D$ are even-even, even-odd, odd-even and
odd-odd blocks respectively.  
Superdeterminant of this matrix is introduced as
\bea
{\rm sdet} M=\int d^8 z_1 d^8 z_2 exp (-z_1 M z_2)
\eea
It is equal to
\bea
{\rm sdet} M=\det A det^{-1}(D-CA^{-1}B)
\eea
And supertrace is equal to $str M=\sum_A (-1)^{\epsilon_A}M^A_A=tr A -
tr D$. As usual, $sdet M=\exp (str \log M)$. 

We can introduce change of variables in superspace. Then, if it has the
form
\bea
x^{'a}=x^{'a}(x,\q,\bar{\q});\, \q^{\a}=\q^{\a}(x,\q,\bar{\q}),\,
\bar{\q}^{\ad}=\bar{\q}^{\ad}(x,\q,\bar{\q})
\eea
the measure of integral is transformed as
\bea
d^4 x' d^4 \q'= d^4 x d^4 \q \ {\rm sdet}(\frac{\partial z'}{\partial z})
\eea
where supermatrix $(\frac{\partial z'}{\partial z})$ is 
\bea
\frac{\pa z'}{\pa z}=\left(\begin{array}{ccc}
\frac{\pa x'}{\pa x}&\frac{\pa x'}{\pa \q}&\frac{\pa x'}{\pa
  \bar{\q}}\\
\frac{\pa \q'}{\pa x}&\frac{\pa \q'}{\pa \q}&\frac{\pa \q'}{\pa
  \bar{\q}}\\
\frac{\pa \bar{\q}'}{\pa x}&\frac{\pa \bar{\q}'}{\pa \q}&
\frac{\pa \bar{\q}'}{\pa \bar{\q}}\\
\end{array}
\right)
\eea 
We also must introduce variational derivative.
In common field theory it is defined as
\bea
\frac{\delta}{\delta A(x)}\int d^4 y f(y) A(y) =f(x)
\eea
if $f(x)$ and $A(x)$ are functionally independent.
Just analogous definition can be introduced for general (not chiral)
superfield:
\bea
\frac{\delta}{\delta V(z)}\int d^8 z'f(z') V(z')=f(z)
\eea
However, for chiral superfields the definition differs. Really, by
definition
chiral superfield $\Phi(z)$ satisfies the condition
$\bar{D}_{\ad}\Phi=0$. Choice of supercovariant derivatives in the form 
(\ref{gend}) allows one make $\Phi$ 
$\bar{\q}$-independent, then the integral from a chiral function is
non-trivial when it is calculated over chiral subspace, i.e. over 
$d^6z=d^4xd^2\q$. Hence we must introduce variational derivative with
respect to chiral superfield $\Phi$ as
\bea
\frac{\delta}{\delta\Phi(z)}\int d^6z'F(z')\Phi(z')=F(z)
\eea
And variational derivative from integral over whole superspace with
respect to chiral superfield can be
introduced as
\bea
\frac{\delta}{\delta\Phi(z)}\int d^8 z' G(z')\Phi(z')=
\frac{\delta}{\delta\Phi(z)}\int d^6 z' (-\frac{1}{4}\bar{D}^2)G(z')\Phi(z')
=-\frac{1}{4}\bar{D}^2G(z)
\eea
Therefore $\frac{\delta\Phi(z)}{\delta\Phi(z')}=
\delta_+(z-z')$ where
$\delta_+(z-z')=-\frac{1}{4}\bar{D}^2\delta^8(z-z')$
is a chiral delta function. It allows us to obtain useful relation
\bea
\frac{\delta^2}{\delta\Phi(z_1)\delta\bar{\Phi}(z_2)}=\frac{1}{16}
\bar{D}^2_1D^2_2\delta^8(z_1-z_2)=(-\frac{1}{4})D^2\delta_+(z_1-z_2)
=(-\frac{1}{4})\bar{D}^2\delta_-(z_1-z_2)
\eea
Here $\delta_-(z_1-z_2)=-\frac{1}{4}D^2\delta^8(z_1-z_2)$ is
antichiral delta function. Note the
relation $D^2_1\delta^8(z_1-z_2)=D^2_2\delta^8(z_1-z_2)$.

If we consider some differential operator $\Delta$ acting on superfields we can
introduce its fuctional supertrace and superdeterminant:
\bea
str \Delta=\int d^8 z_1 d^8 z_2 \delta^8(z_1-z_2) \Delta \delta^8(z_1-z_2)
\eea
If we introduce kernel of $\Delta$ of the form $\Delta(z_1,z_2)$ we can
write
\bea
str\Delta=\int d^8 z \Delta(z,z)
\eea
Superdeterminant is introduced as
\bea
sdet \Delta =\exp str (\log \Delta) 
\eea
Further we will be generally interested in theories describing
dynamics of chiral and real scalar superfield. Note that irreducible
representation of supersymmetry algebra is realized namely on these 
superfields \cite{WB}. The most important examples are Wess-Zumino
model,
general chiral superfield theory \cite{my2}, $N=1$ super-Yang-Mills
theory and four-dimensional dilaton supergravity \cite{my1}.
In this paper we consider application of superfield approach to these models.

\section{Generating functional and Green functions for superfields}
\setcounter{equation}{0}

Now our aim consists of describing a method for calculation of 
generating functional and Green functions for superfields and
following application of this method to calculation of superfield
quantum corrections, i.e. in development of superfield perturbative 
technique. We note that during last years activity in 
development of nonperturbative approaches in superfield quantum theory
stimulated by paper \cite{SW} essentially increased. Nevertheless
perturbative approach is still the leading one, and possibility of using
nonperturbative approach is often based on perturbative one.

The generalization of path integral method for superfield theory turns
to be quite straightforward but a bit formal. Really, generating
functional is defined in terms of path integral which is well-defined
only for some special cases. However, the case of Gaussian path
integral is: (i) well-defined both in standard field theory and in
superfield theory (ii) enough for development of superfield
perturbation technique.

Let us shortly describe introduction of path integral in common field
theory. Let classical action $S[\phi]$ be a local space-time
functional. The equations of motion are
$S_{,i}[\phi]=0|_{\phi=\phi_0}$.
The $\phi_0$ is a solution for this equation. We suppose that Hessian is
non-singular at this point: ${\rm det} S_{ij}[\phi]_{\phi=\phi_0}\neq
0$ (or as is the same equation $S_{ij}|_{\phi=\phi_0}a^j=0$ is
satisfied if and only if $a^j=0$. If Hessian is singular, we add to
the action some term to make it non-zero (in gauge theories such term
is called gauge-fixing one), after adding of this term all 
consideration is just the
same as if the Hessian is non-zero from the very beginning. We suggest
that action $S[\phi]$ is analytic functional, i.e. it can be expanded
into power series in a neighbourhood of $\phi_0$:
\bea
S[\phi]=S[\phi_0]+\sum_{n=2}^{\infty}\frac{1}{n!}S_{,i_1\ldots i_n}
(\phi-\phi_0)^{i_n}\ldots (\phi-\phi_0)^{i_1}
\eea
The term with $n=2$ is called linearized action:
\bea
S_0=\frac{1}{2}\tilde{\phi}^i S_{ij}[\phi_0]\tilde{\phi}^j
\eea
Here and further $\tilde{\phi}^i=\phi^i-\phi_0^i $.
Terms with $n\geq 3$ are called interaction terms $S_{int}$, and the 
action takes
the form
\bea
\label{inact}
S[\phi]=S[\phi_0]+S_0[\tilde{\phi};\phi_0]+S_{int}[\tilde{\phi};\phi_0]
\eea
The Green function $G^{ij}$ is determined on the base of linearized action as
\bea
S_{ij}[\phi_0]G^{jk}=-\delta^k_i;\ G^{ij}S_{jk}[\phi_0]=-\delta_k
\eea
The generating functional of Green functions is introduced as
\bea
\label{GF}
Z[J]=N\int D\phi \exp (\frac{i}{\hbar}(S[\phi]+J\phi))
\eea
The Green functions can be obtained on the base of the generating
functional as
\bea
\label{i}
<\phi(x_1)\ldots\phi(x_n)>=(\frac{1}{i}\frac{\delta}{\delta J (x_1)})
\ldots (\frac{1}{i}\frac{\delta}{\delta J (x_n)})
N\int D\phi \exp (\frac{i}{\hbar}(S[\phi]+J\phi)
\eea
We can calculate the path integral (\ref{GF}). To do it we expand 
$S[\phi]=S_0[\phi]+S_{int}[\phi]$ after changing $\tilde{\phi}\to\phi$
in (\ref{inact}), $S_0[\phi]=\int d^4x \phi\Delta\phi$ where $\Delta$ is
some operator. Of course, path integration is quite formal
operation well-defined only for Gaussian integral and expressions
derived from it. However, both in standard and superfield case we need
mostly Gaussian integrals.
As usual,
\bea
\label{ii}
& &\int D\phi \exp (\frac{i}{\hbar}(S[\phi]+J\phi))=
\int D\phi \exp (\frac{i}{\hbar}(\phi\Delta\phi+S_{int}[\phi]+J\phi))
=\nonumber\\&=&
\exp(\frac{i}{\hbar}S_{int}(\frac{\hbar}{i}\frac{\delta}{\delta J}))
\int D\phi \exp (\frac{i}{\hbar}(\phi\Delta\phi+J\phi))
\eea
And (since this integral is Gaussian-like)
\bea
\label{iii}
\int D\phi \exp (\frac{i}{\hbar}(\phi\Delta\phi+J\phi))=
\exp(-\frac{i}{2}J(\frac{\hbar}{\Delta})J)det^{-1/2}(\frac{\Delta}{\hbar})
\eea
Therefore all dependence of sources is concentrated in 
$\exp(-\frac{i}{2}J(\frac{\hbar}{\Delta})J)$. Construction of Feynman
diagrams from expressions (\ref{i},\ref{ii},\ref{iii}) is quite 
straightforward.

Let us carry out this approach for superfield theory. Our example is
Wess-Zumino model, consideration of other theories is rather
analogous. We do not address specifics of gauge theories in which one
must introduce gauge fixing and ghosts since after their introduction
all procedure is just the same.
The action of Wess-Zumino model with chiral sources is
\bea
\label{actwz}
S_J[\Phi,\bar{\Phi};J,\bar{J}]
=\int d^8z\Phi\bar{\Phi}+(\int d^6z (\frac{\l}{3!}\Phi^3+
\frac{m}{2}\Phi^2+\Phi J)+h.c.)
\eea 
(as usual, conjugated terms to chiral superfields are antichiral ones). 
It can be rewritten in terms of integrals over chiral and antichiral
subspace only:
\bea
\label{only}
S_J[\Phi,\bar{\Phi};J,\bar{J}]
=\int d^6z (\frac{1}{2}\Phi(-\frac{\bar{D}^2}{4})\bar{\Phi}
+\frac{\l}{3!}\Phi^3+
\frac{m}{2}\Phi^2+\Phi J)+h.c.
\eea
The generating functional is
\bea
Z[J,\bar{J}]=\int D\Phi D\bar{\Phi}\exp(iS_J[\Phi,\bar{\Phi};J,\bar{J}])
\eea
The action $S_J$ (\ref{only}) can be represented in matrix form
\bea
S_J&=&\frac{1}{2}\int dz_1 dz_2
\left(\begin{array}{cc}\Phi(z_1)\bar{\Phi}(z_1)\end{array}
\right)
\left(\begin{array}{cc}
m&-\frac{1}{4}\bar{D}^2\\
-\frac{1}{4}D^2 & m
\end{array}\right)
\left(\begin{array}{cc}
\delta_+(z_1-z_2)&0\\
0&\delta_-(z_1-z_2)
\end{array}\right)\times\nonumber\\&\times&
\left(\begin{array}{c}\Phi(z_2)\\ \bar{\Phi}(z_2)
\end{array}\right)+\frac{\lambda}{3!}(\int d^6 z \Phi^3 +h.c.)
\eea
Integration in all terms is assumed with corresponding chirality.
We see that the operator $\Delta$
determining quadratic part of the action (see (\ref{ii},\ref{iii})) looks
like
\bea
\Delta&=&
\left(\begin{array}{cc}
m&-\frac{1}{4}\bar{D}^2\\
-\frac{1}{4}D^2 & m
\end{array}\right)
\eea
The propagator is an operator inverse to this one:
\bea
G=\Delta^{-1}=\frac{1}{\Box-m^2}\left(\begin{array}{cc}
m&\frac{1}{4}\bar{D}^2\\
\frac{1}{4}D^2 & m
\end{array}\right)
\eea
In other words, propagator $G$ satisfies the equation
\bea
\Delta G=-
\left(\begin{array}{cc}
\delta_+(z_1-z_2)&0\\
0&\delta_-(z_1-z_2)
\end{array}\right)
\eea
The matrix ${\bf I}=\left(\begin{array}{cc}
\delta_+(z_1-z_2)&0\\
0&\delta_-(z_1-z_2)
\end{array}\right)$ plays the role of functional unit matrix.

Thus, the generating functional is
\bea
\label{GF2}
Z[J,\bar{J}]&=&
\exp(i\frac{\l}{3!}\int d^6 z (\frac{\delta}{\delta J(z)})^3
+h.c.)det^{-1/2}\Delta\times\nonumber\\&\times&
\exp(-\frac{i}{2}\int dz_1 dz_2
\left(\begin{array}{cc}J(z_1)\bar{J}(z_1)\end{array}
\right)\frac{1}{\Box-m^2}
\left(\begin{array}{cc}
m&\frac{1}{4}\bar{D}^2\\
\frac{1}{4}D^2 & m
\end{array}\right)\times\nonumber\\&\times&
\left(\begin{array}{cc}
\delta_+(z_1-z_2)&0\\
0&\delta_-(z_1-z_2)
\end{array}\right)
\left(\begin{array}{c}J(z_2)\\ \bar{J}(z_2)
\end{array}\right))
\eea
The argument of the exponential function in last expression can be
rewritten as
\bea
-\frac{i}{2}\Big(\int d^6 z J\frac{m}{\Box-m^2} J +2 \int d^6 z
J\frac{1/4\bar{D}^2}{\Box-m^2}\bar{J}+\int d^6 \bar{z} 
\bar{J}\frac{m}{\Box-m^2}\bar{J}
\Big)
\eea
We can introduce two-point Green functions:
\bea
\label{2point}
G_{++}(z_1,z_2)&=&
\frac{1}{i^2}\frac{\delta^2 Z[J]}{\delta J(z_1)\delta J(z_2)}=
(-\frac{1}{4})^2\bar{D}^2_1\bar{D}^2_2K_{++}(z_1,z_2)\nonumber\\
G_{+-}(z_1,z_2)&=&
\frac{1}{i^2}\frac{\delta^2 Z[J]}{\delta J(z_1)\delta \bar{J}(z_2)}=
(-\frac{1}{4})^2\bar{D}^2_1D^2_2K_{+-}(z_1,z_2)\nonumber\\
G_{--}(z_1,z_2)&=&
\frac{1}{i^2}\frac{\delta^2 Z[J]}{\delta \bar{J}(z_1)\delta \bar{J}(z_2)}=
(-\frac{1}{4})^2 D^2_1 D^2_2K_{--}(z_1,z_2)
\eea
Here
$K_{+-}(z_1,z_2)=K_{-+}(z_1,z_2)=-\frac{1}{\Box-m^2}\delta^8(z_1-z_2)$,
$K_{++}(z_1,z_2)=\frac{m\bar{D}^2}{4\Box(\Box-m^2)}\delta^8(z_1-z_2)$,
$K_{--}(z_1,z_2)=\frac{m D^2}{4\Box(\Box-m^2)}\delta^8(z_1-z_2)$,
We note that in theory of standard (not chiral)
superfield the propagator does not carry factors $D^2$,
$\bar{D}^2$. These factors are caused by chirality. F.e. for theory of
real scalar superfield with action $\frac{1}{2}\int d^8 z v\Box v$
the propagator is simply $G(z_1,z_2)=\frac{1}{\Box}\delta(z_1-z_2)$.

Different vacuum expectations can be expressed in terms of the generating 
functional (\ref{GF2}) as
\bea
\label{vacexp}
& &<\phi(x_1)\ldots \phi(x_n)\bar{\phi}(y_1)\ldots\bar{\phi}(y_m)>=\nonumber\\
&=&
(\frac{1}{i}\frac{\delta}{\delta J(x_1)})\ldots
(\frac{1}{i}\frac{\delta}{\delta J(x_n)})
(\frac{1}{i}\frac{\delta}{\delta \bar{J}(y_1)})\ldots
(\frac{1}{i}\frac{\delta}{\delta \bar{J}(y_m)})
\times\nonumber\\&\times&
\exp(i\frac{\l}{3!}\int d^6 z (\frac{\delta}{\delta J(z)})^3
+h.c.)
det^{-1/2}\Delta\times\nonumber\\&\times&
\exp(-\frac{i}{2}\int dz_1 dz_2
\left(\begin{array}{cc}J(z_1)\bar{J}(z_1)\end{array}
\right)\frac{1}{\Box-m^2}
\left(\begin{array}{cc}
m&\frac{1}{4}\bar{D}^2\\
\frac{1}{4}D^2 & m
\end{array}\right)\times\nonumber\\&\times&
\left(\begin{array}{cc}
\delta_+(z_1-z_2)&0\\
0&\delta_-(z_1-z_2)
\end{array}\right)
\left(\begin{array}{c}J(z_2)\\ \bar{J}(z_2)
\end{array}\right))
\eea 
Of course, this expression contains all orders in coupling $\l$. To
obtain vacuum expectations up to some order in couplings we should
expand $\exp(i\frac{\l}{3!}\int d^6 z (\frac{\delta}{\delta J(z)})^3
+h.c.)$ into power series. As a result as usual we arrive at some 
Feynman diagrams. In these diagrams $n+m$ is the number of external
points, and order in $\l$ is the number of internal points. Each vertex
evidently corresponds to integration over $d^6 z$ or $d^6\bar{z}$.
Therefore we have to introduce diagrams for superfield
theory, i.e. Feynman supergraphs. Their value consists of the fact
that they allow one to preserve manifest supersymmetry covariance at any
step of calculations.   

Generating functionals of arbitrary models can be constructed 
by analogy with Wess-Zumino model:
\bea
Z[\vec{J}]=\exp(i(S[\vec{\phi}]+\vec{\phi}\vec{J}))
\eea
Here $\vec{\phi}$ is a column matrix denoting set of all superfields,
$\vec{J}$ is a column matrix denoting set of corresponding sources.  
The Green functions can be determined in analogy with (\ref{vacexp}). 

\section{Feynman supergraphs}
\setcounter{equation}{0}
It is easy to see that the Green functions (\ref{2point}), the generating
 functional (\ref{GF2}) and the vacuum expectations (\ref{vacexp}) lead to
the known supergraph technique. 
Really, any
 $<\phi\bar{\phi}>$-propagator corresponds to $(\Box-m^2)^{-1}$, at a
 chiral vertex each propagator is associated with
 $-\frac{1}{4}\bar{D}^2$, and at an antichiral one -- with
 $-\frac{1}{4}D^2$. However, each chiral (antichiral) vertex
 corresponds to integration over $d^6 z$ ($d^6\bar{z}$). However,
 since we deal with $\delta^8(z_1-z_2)$, for sake of unity it is
 more convenient to represent all contributions in the form of
 integrals over $d^8 z$ via the rule $d^6 z (-\frac{1}{4})\bar{D}^2 F
=\int d^8 z F$. As a result, $\int d^6 z\Phi^n$-vertex is 
associated with $n-1$ $(-\frac{1}{4}\bar{D}^2)$-factors, and 
$\int d^6\bar{z}\bar{\Phi}^m$-vertex  -- with $m-1$
 $(-\frac{1}{4}D^2)$-factors -- of course, in the case when all
 superfields are contracted into propagators.
And the vertex $d^8 z \Phi^m \bar{\Phi}^n$ in the same case -- with $m$
 $(-\frac{1}{4}\bar{D}^2)$-factors and $n$
 $(-\frac{1}{4}D^2)$-factors.
Here and further we refer to superfields contracted into propagators
as to the quantum ones.
We see that the 
number of $D^2$, $\bar{D}^2$-factors for such vertices is number
 of antichiral (chiral) {\bf quantum} superfields associated with this vertex.
There is no $D,\bar{D}$-factors arisen from propagators of non-chiral 
(f.e. real) superfields. The propagator $<\phi\phi>$ 
($<\bar{\phi}\bar{\phi}>$) corresponds to $\frac{m\bar{D}^2}{4\Box(\Box-m^2)}$
($\frac{mD^2}{4\Box(\Box-m^2)}$). However, only quantum fields
 (i.e. those ones contracted into propagators) correspond to $D^2$, 
$\bar{D}^2$-factors. External lines do not carry such a factor, and if
 one, two... $n$ chiral (antichiral) superfields associated with the
 vertex are external the number of $\bar{D}^2$- ($D^2$-) factors
 corresponding to this vertex is less by one, two... $n$ than in the case
 when all superfields are contracted to propagators.

If we consider the theory of $N=1$ super-Yang-Mills (SYM) field, its quadratic
action after gauge fixing looks like
\bea
S=\frac{1}{2}tr\int d^8 z v\Box v
\eea
The propagator is 
\bea
\label{propsym}
G(z_1,z_2)=\frac{1}{\Box}\delta^8(z_1-z_2)
\eea
(note the opposite sign with respect to $<\phi\bar{\phi}>$-propagator).
Here $tr$ is matrix trace (superfield $v$ is Lie-algebra valued).
There is no $D$-factors associated with this propagator but they are
associated with vertices. In pure $N=1$ SYM theory vertices are given
by
\bea
S_{int}=\frac{g}{8}\int d^8 z (\bar{D}^2D^{\alpha}v)[v,D_{\alpha}v]+\ldots
\eea
Here dots denote higher orders in $v$ Moreover, in any order there are two $D$-
and two $\bar{D}$-factors. If we consider interaction of a real
superfield with a chiral one, additional $D$-factors correspond to
propagators of chiral superfield only.
As a result we can formulate Feynman rules:

propagators look like
\bea
<\phi\bar{\phi}>&=&-\frac{1}{\Box-m^2}\delta^8(z_1-z_2)\\
<\phi\phi>&=&\frac{m\bar{D}^2}{\Box(\Box-m^2)}\delta^8(z_1-z_2)\nonumber\\
<vv>&=&\frac{1}{\Box}\delta^8(z_1-z_2)\nonumber
\eea
vertices (here $\phi$, $\bar{\phi}$ are quantum superfields) correspond to
\bea
\int d^6 z\phi^n \to (n-1)(-\frac{1}{4})\bar{D}^2\nonumber\\
\int d^8 z \phi\bar{\phi}v^m \to (-\frac{1}{4})D^2 (-\frac{1}{4})\bar{D}^2
\eea
All derivatives in derivative depending vertices act on the propagators.
Any external chiral (antichiral) fields do not correspond to 
$\bar{D}\, (D)$-factors.

Of course, it is more suitable to make Fourier representation for all 
propagators (note that Fourier transformation is carried out with
respect to bosonic coordinates only) by the rule
\bea
\tilde{f}(k)=\int\frac{d^4k}{(2\pi)^4}f(x)e^{ikx}
\eea
The propagators in momentum representation look like
\bea
\label{primp}
<\phi(1)\bar{\phi}(2)>&=&\frac{1}{k^2+m^2}\delta^4_{12}\\
<\phi(1)\phi(2)>&=&\frac{m\bar{D}^2}{4k^2(k^2+m^2)}\delta^4_{12}\nonumber\\
<v(1)v(2)>&=&-\frac{1}{k^2}\delta^4_{12}
\eea
Here 1, 2 are numbers of arguments, and $\delta^4_{12}\equiv
\delta^4(\q_1-\q_2)=\frac{1}{16}(\q_1-\q_2)^2(\bar{\q}_1-\bar{\q}_2)^2$
is a Grassmann delta function. The $D$-factors are introduced as
above. Note, however, that spinor derivatives depend after Fourier
transform on momentum
of propagator with which they are associated. 
The external superfields also can be represented in the form
of Fourier integral. Each propagator is parametrized by momentum, and any
vertex corresponds to integration over $d^4\q$, coupling and delta
function over incoming momenta. As usual, contribution of supergraph
includes integration over all momenta and combinatoric factor which is
totally analogous to that one in standard quantum field theory.

Essentially new feature of superfield theories is presence of $D$-factors.
To evaluate $D$-algebra we can transport them via integration by
parts, then, we can use the identity 
\bea
\label{loop}
\delta^4_{12}D^2\bar{D}^2\delta^4_{12}=16\delta^4_{12}
\eea
To prove this identity we can use expansion of supercovariant derivatives
(\ref{gend}) and note that due to the evident property\\
$\delta^4_{12}\frac{\pa}{\pa\q^{\a}}\delta^4_{12}=\frac{\pa}{\pa\q^{\a}}
\delta^4_{12}|_{\q_1=\q_2}=\frac{1}{8}(\q_{1\a}-\q_{2\a})
(\bar{\q}_1-\bar{\q}_2)^2|_{\q_1=\q_2}=0$\\ only terms of the form
$\delta^4_{12}(\frac{\pa}{\pa\q})^2(\frac{\pa}{\pa\bar{\q}})^2\delta^4_{12}
=16\delta^4_{12}$ survive.

We can prove the following {\bf theorem}. 

The final result for the contribution of any supergraph should have
the form
of {\bf one} integral over $d^4\q$.

{\bf Proof:} Let us consider propagator with $L$ loops, $V$ vertices and $P$
propagators. Any vertex contains integration over $d^4\q$, i.e. there
are $V$ such integrations.
Then, due to (\ref{primp}) any propagator carries
a delta function over Grassmann coordinates, i.e. there are $P$ delta
functions. Then, in any loop we can reduce the number of delta functions
by one using identity (\ref{loop}), i.e. there are $P-L$ independent
delta functions. As a result we can carry out $P-L$ integrations from
$V$, and after $D$-algebra transformations we stay with $V-(P-L)$
integrations. And $V-(P-L)=1$, therefore the result contains one
integration over $d^4\q$. The theorem is proved \cite{West}. 

This theorem is often called non-renormalization theorem. It means
that all quantum corrections are local in $\q$-space. This theorem is
often naively treated as a proof of absence of chiral corrections
(proportional to integral over $d^2\q$). However, such interpretation
is wrong since any contribution in the form of integral over chiral
subspace can be rewritten as an integral over whole superspace using
identity
\bea
\int d^6z f(\Phi)=\int d^8 z (-\frac{D^2}{4\Box})f(\Phi) 
\eea
(this observation was firstly made in \cite{West2}, its consequences 
will be studied further).

Now let us study evaluation of contributions from supergraphs.
The algorithm of it is the following one.

1. We start with one of loops. 
If the number of $D$-factors in this loop is equal to 4 we turn to step 2.
If it is more than  4,
superfluous $D$-factors can be transported to external lines or another
loops via integration by parts, and some of them are converted into
internal momenta via identities $D^2\bar{D}^2D^2=16\Box D^2, 
\{D_{\alpha},\bar{D}_{\beta}\}=-2i\partial_{\alpha\beta}$. 
As a result we stay with exactly 4 $D$-factors.
If the number of $D$-factor is less than 4 than contribution from supergraph
is zero.

2. We contract this loop into a point using identity (\ref{loop}) and
intergrate over one of $d^4\q$ via delta function free of derivatives.  

3. This procedure is repeated for next loops.

4. We integrate over internal momenta.

However, the best way to study evaluation of supergraphs is in
considering some examples.

{\bf Example 1. One-loop supergraph in Wess-Zumino model.}

%\unitlength=1mm
\begin{center}
\begin{picture}(100,60)
\put(50,30){\circle{40}}
\put(30,30){\line(-1,0){20}}
\put(70,30){\line(1,0){20}}
%\put(15,55){\line(0,-1){10}}
%\put(80,55){\line(0,-1){10}}
%\put(80,40){$\partial_{\alpha\dot{\alpha}}$}
%\put(10,60){$\partial_{\beta\dot{\beta}}$}
%\put(65,40){\line(0,-1){10}}
\put(65,50){\line(0,-1){10}}
\put(70,55){$D^2$}
%\put(70,20){$\bar{D}^{\dot{\alpha}}$}
%\put(35,40){\line(0,-1){10}}
%\put(30,20){$D^{\beta}$}
\put(35,50){\line(0,-1){10}}
\put(30,55){$\bar{D}^2$}
%\put(45,80){$G_{+-}$}
%\put(45,20){$G_{-+}$}
\put(40,-5){Fig.1}
\end{picture}
\end{center}
  
The contribution of this supergraph is equal to
\bea
I_1&=&\frac{1}{2}\lambda^2\int d^4\q_1
d^4\q_2\int\frac{d^4p}{(2\pi)^4}\Phi(-p,\q_1)\bar{\Phi}(p,\q_2)
\delta^4_{12}\frac{\bar{D}^2_1D^2_2}{16}\delta^4_{12}\times\nonumber\\&\times&
\int\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2+m^2)((k+p)^2+m^2)}
\eea
The number of $D$-factors is just 4. $D$-algebra transformations are
trivial: we use identity (\ref{loop}) and write 
$\delta^4_{12}\frac{\bar{D}^2_1D^2_2}{16}\delta^4_{12}=\delta^4_{12}$.
The free delta function $\delta^4_{12}$ allows us to integrate over 
$d^4\q_2$ and denote $\q_1=\q$. As a result we get
\bea
I_1=\frac{1}{2}\lambda^2\int d^4\q
\int\frac{d^4p}{(2\pi)^4}\Phi(-p,\q)\bar{\Phi}(p,\q)
\int\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2+m^2)((k+p)^2+m^2)}
\eea
Integral over $k$ can be calculated via dimensional regularization,
the result for it is
\bea
\int\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2+m^2)((k+p)^2+m^2)}=
\frac{1}{16\pi^2}
(\frac{1}{\epsilon}-\int_0^1 dt\log\frac{p^2t(1-t)+m^2}{\mu^2})
\eea 
As a result, contribution of this supergraph takes the form
\bea
I_1=\frac{1}{2}\lambda^2\int d^4\q
\int\frac{d^4p}{(2\pi)^4}\Phi(-p,\q)\bar{\Phi}(p,\q)
\frac{1}{16\pi^2}
(\frac{1}{\epsilon}-\int_0^1 dt\log\frac{p^2t(1-t)+m^2}{\mu^2})
\eea
However, in higher loops regularization in superfield theory possesses
some peculiarities \cite{Jack}.

{\bf Example 2. Two-loop supergraph in Wess-Zumino model.}

\hspace{5cm}
\unitlength=.7mm
\begin{picture}(12,12)
\put(0,0){\circle{20}}\put(-10,0){\line(1,0){20}}
%\put(-8,0){$|$}\ind(-6,2){\bar{D}^2}
%\ind(6,0){|}\ind(5,2){D^2}
\put(-18,2){$D^2$} \put(-18,-5){$\bar{D}^2$} %\put(10,2){$\bar{D}^2$}
\put(10,-5){$D^2$}
\put(-10,2){-} \put(-10,-3){-} %\put(9,2){-} 
\put(9,-3){-}
\put(1,-16){Fig.2}
\put(-7,0){$|$}%\put(7,0){$|$}
\put(-8,-6){$\bar{D}^2$}%\put(2,-6){$D^2$}
\end{picture}

\begin{picture}(5,5)
\put(1,1){}
\end{picture}

\vspace*{5mm}

The contribution of this supergraph is equal to 
\bea
I_2&=&\frac{\lambda^2}{6}\int\frac{d^4k d^4 l}{(2\pi)^8}\int d^4\q_1 d^4\q_2
(-\frac{D^2_1}{4})\delta^4_{12}\frac{D^2_1\bar{D}^2_2}{16}\delta^4_{12}
(-\frac{\bar{D}^2_2}{4})\delta^4_{12}\times\nonumber\\&\times&
\frac{1}{(k^2+m^2)(l^2+m^2)((k+l)^2+m^2)}
\eea
First we do $D$-algebra transformations: we can write
$$(-\frac{D^2_1}{4})\delta^4_{12}\frac{D^2_1\bar{D}^2_2}{16}\delta^4_{12}
(-\frac{\bar{D}^2_2}{4})\delta^4_{12}=
\delta^4_{12}
\frac{D^2_1\bar{D}^2_2}{16}\delta^4_{12}\frac{D^2_1\bar{D}^2_2}{16}
\delta^4_{12}
$$
Then we use identity (\ref{loop}) two times:
$$
\delta^4_{12}
\frac{D^2_1\bar{D}^2_2}{16}\delta^4_{12}\frac{D^2_1\bar{D}^2_2}{16}
\delta^4_{12}=\delta^4_{12}
$$ 
As a result we can integrate over $\q_2$ using the  delta function.
We get
\bea
I_2&=&\frac{\lambda^2}{6}\int\frac{d^4k d^4 l}{(2\pi)^8}\int d^4\q_1 
%(-\frac{D^2_1}{4})\delta^4_{12}\frac{D^2_1\bar{D}^2_2}{16}\delta^4_{12}
%(-\frac{\bar{D}^2_2}{4})\delta^4_{12}\times\nonumber\\&\times&
\frac{1}{(k^2+m^2)(l^2+m^2)((k+l)^2+m^2)}
\eea
This integral vanishes in the standard case since it is proportional to
an integral over $d^4\q$ from constant. However, if we suppose that $m$
is not a constant but $\q$-dependent superfield this contribution is
not zero. Namely this case is studied when the effective action is studied
and $m$ is suggested to depend on background superfields.

{\bf Example 3. One-loop supergraph in dilaton supergravity.}

\unitlength=.4mm
\begin{center}
\begin{picture}(100,100)
\put(50,50){\circle{40}}
\put(30,50){\line(-1,0){20}}
\put(70,50){\line(1,0){20}}
\put(15,52){$|$}
\put(80,52){$|$}
\put(80,40){$\partial_{\alpha\dot{\alpha}}$}
\put(10,60){$\partial_{\beta\dot{\beta}}$}
\put(65,40){$|$}
\put(65,65){$|$}
%\put(70,75){$D^{\alpha}$}
%\put(70,20){$\bar{D}^{\dot{\alpha}}$}
\put(35,40){$|$}
%\put(30,20){$|$}
\put(35,65){$|$}
\put(30,70){$\bar{D}^{\dot{\beta}}D^2$}
%\put(40,60){$|$}\put(40,35){$|$}
%\put(60,60){$|$}\put(60,35){$|$}
%\put(40,20){$\bar{D}^2$}\put(40,80){$D^2$}
\put(60,20){$D^2\bar{D}^{\ad}$}\put(60,70){$\bar{D}^2D^{\a}$}
\put(45,90){$G(k)$}
\put(45,10){$G(k+p)$}
\put(40,0){Fig.3}
\end{picture}
\end{center} 

The contribution of this supergraph is equal to
\bea
\label{i3}
I_3&=&\frac{\xi_1^2}{2}\int d^4\q_1 d^4\q_2\int\frac{d^4p}{(2\pi)^4}
\frac{d^4k}{(2\pi)^4}(\partial_{\alpha\dot{\alpha}}\sigma(-p,\q_1))
(\partial_{\beta\dot{\beta}}\sigma(p,\q_2))\times\nonumber\\&\times&
\frac{D^{\alpha}\bar{D}^2D^2\bar{D}^{\dot{\beta}}}{16}\delta^4_{12}
\frac{\bar{D}^{\dot{\alpha}}D^2\bar{D}^2 D^{\beta}}{16}\delta^4_{12}
\times\nonumber\\&\times&G(k)G(k+p)
\eea
Here $G(k),G(k+p)$ are functions of momenta which explicit form is not
essential here (they are exactly found in \cite{my1}). The derivatives
$\partial_{\a\dot{\a}},\partial_{\b\dot{\b}}$ are not transported from
external fields $\sigma,\bar{\sigma}$. Our aim here is to obtain terms
proportional to $\partial^m\sigma\partial^n\bar{\sigma}$.
We suggest that spinor derivatives associated with one propagator
depend on momentum $k$, and with another -- to $k+p$.

Using commutation relations (\ref{salg}) we find that
\bea
\label{stru}
\frac{D^{\alpha}\bar{D}^2D^2\bar{D}^{\dot{\beta}}}{16}\delta^4_{12}
\frac{\bar{D}^{\dot{\alpha}}D^2\bar{D}^2 D^{\beta}}{16}\delta^4_{12}=
\frac{2k^{\a\dot{\g}}\bar{D}_{\dot{\g}}D^2\bar{D}^{\bd}}{16}\delta^4_{12}
\frac{2(k+l)^{\g\ad}D_{\g}\bar{D}^2D^{\b}}{16}\delta^4_{12}
\eea
We transport all spinor supercovariant derivatives to one propagator.
As a result we arrive at
\bea
4k^{\a\dot{\g}}(k+p)^{\g\ad}\delta^4_{12}
\frac{\bar{D}^{\bd}D^2\bar{D}_{\dot{\g}}D_{\g}\bar{D}^2 D^{\b}}{256}
\delta^4_{12}
\eea
We can use (\ref{salg}) several times. At the end we get
\bea
4k^{\a\dot{\g}}(k+p)^{\g\ad}(k+p)_{\g\dot{\g}}(k+p)^{\bd\delta}
\delta^4_{12}D_{\delta}\bar{D}^2D^{\beta}\delta^4_{12}
\eea
Equations (\ref{salg}) and (\ref{loop}) allow one to write
$$
\delta^4_{12}D_{\delta}\bar{D}^2D^{\beta}\delta^4_{12}=-\frac{1}{2}
16\delta^{\b}_{\delta}\delta^4_{12}
$$
We substitute this expression in (\ref{stru}). Using identity
$k^{\a\bd}k_{\g\bd}=\delta^{\a}_{\g}k^2$ we obtain the
contribution from (\ref{stru}) in the form
$$
64k^{\a\ad}(k+p)^{\b\bd}(k+p)^2\delta^4_{12}
$$
which after integration over $\q_2$ leads to the 
following contribution for $I_3$:
\bea
I_3&=&64\frac{\xi_1^2}{2}\int d^4\q_1 \int\frac{d^4p}{(2\pi)^4}
\frac{d^4k}{(2\pi)^4}(\partial_{\alpha\dot{\alpha}}\sigma)(-p,\q_1)
(\partial_{\beta\dot{\beta}}\sigma)(p,\q_1)\times\nonumber\\&\times&
k^{\a\ad}(k+p)^{\b\bd}(k+p)^2
\eea
Detailed analysis carried out in \cite{my1} shows that this
correction is divergent.

Calculation of corrections from supergraphs in other superfield
theories is carried out on the base of analogous approach.

We showed that supergraph technique is a very effective method for
consideration of quantum corrections in superfield theories. The next
step of its development is introducing renormalization in these theories.

\section{Superficial degree of divergence. Renormalization.}

\setcounter{equation}{0}

We found that divergent quantum corrections arise in
superfield theories as well as in standard field theories. Therefore
we face two problems:

(i) to classify possible divergences

(ii) to develop a procedure of renormalization in superfield theories.

It turns out that the technique for solving these problems is quite
analogous to that one used in standard field theory. First problem
can be solved on the base of superficial degree of divergence. The
natural way for solving second one is in introducing superfield
counterterms which are quite analogous to standard ones.

First of all let us consider superficial degree of divergence
\cite{Collins}. 

{\bf Example.} The $N=1$ super-Yang-Mills (SYM) theory with chiral matter
(with Wess-Zumino self-interaction). For all other models
consideration is quite analogous.
The action of the theory is
\bea
\label{actsym}
S&=&\int d^8 z\bar{\Phi}_i(e^{gV})^i_j\Phi^j+
(\int d^6 z(\frac{1}{2}m\Phi^2+\frac{\l}{3!}\Phi^3)+h.c.)+\nonumber\\&+&
{\rm tr}\frac{1}{g^2}\int d^8 z (e^{-gV}D^{\alpha}e^{gV})
\bar{D}^2(e^{-gV}D_{\a}e^{gV})
\eea
The vertices in this theory look like
\bea
\label{vert}
\int d^6z\Phi^3,\ \int d^6\bar{z}\bar{\Phi}^3,\ \int d^8 z \bar{\Phi}_i
{(V^n)}_j^i \Phi^j, \int d^8 z (\bar{D}^2 D^{\alpha} V) [V,D_{\alpha} V]
\eea
and higher ones. Indices $i,j$ are matrix indices since $\Phi_i$ is an
isospinor.
However, all vertices corresponding to pure SYM 
self-interaction contain exactly two chiral and two antichiral
derivatives. We have proved already that all corrections should be
proportional to one integral over $d^4\q$. 

As usual, superficial degree of divergence (SDD) is the order of 
the integral over
internal momenta for corresponding contribution, or, as is the same,
as a degree of homogeneity of diagram in momenta, considered after
evaluation of $D$-algebra transformations \cite{BK0}. 
The only difference of SDD in our case is the additional contribution 
from $D$-factors.

It is easy to see that contributions to SDD are generated by 
momentum depending factors in propagators and vertices
(as usual, any internal momentum $k$ gives contribution 1), loop
integrations, or, in other words, by manifest momentum dependence
which is associated with propagators and loop integration, and by
$D$-factors which are associated with propagators and vertices
(note that due to identities $D^2\bar{D}^2D^2=16\Box D^2$,
$\{D_{\a},\bar{D}_{\ad}\}=-2i\pa_{\a\ad}$ chiral derivative
combined with antichiral one can be converted to momentum; therefore
any $D$-factor contribute to SDD with $1/2$). If not all
spinor derivatives are converted to internal momenta SDD from
supergraph evidently decreases. 

Let us consider arbitrary supergraph with $L$ loops, $V$ vertices, $P$
propagators ($C$ of them are $<\phi\phi>$, $<\bar{\phi}\bar{\phi}>$
-propagators) and $E$ external lines ($E_c$ of them are chiral). 
We denote SDD as $\omega$.

Any integration over internal momentum (i.e. over $d^4k$) contributes
to SDD with 4. Since the number of integrations over internal momenta is
the number of loops, the  total 
contribution from all such integrations is $4L$.  
Any propagator includes $\frac{1}{k^2+m^2}$ or $\frac{1}{k^2}$ 
(\ref{primp}), hence contribution of all propagators is equal to $-2P$. Since
$<\Phi\Phi>,<\bar{\Phi}\bar{\Phi}>$-propagator contains additional 
$\frac{1}{k^2}$ these propagators give additional contribution $-2C$.
Therefore manifest dependence of momenta gives contribution to
$\omega$ equal to $4L-2P-2C$.

Now let us consider contribution of $D$-factors to SDD.
Each (both pure gauge and that one containing chiral superfields) {\bf vertex}
without external chiral (antichiral) lines contains four 
$D$-factors (\ref{vert}) since any superfield $\phi$ (contracted to
propagator) corresponds to $\bar{D}^2$, and $\bar{\phi}$ -- to $D^2$.
Therefore each vertex gives contribution 2.
However, external chiral (antichiral) lines do not correspond to
$D$-factors. As a result, any external line decreases $\omega$ by 1, 
Each $<\phi\phi>,<\bar{\phi}\bar{\phi}>$-propagator contains a factor
$\bar{D}^2$ ($D^2$) with contribution 1. Then, due to identity
(\ref{loop}) contraction of any loop into a point decreases the number of
$D$-factors which can be converted to internal momenta by 4, and 
$\omega$ -- by 2.
As a result the total contribution of $D$-factors to $\omega$ is equal to
$2V-E_c-2L+C$ (remind that  $D$-factors contribute to $\omega$ with
$1/2$). 

Therefore SDD is equal to
\bea
\omega=4L-2P-2C+2V-E_c-2L+C=2L-2P+2V-C-E_c
\eea
Using known topological identity $L+V-P=1$ we have
\bea
\label{ind}
\omega=2-C-E_c
\eea
Really, SDD can be lower than (\ref{ind}) if some of $D$-factors are
transported to external lines and do not generate internal momenta. If $N_D$
D-factors are moved to external lines the $\omega$ is equal to
\bea
\omega=2-C-E_c-\frac{1}{2}N_D.
\eea
 It is final expression for SDD. As usual, at $\omega\geq 0$
 supergraph diverges, and at $\omega <0$ -- converges.
We note that:\\ 
1. $\omega\leq 2$ hence SDD is restricted from above.\\
2. As the number of external lines grows $\omega$ decreases. Therefore
the number of divergent structures is essentially restricted -- it is
finite (really, there can be no more than two external chiral legs
and no more than two $<\F\F>,<\bar{\F}\bar{\F}>$ propagators).
And if the number of divergent structures is finite the theory is
renormalizable. Hence we shown that the theory including chiral
superfields with Wess-Zumino-type interaction and gauge superfields
with action (\ref{actsym}) is renormalizable.
This is quite natural since the mass dimension of all couplings in this
theory is zero.

However, non-renormalizable superfield theories also exist.

{\bf Example.} General chiral superfield model \cite{my2}.

The action of the model is
\bea
\label{actgen}
S&=& \int d^8 z K(\Phi,\bar{\Phi})+(\int d^6 z W(\Phi) +h.c.)=\nonumber\\
&=&\int d^8 z \Phi\bar{\Phi}+
[\int d^6 z (\frac{1}{2}m\F^2+\frac{\l}{3!}\F^3)+h.c.]+\\&+&
\int d^8 z
[K_{12}\F\bar{\F}^2+K_{21}\bar{\F}\F^2+\sum_{m,n=2}^{\infty}
 \frac{1}{m!n!}K_{mn}\F^n\bar{\F}^m ]+(\int d^6
 z\sum_{l=4}^{\infty}\frac{W_n}{n!}
\Phi^n+h.c.)\nonumber
\eea
Here $K_{ij}, W_l$ are constants.

Propagators in the theory are just (\ref{primp}), their contribution
to SDD is equal to $4L-2P-2C$ as above. However, the contribution from
$D$-factors differs. Any vertex $K_{nm}\Phi^n\bar{\F}^m$ corresponds
to $n$ $\bar{D}^2$-factors and $m$ $D^2$-factors.
The total contribution to $\omega$ from all such vertices is
$\sum_{V_t}(n_v+m_v)$, i.e. sum of $n$ and $m$ over all vertices 
corresponding to integral over total superspace.
Any vertex $W_l\F^l$ contains an integral over $d^6 z$ and
effectively corresponds to $(l-1)$ $\bD^2$-factors. Total
contribution from such vertices is $\sum_{V_c}(l_c-1)$ (i.e. sum over
all purely chiral or antichiral vertices). Again external lines
decrease the number of $D^2 \ (\bD^2)$-factors by $2E_c$ ($E_c$ is a
number of external lines), each $<\F\F>,<\bar{F}\bar{F}>$-propagators
carries one $\bD^2$ ($D^2$)-factor. Contraction of each loop to a
point decreases the number of $D$-factors by 4.
Hence the total number of $D$-factors is
\bea
2\sum_{V_t}(n_v+m_v)+2\sum_{V_c}(l_c-1)-2E_c-4L+2C
\eea 
Contribution to SDD from $D$-factors is their number divided by two.
Therefore total SDD is equal to
\bea
\omega&=&4L-2P-2C+\frac{1}{2}
(2\sum_{V_t}(n_v+m_v)+2\sum_{V_c}(l_c-1)-2E_c-4L+2C)=
\nonumber\\&=&
2-2V-C-2E_c+[\sum_{V_t}(n_v+m_v)+\sum_{V_c}(l_c-1)]
\eea
Here we used $2L-2P=2-2V$. However, any vertex gives contribution
$-2$ to term $-2V$ and $l_c-1$ or $n_v+m_v$ to other terms of
$\omega$. It is evidently that either $l_c-1$ or $n_v+m_v$ 
can be more than 2 since either $l_c\geq 3$ or $n_v+m_v\geq
3$.
Hence in general case $\sum_{V_t}(n_v+m_v)+\sum_{V_c}(l_c-1)-2V\geq
0$,
the number of divergent structures is not restricted, and the theory is
non-renormalizable. This is quite natural since constants $K_{ij}$
(if i or j no less than 2) and $W_l$ (if $l\geq 4$) have negative mass
dimension.

The next problem is introduction of regularization.
The most natural way of introducing regualarization in supersymmetric
theories is dimensional regularization. It can be introduced as usual:
integral
$$
\int\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2+m^2)^N}
$$
is replaced by
$$
\int\frac{d^{4+\epsilon}k}{(2\pi)^{4+\epsilon}}\frac{1}{(k^2+m^2)^N}
$$
All divergences corresponds to poles in $\epsilon$ (no more than
$\frac{1}{\epsilon^L}$ for $L$-loop correction).

However, there are some peculiarities. First of all, at component
level any supersymmetric action includes spinors and hence
$\gamma$-matrices which are well defined if and only if the dimension of
space-time is integer. Therefore we must use some modification of the
dimensional regularization called dimensional reduction. According to
it all objects with well behaviour only at separate dimensions (such
as spinors and $\gamma$-matrices) are evaluated at these dimensions
(or namely at dimension equal to 4), and integrals over momenta -- at
arbitrary dimension (this procedure was introduced by W. Siegel).
However, dimensional reduction leads to some difficulties in
calculation of higher loop corrections since many supergraphs involve
contractions of essentially four-dimensional objects, such as
Levi-Civita tensor $\epsilon^{abcd}$, with $d$-dimensional objects,
and such contractions need additional definition. As a result often
some ambiguities arise. However, such phenomena are observed only
beyond two loops.

We also can use analytic regularization which corresponds to change
$$
\int\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2+m^2)^n}\to
\int\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2+m^2)^{n+\epsilon}}
$$
However, this regularization also leads to some difficulties
(see discussion of questions connected to regularization in
supersymmetric theories in \cite{Jack}).

Technique for renormalization in superfield theories is quite
analogous to that one in common QFT. It is carried out via
introduction of counterterms.

{\bf Example.} Consider one-loop contribution to the kinetic term in
Wess-Zumino model. Corresponding supergraph is given by Fig. 1 (see
above), its contribution is equal to
\bea
I_1=\frac{1}{2}\lambda^2\int d^4\q
\int\frac{d^4p}{(2\pi)^4}\Phi(-p,\q)\bar{\Phi}(p,\q)
\frac{1}{16\pi^2}
(\frac{1}{\epsilon}-\int_0^1 dt\log\frac{p^2t(1-t)+m^2}{\mu^2})
\eea 
We see that this divergence has the form of pole part proportional to 
$\frac{1}{\epsilon}$. To cancel it we must add to the initial kinetic
term
\bea
S=\int d^8 z \Phi(x,\q)\bar{\Phi}(x,\q) 
\eea
(which is just $\int \frac{d^4 p}{(2\pi)^4} d^4 \q \Phi (-p,\q)
\bar{\Phi}(p,\q)$) a counterterm
\bea
\Delta S_{countr}=-\frac{\lambda^2}{32\pi^2\epsilon}\int d^8 z
\Phi(z)\bar{\Phi}(z)
\eea
which corresponds to the replacement of $\int d^8 z \Phi\bar{\Phi}$ in
the classical action by $\int d^8 z Z\Phi(z)\bar{\Phi}(z)$ where
\bea
Z=1-\frac{\lambda^2}{32\pi^2\epsilon}
\eea
is a wave function renormalization.

The essential peculiarity of superfield theories is the fact that
number of counterterms in these theories is less than in their
non-supersymmetric analogs. For example, Wess-Zumino model is a
supersymmetric generalization of $\phi^4$-theory, but it possesses
only renormalization of kinetic term and no renormalization of
couplings. The conclusion about absence of divergent correction to
coupling the $\lambda$ (or as is the same -- to chiral potential) is also
called non-renormalization theorem. However, this theorem does not
forbid {\bf finite} corrections to superpotential which present in
massless Wess-Zumino model \cite{West3,West4,my3,my4}.

Theare are also some interesting properties of renormalization in
superfield theories.

First, all tadpole-type contributions in Wess-Zumino model vanish:
supergraph

\unitlength=.4mm
\hspace*{5cm}
\begin{picture}(100,100)
\put(50,50){\circle{40}}
\put(32,50){\line(-1,0){10}}
\put(31,53){-}
\put(22,54){$D^2$}
\end{picture}

has contribution proportional to
\noindent $D^2\delta_{11}=\delta^{12}D^2\delta^{12}=0$.
However, in theories including vertices proportional to integral over
whole superspace (f.e. dilaton supergravity) tadpole contributions are
not equal to zero \cite{my1}.

Second, all contributions from vacuum supergraphs are proportional to
$\int d^4\q c$ (with $c$ is a constant) and also vanish. However, this
statement is not true for background dependent propagators. Using of
background dependent propagators are very important method for
calculation of effective action. Now we turn to its studying.

\section{Effective action and loop expansion}
\setcounter{equation}{0}

Effective action is a central object of quantum field theory. Studying
of effective action allows to investigate problems of vacuum stability,
Green functions, spontaneous symmetry breaking, anomalies and
many other problems.

Effective action in superfield theory is defined as usual as a
generating functional of one-particle-irreducible Green functions. It
is obtained as a Legendre transform for generating functional of
connected
Green functions:
\bea
\Gamma[\Phi]=W[J]-\int dz J(z)\Phi(z)
\eea
Here $\Gamma[\Phi]$ is an effective action, $dz$ denotes integral over
the corresoponding subspace ($d^6z$ for
chiral sources, $d^8z$ for general ones), $\Phi$ is a set of all
superfields, $\Phi(z)=\frac{\delta W[J]}{\delta J(x)}$ is so called
mean field or background field, 
$W[J]=\frac{1}{i}\log Z[J]$ is a generating functional of
the connected Green functions. As usual, $\Gamma[\Phi]$ satisfies the
equation
$$
\frac{\delta\Gamma[\Phi]}{\delta\Phi(x)}=-J(x)
$$
The effective action can be expressed in the form of path integral
\cite{BO}:
\bea
\label{efact}
e^{\frac{i}{\hbar}\Gamma[\Phi]}=\int D\phi e^{\frac{i}{\hbar}
(S[\phi]+\phi J-\Phi J)}
\eea
Here $S[\phi]$ is a classical action of the corresponding theory. Note
that $\phi$ is a variable of integration, and $\Phi$ is a function of 
classical source $J$ which does not depend on $\phi$. We introduced
$\hbar$
by dimensional reasons and to obtain loop  expansion \cite{BO}.
To calculate this integral we make change of variables of integration:
$$
\phi\to\Phi+\sqrt{\hbar}\phi
$$
If we have several fields we can unite them into a column vector, and
all consideration is quite analogous.
The integral (\ref{efact}) after this change takes the form
\bea
\label{intmov}
e^{\frac{i}{\hbar}\Gamma[\Phi]}=\int D\phi e^{\frac{i}{\hbar}
(S[\Phi+\sqrt{\hbar}\phi]+\sqrt{\hbar}\phi J)}
\eea
Our aim consists here of the expansion of $\Gamma[\phi]$ in power series
in $\hbar$ following the approach in \cite{BO}.

First, we expand factor in the exponent into power series in $\hbar$:
\bea
& &\frac{i}{\hbar}S[\Phi+\sqrt{\hbar}\phi]+\frac{i}{\sqrt{\hbar}}\phi J=
\frac{i}{\hbar}\Big(S[\Phi]+S'[\Phi]\sqrt{\hbar}\phi+
\frac{\hbar}{2}S^{''}[\Phi]\phi^2+\ldots+\nonumber\\&+&
\frac{\hbar^{n/2}}{n!}S^{(n)}[\Phi]
\phi^n+\ldots
\Big)
\eea
Here $S^{(n)}[\Phi]$ denotes $n$-th variational derivative of
the classical action with respect to $\Phi$ (integration over
corresponding space is assumed).
This expansion can be substituted into (\ref{intmov}). We introduce 
$\bar{\Gamma}[\Phi]=\Gamma[\Phi]-S[\Phi]$ which is a quantum
contribution to effective action that can be expanded into power
series in $\hbar$:
$\bar{\Gamma}=\sum_{n=1}^{\infty}\hbar^n\Gamma^{(n)}$.
As a result we have
\bea
e^{\frac{i}{\hbar}\bar{\Gamma}[\Phi]}=\int D\phi \exp\Big[
\frac{i}{\hbar}\Big(S[\Phi]+S'[\Phi]\sqrt{\hbar}\phi+
\frac{\hbar}{2}S^{''}[\Phi]\phi^2+\ldots+\frac{\hbar^{n/2}}{n!}S^{(n)}[\Phi]
\phi^n+\ldots
\Big]
\eea
Then, the block $\frac{i}{\sqrt{\hbar}}(S'[\Phi]+J)\phi$ can lead only to
one-particle-reducible supergraphs since its contribution with one
quantum field $\phi$ can form only one propagator. Hence we can omit
this term. Then we can expand the exponent into power series in $\hbar$:
\bea
\label{exph}
& &e^{\frac{i}{\hbar}\bar{\Gamma}[\Phi]}=\int D\phi \exp\Big[
\frac{1}{2}S^{''}[\Phi]\phi^2\Big(
1+\frac{i\sqrt{\hbar}}{3!}S^{(3)}[\Phi]\phi^3+\frac{i\hbar}{4!}S^{(4)}
[\Phi]\phi^4+\nonumber\\&+&
(\frac{i\sqrt{\hbar}}{3!})^2(S^{(3)}[\Phi]\phi^3)^2+\ldots
\Big)
\Big]
\eea 
At the same time substituting the expansion of
$\bar{\Gamma}$ into power series in $\hbar$ we have 
$\exp(\frac{i}{\hbar}\bar{\Gamma}[\Phi])=\exp(i\Gamma^{(1)}[\Phi])
(1+i\hbar\Gamma^{(2)}[\Phi]+\ldots)$ (here we suppose that $\hbar$
is a small parameter). Substituing this expansion into
(\ref{exph}) and comparing equal powers of $\hbar$ we see that any 
correction $\Gamma^{(n)}$ corresponds to some correlator. For example,
one-loop correction is defined from equation
\bea
\label{1loop}
\exp(i\Gamma^{(1)}[\Phi])=\int D\phi \exp(\frac{i}{2}S^{''}[\Phi]\phi^2)
\eea 
and two-loop one -- from equation
\bea
\label{2loop}
\Gamma^{(2)}=\frac{1}{i}\frac{
\int D\phi\exp(\frac{i}{2}S^{''}[\Phi]\phi^2)\Big(
\frac{i}{4!}S^{(4)}[\Phi]\phi^4-\frac{1}{(3!)2}(S^{(3)}[\Phi]
\phi^3)^2\Big) }{\int D\phi \exp(\frac{i}{2}S^{''}[\Phi]\phi^2)}
\eea
Here, as usual, integration over coordinates in expressions of the form
$S^{(n)}[\Phi]\phi^n$ is assumed.

We can see that:

(i) All odd orders in $\sqrt{\hbar}$ vanish since they correspond to 
$\int D\phi\phi^{2n+1}\exp(\frac{i}{2}S^{''}[\Phi]\phi^2)$. Due to symmetrical
properties this integral is equal to zero.

(ii) All terms beyond first order in $\hbar$ are expressed in the form of
some correlators.

(iii) One-loop correction (\ref{1loop}) can be expressed in the form of
functional determinant since
\bea
\int D\phi\exp(\frac{i}{2}S^{''}[\Phi]\phi^2)=Det^{-1/2}S^{''}[\Phi]
\eea
which leads to
\bea
\Gamma^{(1)}=\frac{i}{2} Tr\log S^{''}[\Phi]
\eea
And $S^{''}[\Phi]$ (further we denote it as $\Delta$) is some
operator. In many cases it has the form $\Delta=\Box+\ldots$. 
We can express one-loop effective action in terms of functional (super)trace
\bea
\Gamma^{(1)}=\frac{i}{2} {\rm Tr} \int_0^{\infty}\frac{ds}{s}e^{is\Delta}
\eea  
This expression is called Schwinger representation. Sign ${\rm Tr}$
denotes both matrix trace ${\rm tr}$ (if $\Delta$ possesses matrix indices) and
functional trace, i.e.
$$
{\rm Tr} e^{is\Delta}={\rm tr} \int d^8 z_1 d^8 z_2 \delta^8(z_1-z_2)
e^{is\Delta}\delta^8(z_1-z_2)
$$
Calculation of $e^{is\Delta}$ in field theories is carried out
with use of a special procedure called Schwinger-De Witt method or
proper time method \cite{BSD}.

Let us consider higher loop corrections. From (\ref{exph}) it is easy
to see that all loop corrections beyond one-loop order have the form of
some correlators, i.e. they include
\bea
\int D\phi \exp(\frac{i}{2}S^{''}[\Phi]\phi^2)\prod_n(S^{(n)}[\Phi]\phi^n)
\eea
This correlators can be calculated in the way analogous to standard
theory of perturbations. We can use expression
\bea
\int D\phi \phi^n e^{i\phi\Delta\phi}=(\frac{1}{i}
\frac{\delta}{\delta j})^n\int D\phi e^{i(\phi\Delta\phi+j\phi)}|_{j=0}
\eea
which allows to introduce diagram technique in which the role of vertices
is played by $\frac{S^{(n)}[\Phi]\phi^n}{n!}$, and role of propagators
-- by $\Delta^{-1}$. However since $\Delta=S^{''}[\Phi]$ is background
dependent (see above)
we arrive at background dependent propagators $<\phi(z_1)\phi(z_2)>=
\Delta^{-1}\delta^8(z_1-z_2)$. These propagators can be found exactly
only in some special cases, the most important of them are: first,
constant in space-time background superfields, second, the
background superfields are only chiral. Further we consider
examples.

Let us turn again to (\ref{exph}). We see that each quantum superfield
corresponds to $\hbar^{-1/2}$, and each vertex -- to $\hbar^{-1}$ (which
provides $\hbar^{n/2-1}S^{(n)}[\Phi]\phi^n$). Arbitrary (super)graph
with $P$ propagators and $V$ vertices contain $2P$ quantum superfields
(each propagator is formed by contraction of two
superfields). Therefore if this (super)graph contain vertices 
$S^{n_1}[\Phi]\phi^{n_1},S^{n_2}[\Phi]\phi^{n_2},\ldots,
S^{n_V}[\Phi]\phi^{n_V}$ its power in $\hbar$ is $\sum_{i=1}^V
(\frac{n_i}{2}-1)=\frac{1}{2}\sum_{i=1}^V n_i -V$. However,
$\sum_{i=1}^V n_i$ is just the number of quantum fields associated with
all vertices which is equal to $2P$. Therefore the correlator described by
this (super)graph
has power of $\hbar$ equal to $P-V=L-1$, with $L$ is number of
loops. But any correlator of the
form (\ref{exph}) is a contrbution to $\frac{\Gamma}{\hbar}$, hence 
contribution from $L$-loop (super)graph to $\Gamma$ is proportional to 
$\hbar^L$. Hence we found that the order in $\hbar$ from an arbitrary
(super)graph is just the number of loops in it, and the expansion in powers of
$\hbar$ is called loop expansion. As a result we see that loop
corrections can be calculated on the base of special (super)field technique.

Let us make some comments. One of the most often questions is: how is the
definition of (one-loop) correction in effective action in terms of
trace of logarithm related to expression of the same correction in
terms of supergraphs? 

To clear this relation we give an example. One-loop effective action
in Wess-Zumino model is given by \cite{Buch1,BK0}
\bea
\label{olwz}
\Gamma^{(1)}=\frac{i}{2}{\rm Tr}\log (\Box-\frac{1}{4}\Psi\bar{D}^2
-\frac{1}{4}\bar{\Psi}D^2)
\eea
Here $\Psi$ is background chiral superfield. This expression
can be rewritten as
\bea
\Gamma^{(1)}=\frac{i}{2}{\rm Tr}\log[\Box(1-
\frac{1}{4\Box}(\Psi\bar{D}^2+\bar{\Psi}D^2)
)]
\eea
Expansion of the logarithm into power series leads to
\bea
\Gamma^{(1)}=\frac{i}{2}{\rm Tr}\sum_{n=1}^{\infty}\frac{1}{n}
[\frac{1}{4\Box}(\Psi\bar{D}^2+\bar{\Psi}D^2)]^n
\eea
This expression evidently corresponds to the following supergraphs

\hspace{0.5cm}
\unitlength=.6mm
%\thicklines
%\GRAPH(hsize=3){%\ind(50,-30){.1.}
\begin{picture}(20,20)
\put(0,10){\circle{20}}\put(-10,10){\line(-1,0){5}}
%\put(-10,8.5){\line(-1,0){5}}
\put(10,10){\line(1,0){5}}%\put(10,8.5){\line(1,0){5}}
%\ind(24,0){W''}\ind(-26,0){\bar{W''}}
%\put(4.8,0){\GRAPH(hsize=3){
\end{picture}
\hspace{2cm}
\begin{picture}(20,20)
\put(0,10){\circle{20}}\put(-10,10){\line(-1,0){5}}
%\put(-10,8.5){\line(-1,0){5}}
\put(10,10){\line(1,0){5}}%\put(10,8.5){\line(1,0){5}}
\put(0,20){\line(0,1){5}}%\put(-1,20){\line(0,1){5}}
\put(0,0.5){\line(0,-1){5}}
%\put(-1,0){\line(0,-1){5}}
%}}
%\ind(-10,20){W''}\ind(-10,-20){W''}
%\ind(-49,0){\bar{W}''}\ind(5,0){\bar{W''}}
%\put(30,0){\ldots}
\put(0,-12){Fig.4}
\end{picture}
%}}}}
\hspace{2cm}
\begin{picture}(30,30)
\put(0,10){\circle{20}}
\put(-10,10){\line(-1,0){5}}%\put(-10,8.5){\line(-1,0){5}}
\put(10,10){\line(1,0){5}}%\put(10,8.5){\line(1,0){5}}
\put(-9,5){\line(-1,-1){6}}%\put(-8,4){\line(-1,-1){6}}
\put(9,5){\line(1,-1){6}}%\put(8,4){\line(1,-1){6}}
\put(-9,15){\line(-1,1){6}}%\put(-8,16){\line(-1,1){6}}
\put(9,15){\line(1,1){6}}%\put(8,16){\line(1,1){6}}
\put(30,10){\ldots}
\end{picture}

\vspace*{5mm}

External lines are for alternating $\Psi\bar{D}^2/4$ and
$\bar{\Psi}D^2/4$,
and internal ones are for $\Box^{-1}$. At the same time, if we consider
theory of real scalar superfield $u$ in external chiral superfield
$\Psi$ with action
\bea
S=\int d^8 z u (\Box-\frac{1}{4}\Psi\bar{D}^2-\frac{1}{4}\bar{\Psi}D^2)u
\eea
it leads just to these supergraphs (if $\int d^8 z u(-\frac{1}{4}\Psi
\bar{D}^2) u$ and the conjugated term are treated as vertices), and
one-loop effective action is given by (\ref{olwz}).

We can see that the expression of one-loop effective action in
theform of trace of logarithm allows to use some special technique which
is equivalent to supergraph approach, but more convenient in many
cases.
This technique is called proper-time technique.

\section{Superfield proper-time technique}
\setcounter{equation}{0}

As we have already proved, if the quadratic action of a 
quantum (super)field $\phi$
on classical background $\Phi$ has the form $\int dx \phi \Delta[\Phi]\phi$
($\int dx$ here denotes integral over all (super)space), one-loop effective
action in this theory is $\Gamma^{(1)}=\frac{i}{2}Tr\int_0^{\infty}
\frac{ds}{s}e^{is\Delta}$. Therefore we face the problem of calculating
the operator $e^{is\Delta}$. In most important cases
$\Delta=\Box+\ldots$
where dots denote background dependent terms.
It is known \cite{BSD} that the best way
to find this operator in the case of common field theory is as follows.
We introduce $U(x,x'|s)=e^{is\Delta}\delta^4(x-x')$ called Schwinger
kernel. Of course, $U$
depends on background superfields. It satisfies the equation:
\bea
i\frac{\partial U}{\partial s}=-U\Delta
\eea
The $\Delta$ is supposed to have form of power series in
derivatives. And $U$ satisfies initial condition 
$$U(x,x')|_{s=0}=\delta^4(x-x').$$  
In general case $U$ is represented in the form of infinite power
series in parameter $s$ (called proper time) as 
\bea
U=-\frac{i}{(4\pi s)^2}\exp(\frac{i}{4s}(x-x')^2)
\sum_{n=0}^{\infty}a_n (is)^n
\eea 
\cite{BSD}. (Ultraviolet) divergences correspond to {\bf lower}
orders of this expansion (note that ultraviolet case corresponds to
$s\to 0$, infrared one -- to $s\to\infty$).
Coefficients $a_n$ depend on background superfields and their derivatives.
We note that if background superfields are put to zero, we arrive at
\bea
\label{u0}
U^{(0)}(x,x';s)=e^{is\Box}\delta^4(x-x')=-\frac{i}{(4\pi s)^2}
\exp(\frac{i}{4s}(x-x')^2)
\eea
which satisfies condition
\bea
i\int_0^{\infty} ds U^{(0)}(x,x';s)=\frac{1}{\Box}\delta^4(x-x')
\eea
The approach in case of superfield theories is quite
analogous. However, it possesses essential advantage. In this case it is more
convenient to expand Schwinger kernel $U(x,x';s)$ not in infinite
power series in $s$ but in a {\bf finite} power series in spinor
supervcovariant derivatives (these series are finite due to
anticommutation of spinors).

Really, in most cases operator $\Delta$ in superfield theories looks
like
\bea
\Delta=\Box+\sum A_{nm}(D^{\a})^n(\bar{D}^{\ad})^m\equiv \Box+\tilde{\Delta}
\eea
with $\tilde{\Delta}$ is some background dependent operator
(in most cases it contains only even orders in spinor derivatives,
here we consider this case), 
$A_{nm}$ are background dependent coefficients.
We introduce the structure
\bea
\label{u}
U(z,z';s)=\exp(is\Delta)\delta^8(z-z')\equiv
\exp(is\tilde{\Delta})\exp(is\Box)\delta^8(z-z')
\eea
(last identity is valid in the case of contributions which do not
depend
on space-time derivatives of superfields; however, in most cases
namely such contributions are studied). 
We substitute natural initial condition 
$$
U(z,z';s)|_{s=0}=\delta^8(z-z')
$$ 
And
$\exp(is\Box)\delta^8(z-z')=\delta^4(\theta-\theta')U^{(0)}(x,x';s)$
where $U^{(0)}(x,x';s)$ is given by (\ref{u0}). Hence
\bea
U(z,z';s)=\exp(is\tilde{\Delta})U^{(0)}(x,x';s)\delta^4(\theta-\theta')
\eea 
Hence we face the problem of calculating
$\tilde{U}=\exp(is\tilde{\Delta})$.
The $\tilde{U}$ satisfies the equation
\bea
\label{etu}
i\frac{\partial\tilde{U}}{\partial s}=-\tilde{U}\tilde{\Delta}
\eea
It is easy to see that $\tilde{U}|_{s=0}=1$.
We expand $\tilde{U}$ into power series in spinor supercovariant
derivatives:
\bea
\label{tu}
\tilde{U}&=&1+\frac{1}{16}A(s)D^2\bar{D}^2+\frac{1}{16}\tilde{A}(s)
\bar{D}^2D^2+\frac{1}{8}B^{\a}(s)D_{\a}\bar{D}^2+\frac{1}{8}\tilde{B}_{\ad}
\bar{D}^{\ad}D^2+\nonumber\\&+&
\frac{1}{4}C(s)D^2+\frac{1}{4}\tilde{C}(s)\bar{D}^2
\eea
We substitute (\ref{tu}) into equation (\ref{etu}). As a result we
obtain in right-hand side some power series in spinor derivatives.
Comparing coefficients at analogous derivatives in right-hand side and
left-hand side of identity we get
\bea
\label{sys0}
\frac{1}{16}\dot{A}&=&\tilde{U}\tilde{\Delta}|_{D^2\bar{D^2}}\nonumber\\
\frac{1}{8}\dot{B}^{\alpha}&=&\tilde{U}\tilde{\Delta}|_{D_{\alpha}\bar{D^2}}
\nonumber\\
\frac{1}{4}\dot{C}&=&\tilde{U}\tilde{\Delta}|_{D^2}
\eea
and analogous equations for
$\tilde{A},\tilde{B}_{\ad},\tilde{C}$. Here dot denotes
$\frac{\partial}{\partial is}$, and $|_{D^2}$ etc. denotes coefficient
at $D^2$ etc. in $\tilde{U}\tilde{\Delta}$.
As a result we havce system of first-order differential equations on
coefficients determining structure of operator $\tilde{U}$.
Since $\tilde{U}|_{s=0}=1$ we have natural initial conditions
\bea
\label{con}
A+\tilde{A}=B^{\a}=\tilde{B}_{\ad}=C=\tilde{C}|_{s=0}=0
\eea
The system (\ref{sys0}) with initial conditions (\ref{con}) can be
solved like common system of differential equations (note however,
that this solution is mostly found in special cases, such as
independence on spinor derivatives of background superfields, or
dependence on chiral background superfields only etc.)

Then, $\tilde{U}(s)$ (often called heat kernel) can be used for
calculation of Green function as
\bea
G(z_1,z_2)=i\int_0^{\infty}ds \tilde{U}U^{(0)}(x,x';s)\delta^4(\q-\q') 
\eea
(note that $\tilde{U}$ is a differential operator in superspace) and
for calculation of one-loop effective action as
\bea
\Gamma^{(1)}=\frac{i}{2}\int_0^{\infty}\frac{ds}{s}\int d^8 z d^8 z'
\delta^8(z-z')\tilde{U}U^{(0)}(x,x';s)\delta^4(\q-\q')
\eea
As usual, $\int d^8 z =\int d^4 x d^4\q$, we also use definition
(\ref{u0}).
Then, it is known that 
$\delta^4(\q-\bar{\q})D^2\bar{D}^2\delta^4(\q-\bar{\q})=
16\delta^4(\q-\bar{\q})$, and all products of less number of spinor
derivatives give zero trace. Hence only coefficients of 
(\ref{tu}) giving non-zero contribution to one-loop effective action
are
$A$ and $\tilde{A}$. And one-loop effective action looks
like
\bea
\Gamma^{(1)}=\frac{i}{2}\int_0^{\infty}\frac{ds}{s}\int d^4 x d^4\theta
(A(s)+\tilde{A}(s))U^{(0)}(x,x';s)|_{x=x'}
\eea
As a result we developed technique for calculating background
dependent propagators and one-loop effective action. Application of
this technique will be further considered on examples of several
theories. There is modification of this method for
supergauge theories \cite{McA}.

\section{Problem of superfield effective potential}
\setcounter{equation}{0}

Effective potential in standard quantum field theory is defined as
the effective Lagrangian considered at constant values of scalar fields,
and other fields are put to zero. The effective potential is used for
studying of spontaneous symmetry breaking and vacuum stability \cite{CW}. 

First, let us shortly describe effective potential in common quantum
field theory. The effective action has the form
\bea
\Gamma[\phi]=\int d^4 x
(-V_{eff}(\phi)-\frac{1}{2}Z(\phi)\pa_m\phi\pa^m\phi+
\ldots)
\eea
where $Z(\phi)$ is a some function of $\phi$, and $V_{eff}(\phi)$ is
effective potential. For slowly varying fields, therefore,
$$
\Gamma[\phi]=-\int d^4 x V_{eff}(\phi)
$$
therefore effective potential is a low-energy leading term. It can be
represented in the form of loop expansion
\bea
V_{eff}(\phi)=V(\phi)+\sum_{n=1}^{\infty}\hbar^n V^{(n)}(\phi)
\eea
For example, consider the theory with action
\bea
S=\int d^4x (\frac{1}{2}\phi\Box\phi+V(\phi))
\eea
After background-quantum splitting $\phi\to\Phi+\chi$ where $\Phi$ is
background superfield and $\chi$ is quantum one, we find the quadratic
action of quantum superfields
\bea
S_2=\int d^4 x [\frac{1}{2}\chi(\Box+V^{''}(\Phi))\chi]
\eea
which leads to one-loop effective action $\Gamma^{(1)}[\Phi]$ of the
form
\bea
\Gamma^{(1)}[\Phi]=\frac{i}{2}Tr\log(\Box+V^{''}(\Phi))
\eea
Following Section 5, we can express this trace of logarithm in the
form of diagrams:

\hspace{0.5cm}
\unitlength=.6mm
%\thicklines
%\GRAPH(hsize=3){%\ind(50,-30){.1.}
\begin{picture}(20,20)
\put(0,10){\circle{20}}\put(-10,10){\line(-1,0){5}}
%\put(-10,8.5){\line(-1,0){5}}
%\put(10,10){\line(1,0){5}}%\put(10,8.5){\line(1,0){5}}
%\ind(24,0){W''}\ind(-26,0){\bar{W''}}
%\put(4.8,0){\GRAPH(hsize=3){
\end{picture}
\hspace{2cm}
\begin{picture}(20,20)
\put(0,10){\circle{20}}%\put(-10,10){\line(-1,0){5}}
%\put(-10,8.5){\line(-1,0){5}}
\put(10,10){\line(1,0){5}}%\put(10,8.5){\line(1,0){5}}
\put(-10,10){\line(-1,0){5}}%\put(-1,20){\line(0,1){5}}
%\put(0,0.5){\line(0,-1){5}}
%\put(-1,0){\line(0,-1){5}}
%}}
%\ind(-10,20){W''}\ind(-10,-20){W''}
%\ind(-49,0){\bar{W}''}\ind(5,0){\bar{W''}}
%\put(30,0){\ldots}
\end{picture}
%}}}}
\hspace{2cm}
\begin{picture}(30,30)
\put(0,10){\circle{20}}
%\put(-10,10){\line(-1,0){5}}%\put(-10,8.5){\line(-1,0){5}}
%\put(10,10){\line(1,0){5}}%\put(10,8.5){\line(1,0){5}}
%\put(-9,5){\line(-1,-1){6}}%\put(-8,4){\line(-1,-1){6}}
\put(9,15){\line(1,1){6}}%\put(8,4){\line(1,-1){6}}
\put(-9,15){\line(-1,1){6}}%\put(-8,16){\line(-1,1){6}}
\put(0,5){\line(0,-1){5}}%\put(8,16){\line(1,1){6}}
\put(30,10){\ldots}
\put(0,-20){Fig. 5}
\end{picture}

\vspace*{1cm}

\noindent where external lines are $V^{''}[\Phi]$. Internal lines
correspond to $\frac{1}{\Box}\delta^4(\q_1-\q_2)$.

The sum of contributions from these supergraphs is
\bea
S=\sum_{n=1}^{\infty}\frac{1}{n}\int\frac{d^4k}{(2\pi)^4}
(-\frac{V^{''}(\Phi)}{k^2})^n=-\int \frac{d^4k}{(2\pi)^4}
\log(1+\frac{V^{''}(\Phi)}{k^2})
\eea
which after integration over $d^4k$ and extraction of divergences is
equal to \cite{BK0}
\bea
\frac{1}{64\pi^2}(V^{''}(\Phi))^2(\log\frac{V^{''}(\Phi)}{\mu^2}+C)
\eea
where $C$ is some constant. The same result can be obtained via
proper-time method (see calculation f.e. in \cite{BK0}).

Now we turn to superfield case. Let $\Gamma[\Phi,\bar{\Phi}]$ be the
renormalized effective action for a theory of chiral and antichiral
superfields. We can represent it as
\bea
%\begin{equation}
 \Gamma[\bar{\Phi},\Phi] = \int d^8z {\cal L}_{eff}
(\Phi,D_A\Phi,D_A D_B\Phi;\bar{\Phi},D_A\bar{\Phi},D_A D_B\bar{\Phi})
+ (\int d^6z {\cal L}^{(c)}_{eff}(\Phi) + h.c.) +\ldots
%\end{equation}
\eea 
Here $D_A\Phi,D_AD_B\Phi,\ldots$ are possible supercovariant
derivatives of superfields $\Phi,\bar{\Phi}$. Term ${\cal L}_{eff}$ is
called general effective Lagrangian, and ${\cal L}^{(c)}_{eff}$ is
called chiral effective Lagrangian. Both these effective Lagrangians
can be expanded into power series in supercovariant derivatives of
background superfields. Dots denote terms depending on
space-time derivatives of $\Phi$, $\bar{\Phi}$.
We note that since chiral effective Lagrangian by definition
depends only on $\Phi$ but not on $\bar{D}^2\bar{\Phi}$ all
terms of the form
$$
\int d^6 z \Phi^n (\bar{D}^2\bar{\Phi})^m
$$
using relation $\int d^6z (-\frac{\bar{D}^2}{4})=\int d^8 z$ can be
rewritten as
$$
\int d^8 z \Phi^n\bar{\Phi} (\bar{D}^2\bar{\Phi})^{m-1} 
$$
i.e. in the form corresponding to general effective
Lagrangian. Therefore here and further we consider all chiral
expressions including $(\bar{D}^2\bar{\Phi})^m$ as contributions to
general effective Lagrangian.

We note that all chiral contributions can be also represented as
integral over whole superspace:
\bea
\int d^6 z G(\Phi)=\int d^8 z (-\frac{D^2}{4\Box})G
\eea
Further, in component approach we must put scalar
component fields to constants, and spinor ones -- to zero, f.e. 
in Wess-Zumino model we write
$$
A=const,\ F=const, \ \psi_{\a}=0.
$$
However, this condition is not supersymmetric, therefore we use
condition of superfield {\bf constant in space-time}:
\bea
\pa_a\Phi=0
\eea
Since $\pa_a$ commutes with all generators of supersymmetry, this
condition is supersymmetric.

Effective potential is introduced as
\bea
\label{ep1}
V_{eff}=\Big\{-\int d^4\q {\cal L}_{eff}-(\int d^2\q {\cal
  L}^{(c)}_{eff}+h.c.)
\Big\}|_{\pa_a\Phi=\pa_a\bar{\Phi}=0}
\eea
Minus is put by convention. We can introduce general effective
potential
${\cal L}_{eff}|_{\pa_a\Phi=\pa_a\bar{\Phi}=0}$ and chiral effective
potential
${\cal L}^{(c)}_{eff}|_{\pa_a\Phi=0}$.
It is easy to see that the general effective potential can be expressed as
\bea
{\cal L}_{eff}={\bf K}(\Phi,\bar{\Phi})+
{\bf F}(D_{\a}\Phi,
\bar{D}_{\ad}\bar{\Phi},D^2\Phi,\bar{D}^2\bar{\Phi};\Phi,\bar{\Phi})
\eea
with 
$
{\bf F}|_{D_{\a}\Phi,
\bar{D}_{\ad}\bar{\Phi},D^2\Phi,\bar{D}^2\bar{\Phi}=0}=0
$.
The ${\bf K}$ is called k\"{a}hlerian effective potential, and ${\bf
  F}$ is called auxiliary fields' effective potential, it is at least
of third order in auxiliary fields of $\Phi$ and $\bar{\Phi}$.
These objects can be represented in the form of loop expansion:
\bea
{\bf
  K}(\Phi,\bar{\Phi})&=&K_0(\Phi,\bar{\Phi})+\sum_{L=1}^{\infty}\hbar^L
K_L(\Phi,\bar{\Phi})\\
{\bf F}&=&\sum_{L=1}^{\infty}\hbar^L F_L
\eea
(zeroth term in last expression is absent for theories which do not
include derivative depending terms in classical action), and
\bea
\label{ep5}
{\cal L}^{(c)}_{eff}(\Phi)={\cal L}^{(c)}(\Phi)+\sum_{L=1}^{\infty}
\hbar^L{\cal L}^{(c)}_L (\Phi)
\eea
Here $K_L,F_L,{\cal L}^{(c)}_L$ are quantum corrections.
For Wess-Zumino model ${\cal L}^{(c)}_1=0$, however, in some quantum
theories (f.e. in $N=1$ super-Yang-Mills theory with chiral matter
one-loop contribution to chiral effective potential exists
\cite{West4}).

The expansion of the effective potential by the rules
(\ref{ep1}--\ref{ep5})
can be applied for all superfield theories including noncommutative
ones. However, we note that effective potential in theories including
gauge superfields must depend on them in a special way. Really,
effective action in such theories should be expressed in terms of
some gauge convariant constructions, f.e. in background field method
gauge superfield is either incorporated to chiral superfields or
presents in supercovariant derivatives and gauge invariant superfield
strengths \cite{GRS,BK0}.

Let us give a few remarks about the method of calculating effective
potential. The best way for it is, of course, using of background
dependent propagators which are expressed in terms of common
propagators and background superfields. Background dependent
propagators can be in certain cases exactly found. To calculate
k\"{a}hlerian effective potential and auxiliary fields' effective
potential one can straightforwardly omit all space-time derivatives,
moreover, to study k\"{a}hlerian effective potential one can omit {\bf
  ALL} supercovariant derivatives and treate background superfields as
constants until final integration. The calculation of chiral effective
potential, however, is characterized by some difficulties. We will study
an approach to it on example of Wess-Zumino model.

\section{Wess-Zumino model and problem of chiral effective potential}
\setcounter{equation}{0}

Now we turn to consideration of superfield effective potential in
Wess-Zumino model. Here we follow the papers \cite{my3,my4,Buch1} and
book \cite{BK0}.

The superfield action of Wess-Zumino model is given by
(\ref{actwz}). Following loop expansion approach we carry out
background-quantum splitting by the rule
\bea
\Phi&\to&\Phi+\sqrt{\hbar}\phi;\nonumber\\
\bar{\Phi}&\to&\bar{\Phi}+\sqrt{\hbar}\bar{\phi}
\eea
The expression (\ref{exph}) defining effective action under such
changes takes the form (here
$\bar{\Gamma}=\sum_{L=1}^{\infty}\hbar^L\Gamma_L$)
\bea
e^{\frac{i}{\hbar}\bar{\Gamma}[\Phi,\bar{\Phi}]}&=&
\int D\phi
D\bar{\phi}
\exp\Big(\frac{i}{2}
\left(\begin{array}{cc}\phi\bar{\phi}\end{array}
\right)
\left(\begin{array}{cc}
\psi&-\frac{1}{4}\bar{D}^2\\
-\frac{1}{4}D^2 & \psi
\end{array}\right)
%\times\nonumber\\&\times&
\left(\begin{array}{c}\phi\\ \bar{\phi}
\end{array}\right)+\nonumber\\&+&
\frac{i\sqrt{\hbar}}{3!}(\frac{\l}{3!}\phi^3+h.c.)
\Big)
\eea
The quadratic action of quantum superfields looks like
\bea
S^{(2)}=\frac{1}{2}
\left(\begin{array}{cc}\phi\bar{\phi}\end{array}
\right)
\left(\begin{array}{cc}
\psi&-\frac{1}{4}\bar{D}^2\\
-\frac{1}{4}D^2 & \psi
\end{array}\right)
%\times\nonumber\\&\times&
\left(\begin{array}{c}\phi\\ \bar{\phi}
\end{array}\right)
\eea
And matrix superpropagator is an operator inverse to
\bea
\left(\begin{array}{cc}
\psi&-\frac{1}{4}\bar{D}^2\\
-\frac{1}{4}D^2 & \psi
\end{array}\right)
\left(\begin{array}{cc}
\delta_+&0\\
0&\delta_-
\end{array}
\right)
\eea
We can see that matrix superpropagator can be represented in the form
\bea
G(z_1,z_2)=\left(
\begin{array}{cc}
G_{++}(z_1,z_2)&G_{+-}(z_1,z_2)\\
G_{-+}(z_1,z_2)&G_{--}(z_1,z_2)
\end{array}
\right)
\eea
where $+$ denotes chirality with respect to corresponding argument,
and $-$ correspondingly -- antichirality.

It turns to be that in Wess-Zumino model this matrix looks like
\bea
\label{gremat}
G(z_1,z_2)=
\frac{1}{16}\left(\begin{array}{cc}
\bar{D}^2_1\bar{D}^2_2G^{\psi}_v(z_1,z_2)&\bar{D}^2_1 D^2_2G^{\psi}_v(z_1,z_2)
\\
D^2_1\bar{D}^2_2G^{\psi}_v(z_1,z_2)& D^2_1 D^2_2 G^{\psi}_v(z_1,z_2)
\end{array}
\right)
\eea
where $G^{\psi}_v(z_1,z_2)=(\Box+\frac{1}{4}\psi\bar{D}^2+
\frac{1}{4}\bar{\psi}D^2)^{-1}\delta^8(z_1-z_2)$.
Really, consider relation
\bea
\left(\begin{array}{cc}
\psi&-\frac{1}{4}\bar{D}^2\\
-\frac{1}{4}D^2 & \psi
\end{array}\right)
\frac{1}{16}\left(\begin{array}{cc}
\bar{D}^2_1\bar{D}^2_2G^{\psi}_v(z_1,z_2)&\bar{D}^2_1 D^2_2G^{\psi}_v(z_1,z_2)
\\
D^2_1\bar{D}^2_2G^{\psi}_v(z_1,z_2)& D^2_1 D^2_2 G^{\psi}_v(z_1,z_2)
\end{array}
\right)=-\left(\begin{array}{cc}
\delta_+&0\\
0&\delta_-
\end{array}
\right)
\eea
and act on both parts of this relation with the operator
$$
\left(
\begin{array}{cc}
0&-\frac{1}{4}\bar{D}^2\\
-\frac{1}{4}D^2& 0
\end{array}
\right).
$$
We get the following system of equations on components of matrix
superpropagator $G$:
\bea
\Box G_{++}&-&\frac{1}{4}\bar{D}^2_1(\bar{\psi}G_{-+})=0\nonumber\\
\Box G_{-+}&-&\frac{1}{4}D^2_1\psi
G_{++})=\frac{1}{16}D^2_1\bar{D}^2_2\delta^8(z_1-z_2)
\nonumber\\
\Box G_{--}&-&\frac{1}{4}D^2_1(\psi G_{+-})=0\nonumber\\
\Box G_{+-}&-&\frac{1}{4}\bar{D}^2_1(\bar{\psi}G_{--})=
\frac{1}{16}\bar{D}^2_1 D^2_2\delta^8(z_1-z_2)
\eea
Straightforward checking shows that components
$G_{++},G_{+-},G_{-+},G_{--}$
given by (\ref{gremat}) satisfy this equation. 
Thus, we found matrix superpropagator (\ref{gremat}) which
will be used for calculation of loop corrections. 

Consider one-loop effective action. Formally it has the form
$$
\Gamma^{(1)}=-\frac{i}{2}{\rm Tr}\log G
$$
where matrix superpropagator $G$ is given by (\ref{gremat}). However,
straightforward calculation of this trace is very complicated since the
elements of this matrix are defined in different subspaces.
The one-loop effective action $\Gamma^{(1)}$
can be obtained from relation
\bea
\label{eag}
e^{i\Gamma^{(1)}}=
\int D\phi
D\bar{\phi}
\exp\Big(\frac{i}{2}
\left(\begin{array}{cc}\phi\bar{\phi}\end{array}
\right)
\left(\begin{array}{cc}
\psi&-\frac{1}{4}\bar{D}^2\\
-\frac{1}{4}D^2 & \psi
\end{array}\right)
%\times\nonumber\\&\times&
\left(\begin{array}{c}\phi\\ \bar{\phi}
\end{array}\right)
\Big)
\eea
To take this integral we introduce a trick \cite{Buch1} which is used
in many theories describing dynamics of chiral superfields.

We consider theory of real scalar superfield with action
\bea
S=\frac{1}{16}\int d^8 z v D^{\a}\bar{D}^2 D_{\a}v
\eea
The action is invariant under gauge transformations $\delta v=\Lambda-
\bar{\Lambda}$ (here $\Lambda$ is chiral, and $\bar{\Lambda}$ is
antichiral).
According to Faddeev-Popov approach, the effective action $W$ for this
theory can be introduced as
\bea
\label{eaw}
e^{iW}=\int Dv e^{\frac{i}{16}\int d^8 z v D^{\a}\bar{D}^2 D_{\a}v}
\delta(\chi)
\eea
Here $\delta(\chi)$ is a functional delta function, and $\chi$ is a
gauge-fixing function. We choose $\chi$ in the form of column matrix
$$
\chi=\left(
\begin{array}{c}
\frac{1}{4}D^2v-\bar{\phi}\\
\frac{1}{4}\bar{D}^2 v-\phi
\end{array}
\right)
$$
Note that since supercovariant derivatives are not real we must impose
two conditions, and (\ref{eaw}) takes the form
\bea
\label{eaw1}
e^{iW}=\int Dv e^{\frac{i}{16}\int d^8 z v D^{\a}\bar{D}^2 D_{\a}v}
\delta(\frac{1}{4}D^2v-\bar{\phi})\delta(\frac{1}{4}\bar{D}^2v-\phi)
{\rm det}\Delta
\eea
where
$$
\Delta=\left(
\begin{array}{cc}
-\frac{1}{4}\bar{D}^2&0\\
0&-\frac{1}{4}D^2
\end{array}
\right)
$$
is a Faddeev-Popov matrix. We note that $W$ is constant by construction.
We multiply left-hand sides and right-hand sides of (\ref{eag}) and 
(\ref{eaw1}) respectively, as a result we arrrive at
\bea
e^{i\Gamma^{(1)}+W}&=&
\int D\phi
D\bar{\phi} Dv
\exp\Big(\frac{i}{2}
\left(\begin{array}{cc}\phi\bar{\phi}\end{array}
\right)
\left(\begin{array}{cc}
\psi&-\frac{1}{4}\bar{D}^2\\
-\frac{1}{4}D^2 & \psi
\end{array}\right)
%\times\nonumber\\&\times&
\left(\begin{array}{c}\phi\\ \bar{\phi}
\end{array}\right)
+\frac{i}{16}vD^{\a}\bar{D}^2D_{\a}v
\Big)\times\nonumber\\&\times&
\delta(\frac{1}{4}D^2v-\bar{\phi})\delta(\frac{1}{4}\bar{D}^2v-\phi)
{\rm det}\Delta
\eea
Integration over $\phi,\bar{\phi}$ with use of delta functions leads
to
\bea
e^{i\Gamma^{(1)}+W}&=&
\int D\phi
D\bar{\phi}
\exp\Big(\frac{i}{2}v(\Box-\frac{1}{4}\psi\bar{D}^2-\frac{1}{4}\bar{\psi}D^2)v
\Big)
{\rm det}\Delta
\eea
However, $W$ and ${\rm det}\Delta$ are constants which can be
omitted. We also took into account
that
$\frac{1}{16}\{D^2,\bar{D}^2\}+\frac{1}{8}D^{\a}\bar{D}^2D_{\a}=\Box$
\cite{West}, Hence the one-loop effective action is equal to
\bea
\Gamma^{(1)}=\frac{i}{2}{\rm Tr}\log(
\Box-\frac{1}{4}\psi\bar{D}^2-\frac{1}{4}\bar{\psi}D^2)
\eea
Here as usual $\psi=m+\lambda\Phi,\bar{\psi}=m+\lambda\bar{\Phi}$.
The one-loop effective action can be expressed in form of Schwinger
expansion:
\bea
\label{gl0}
\Gamma^{(1)}=\frac{i}{2}{\rm Tr}\int_0^{\infty}\frac{ds}{s}\exp(
\Box-\frac{1}{4}\psi\bar{D}^2-\frac{1}{4}\bar{\psi}D^2) 
\eea
or, after manifest writing the trace,
\bea
\label{g11}
\Gamma^{(1)}=\frac{i}{2}\int d^8 z_1 d^8 z_2\int_0^{\infty}\frac{ds}{s}
\delta^8(z_1-z_2)
\exp(is(-\frac{1}{4}\psi\bar{D}^2-\frac{1}{4}\bar{\psi}D^2))
e^{is\Box}
\delta^8(z_1-z_2)
\eea
We consider kernel 
$\tilde{U}(\psi|s)=\exp(is(-\frac{1}{4}\psi\bar{D}^2-\frac{1}{4}\bar{\psi}D^2))
\equiv e^{is\Delta}$.
It evidently satisfies the equation
$$
\frac{\partial \tilde{U}}{\partial s}=i\tilde{U}\Delta
$$
It turns to be that if we calculate k\"{a}hlerian effective potential
and all supercovariant derivatives from background superfields 
$\psi,\bar{\psi}$ are omitted this equation can be easily solved.
We express $\tilde{U}$ in the form (\ref{tu}).
Then $\tilde{U}\Delta$ is equal to
\bea
\tilde{U}\Delta&=&-\frac{1}{4}\psi\bar{D}^2-\frac{1}{4}\bar{\psi}D^2-
\nonumber\\
&-&\frac{1}{4}\psi\tilde{A}\Box\bar{D}^2-\frac{1}{4}\bar{\psi}A\Box D^2+
\nonumber\\
&+&\frac{1}{4}\tilde{B}_{\ad}\psi\pa^{\a\ad}D_{\a}\bar{D}^2
-\frac{1}{4}B_{\a}\bar{\psi}\pa^{\a\ad}\bar{D}_{\ad}D^2-\nonumber\\
&-&\frac{1}{16}\bar{\psi}\bar{C}\bar{D}^2 D^2-\frac{1}{16}\psi C D^2\bar{D}^2
\eea
Comparing coefficients at analogous derivatives in $\frac{1}{i}
\frac{\pa\tilde{U}}{\pa s}$ and $\tilde{U}\Delta$ we get the following
system of equations
\bea
\label{sys}
\dot{A}&=&-\psi C\nonumber\\
\dot{B}^{\a}&=&2i\tilde{B}_{\ad}\psi\pa^{\a\ad}\nonumber\\
\dot{C}&=&-\bar{\psi}-\bar{\psi}A\Box
\eea
System for $\tilde{A},\tilde{B},\tilde{C}$ has the analogous form with
changing $\psi\to\bar{\psi},A\to\tilde{A}$ etc.
Here dot denotes $\frac{1}{i}\frac{\pa}{\pa s}\equiv\frac{\pa}{\pa\tilde{s}}$.
Since $\tilde{U}|_{s=0}=1$, and all terms in expansion of $\tilde{U}$
(\ref{tu}) are evidently linearly independent, natural initial
conditions are
\bea
\label{init}
A=\tilde{A}=B^{\a}=\tilde{B}_{\ad}=C=\tilde{C}|_{s=0}=0.
\eea
We find that the system of equations for $B^{\a}$ and $\tilde{B}_{\ad}$ is
closed
(it is separated from whole system (\ref{sys})) and
homogeneous. Initial conditions above make its only solution to be
zero,
$B^{\a},\tilde{B}_{\ad}=0$.
The remaining from (\ref{sys}) system for $A$ and $C$ (and analogous one 
for $\tilde{A}$ and $\tilde{C}$) can be easily solved like standard 
system of common first-order differential equations.
Its solution looks like
\bea
C&=&-\sqrt{\frac{\bar{\psi}\Box}{\psi}}(A^1_0 \exp (i\omega s)-A^2_0
\exp (-i\omega s))\nonumber\\
A&=&A^1_0 \exp (i\omega s)+A^2_0 \exp (-i\omega s)-\frac{1}{\Box}
\eea
Here $\omega=\sqrt{\psi\bar{\psi}\Box}$.
Imposing initial conditions (\ref{init}) allows to fix coefficients
$A^1_0,A^2_0$. As a result we get
\bea
C&=&-\sqrt{\frac{\bar{\psi}}{\psi\Box}}\sinh(is\sqrt{\psi\bar{\psi}\Box})
\nonumber\\
A&=&\frac{1}{\Box}[\cosh(is\sqrt{\psi\bar{\psi}\Box})-1]
\eea
Since $A$ is symmetric with respect to change $\psi\to\bar{\psi}$ we
find that $A=\tilde{A}$. We note that only $A$ and $\tilde{A}$
contribute to trace in (\ref{g11}).
Therefore one-loop k\"{a}hlerian contribution to effective action is equal to
\bea
\label{ktrace}
K^{(1)}=\frac{i}{2}\int
d^4 x d^4\q\int_0^{\infty}\frac{d\tilde{s}}{\tilde{s}}
\frac{1}{\Box}[\cosh(\tilde{s}\sqrt{\psi\bar{\psi}\Box})-1]U_0(x,x';s)|_{x=x'}
\eea
Here $U_0(x,x';s)$ is given by (\ref{u0}). This function satisfies the
equation (see section 6):
$$
\Box^n U_0(x,x';s)|_{x=x'}=
(\frac{\pa}{\pa \tilde{s}})^n\frac{-i}{16\pi^2\tilde{s}^2}
$$
We expand (\ref{ktrace}) into power series:
$$
\frac{1}{\Box}[\cosh(\tilde{s}\sqrt{\psi\bar{\psi}\Box})-1]=
\sum_{n=0}^{\infty}\tilde{s}^{2n+2}\frac{(\psi\bar{\psi})^{n+1}}{(2n+2)!}\Box^n
$$
And 
\bea
K^{(1)}&=&
\frac{i}{2}\int
d^4 x d^4\q\int_0^{\infty}\frac{d\tilde{s}}{\tilde{s}}
\sum_{n=0}^{\infty}\tilde{s}^{2n+2}\frac{(\psi\bar{\psi})^{n+1}}{(2n+2)!}
\Box^nU_0(x,x';s)|_{x=x'}=\nonumber\\&=&
-\frac{i}{2}\int
d^4 x d^4\q\int_0^{\infty}\frac{d\tilde{s}}{\tilde{s}}
\sum_{n=0}^{\infty}\tilde{s}^{2n+2}\frac{(\psi\bar{\psi})^{n+1}}{(2n+2)!}
(\frac{\pa}{\pa \tilde{s}})^n\frac{-i}{16\pi^2\tilde{s}^2}=\nonumber\\
&=&
-\frac{1}{32\pi^2}\int d^8 z\int_{L^2}^{\infty}\frac{d\tilde{s}}{\tilde{s}^2}
\sum_{n=0}^{\infty}{(-1)}^n\frac{(\tilde{s}\psi\bar{\psi})^{n+1}(n+1)!}
{(2n+2)!}
\eea
Here we cut integral at lower limit by introducing $L^2$ for
regularization.
We make the change $\tilde{s}\psi\bar{\psi}=t$. As a result, one-loop
k\"{a}hlerian contribution to effective action takes the form
\bea
K^{(1)}=-\frac{1}{32\pi^2}\int d^8 z
\psi\bar{\psi}\int_{\psi\bar{\psi}L^2}^{\infty} dt
\sum_{n=0}^{\infty}\frac{(n+1)!t^{n+1}(-1)^n}{(2n+2)!}
\eea
Then, $\sum_{n=0}^{\infty}\frac{(n+1)!t^{n+1}(-1)^n}{(2n+2)!}=
t\int_0^1 due^{-\frac{t}{4}(1-u^2)}$.
Hence 
\bea
K^{(1)}=-\frac{1}{32\pi^2}\int d^8 z
\psi\bar{\psi}\int_{\psi\bar{\psi}L^2}^{\infty}\frac{dt}{t}
\int_0^1 due^{-\frac{t}{4}(1-u^2)}
\eea
At $L^2\to 0$ this integral tends to
\bea
K^{(1)}=-\frac{1}{32\pi^2}\psi\bar{\psi}\log(\mu^2
L^2)
-\frac{1}{32\pi^2}\psi\bar{\psi}(\log\frac{\psi\bar{\psi}}{\mu^2}-\xi)
\eea 
where $\xi$ is some constant which can be absorbed into redefinition
of $\mu$. We can add the counterterm
$\frac{1}{32\pi^2}\psi\bar{\psi}\log(\mu^2 L^2)$ to
cancel the divergence. Such 
counterterm corresponds to renormalization of kinetic
term by the rule
\bea
\Phi\to Z^{1/2}\Phi;\ Z=1+\frac{\lambda^2}{32\pi^2}\log(\mu^2 L^2)
\eea
And the renormalized k\"{a}hlerian effective potential is
\bea
\label{kren}
K^{(1)}=-\frac{1}{32\pi^2}\psi\bar{\psi}(\log\frac{\psi\bar{\psi}}{\mu^2}-\xi)
\eea
Another way for calculating of k\"{a}hlerian effective potential is  
summarizing of contributions from supergraphs given by Fig. 4.
Sum of these contributions looks like \cite{PW}
\bea
K^{(1)}=\int \frac{d^4k}{(2\pi)^4}
\int d^4\q_1\ldots d^4\q_{2n}
\sum_{n=1}^{\infty}\frac{1}{2n}
(\frac{\psi\bar{\psi}}{k^4})\frac{D^2}{4}\delta_{12}\frac{\bar{D}^2}{4}
\delta_{23}\ldots\frac{D^2}{4}\delta_{n-1,n}\frac{\bar{D}^2}{4}\delta_{n1}
\eea
which after $D$-algebra transformations and summation looks like
\bea
K^{(1)}=\int\frac{d^{4-\epsilon} k}{(2\pi)^{4-\epsilon}}\frac{1}{2k^2}
\log(1+\frac{\psi\bar{\psi}}{k^2})
\eea
(here we carried out dimensional regularization by introducing
parameter $\epsilon$). Integration leads to
\bea
K^{(1)}=\frac{1}{32\pi^2}[\frac{\psi\bar{\psi}}{\epsilon}-\psi\bar{\psi}\log
\frac{\psi\bar{\psi}}{e\mu^2}]
\eea
where $e=\exp(1)$. Subtraction of divergence and redefinition of $\mu$
leads to result (\ref{kren}).

Now we turn to calculation of chiral effective potential.
It is not equal to zero for massless theories. Really, as it was noted
by West \cite{West2} the mechanism of arising chiral corrections is
the following one. If the theory describes dynamics of chiral and
antichiral superfields, then quantum correction of the form
\bea
\label{n1}
\int d^8 z f(\Phi)(-\frac{D^2}{4\Box})g(\Phi)
\eea
can be rewritten as
\bea
\int d^6 z f(\Phi)g(\Phi).
\eea
Here we used properties $\int d^8 z =\int d^6 z (-\frac{D^2}{4})$ and
$\bar{D}^2D^2\Phi=16\Box\Phi$ (last identity is true for any chiral
superfield $\Phi$), and $f(\Phi),\, g(\Phi)$
are arbitrary functions of chiral superfield $\Phi$. However, presence
of factor $\Box^{-1}$ is
characteristic for massless theories, in massive theories where we
have $(\Box-m^2)^{-1}$ instead of $\Box^{-1}$, and this mechanism of
arising contributions to chiral effective potential is not valid.
In the case of massless theory we can find matrix superpropagator
exactly: first,
\bea
G^{\psi}_v(z_1,z_2)&=&(\Box+\frac{1}{4}\psi\bar{D}^2)^{-1}\delta^8(z_1-z_2)=
\frac{1}{\Box_1}\delta^8(z_1-z_2)-\nonumber\\&-&
\frac{1}{4\Box_1}\psi(z_1)
\frac{\bar{D}^2_1}{4\Box_1}\delta^8(z_1-z_2)
\eea
(further terms in this expansion are equal to zero because they are
proportional to $\bar{D}^2\psi=0$ or $\bar{D}^4=0$).
Therefore components of matrix superpropagator look like
\bea
G_{++}&=&0;
G_{+-}=G^*_{-+}=\frac{\bar{D}^2_1D^2_2}{16\Box}\delta^8(z_1-z_2)
\nonumber\\
G_{--}&=&-\frac{D^2_1}{4\Box_1}[\psi(z_1)\frac{\bar{D}^2_1
  D^2_2}{16\Box}\delta^8(z_1-z_2)]
\eea
Here $*$ denotes complex conjugation. We note that background chiral
superfield $\psi$ is not constant, otherwise when we arrive at
expression proportional to $D^2\psi$ we get singularity $\frac{0}{0}$
\cite{my3}.
The only two-loop contribution to chiral effective potential is given
by the following supergraph

\hspace{4.5cm}
\Lengthunit=1.5cm
%\Linewidth{1.2pt}
\GRAPH(hsize=3){\ind(0,-16){Fig.6}%\thicklines
\mov(.5,0){\Circle(2)\mov(-1,0){\lin(2,0)}
\ind(-2,10){|}
\ind(-2,-3){\bar{D}^2}\ind(-2,0){|}\ind(-2,-10){|}\ind(-2,-13){\bar{D}^2}
\ind(-2,7){\bar{D}^2}
\ind(-9,2){D^2} \ind(-10,-2){D^2} \ind(8,2){D^2} \ind(8,-2){D^2}
\ind(-18,2){-} \ind(-18,-2){-} \ind(0,2){-} \ind(0,-2){-}
%\Linewidth{0.3pt}
\mov(-1,1){\lin(-.7,.7)}%\mov(-1.1,1){\lin(-.7,.7)}
\mov(-1,-1){\lin(.7,-.7)}%\mov(-1.1,-1){\lin(.7,-.7)}
\mov(-1,0){\lin(-.7,.7)}%\mov(-1.1,0){\lin(-.7,.7)}}
}}

\vspace*{2mm}

\noindent External lines are chiral. We use representation in which
$\Phi(z)=\Phi(x,\theta)$, and
$\bar{D}_{\ad}=-\frac{\pa}{\pa\bar{\q}^{\ad}}$.
Remind that in this case $\psi=\lambda\Phi$.

Contribution of the supergraph given in Fig.6 looks like
\begin{eqnarray}
\label{I1}
I&=&\frac{\lambda^5}{12}
\int \frac{d^4p_1 d^4p_2}{{(2\pi)}^8}\frac{d^4k d^4l}{{(2\pi)}^8}
\int d^4\theta_1 d^4\theta_2 d^4\theta_3 d^4\theta_4 d^4\theta_5
\Phi(-p_1,\theta_3)
\Phi(-p_2,\theta_4)\times\nonumber\\&\times&
\Phi(p_1+p_2,\theta_5)
\frac{1}{k^2 l^2 {(k+p_1)}^2{(l+p_2)}^2{(l+k)}^2{(l+k+p_1+p_2)}^2}
\times\nonumber\\&\times&
\delta_{13}\frac{\bar{D}^2_3}{4}\delta_{32}
\frac{D^2_1 \bar{D}^2_4}{16}\delta_{14}\delta_{42}
\frac{D^2_1 \bar{D}^2_5}{16}\delta_{15}\delta_{52}
\end{eqnarray}
After $D$-algebra transformation this expression can be written as
\begin{eqnarray}
\label{app}
I&=&\frac{\lambda^5}{12}
\int \frac{d^4p_1 d^4p_2}{{(2\pi)}^8}\frac{d^4k d^4l}{{(2\pi)}^8}
\int d^2\theta
\Phi(-p_1,\theta)
\Phi(-p_2,\theta)%\times\nonumber\\&\times&
\Phi(p_1+p_2,\theta)
\times\nonumber\\&\times&
\frac{k^2 p_1^2+ l^2 p_2^2 +2 (k l)(p_1 p_2)}
{k^2 l^2 {(k+p_1)}^2{(l+p_2)}^2{(l+k)}^2{(l+k+p_1+p_2)}^2}
\end{eqnarray}

Here we made transformation $\int d^4\q=\int
d^2\q(-\frac{1}{4}\bar{D}^2)$
and took into account that\\ $\bar{D}^2D^2\Phi(p,\q)=-16p^2\Phi(p,\q)$.
Note that if $\Phi=const$ we get $\bar{D}^2D^2\Phi=0$, hence we cannot
consider $\Phi$ as constant.

As we know the effective potential is the
effective lagrangian for superfields slowly varying in space-time.
Let us study behaviour of the expression (\ref{app}) in this case.
The contribution (\ref{app}) can be expressed as
\begin{eqnarray}
\label{cont}
I&=&\frac{\lambda^5}{12}\int d^2\theta \int \frac{d^4p_1 d^4p_2}{{(2\pi)}^8}
\Phi(-p_1,\theta)
\Phi(-p_2,\theta)%\times\nonumber\\&\times&
\Phi(p_1+p_2,\theta)
S(p_1, p_2)
\end{eqnarray}
Here $p_1, p_2$ are external momenta. The expression $S(p_1, p_2)$ here
is equal to
$$ \int\frac{d^4 k d^4 l}{{(2\pi)}^8}\frac{k^2 p_1^2+ l^2
p_2^2 +2 (kl)(p_1 p_2)} {k^2 l^2
{(k+p_1)}^2{(l+p_2)}^2{(l+k)}^2{(l+k+p_1+p_2)}^2} $$
After Fourier transform eq. (\ref{cont}) has the form
\begin{eqnarray}
\label{cont1}
I&=&\frac{\lambda^5}{12}\int d^2\theta \int d^4 x_1 d^4x_2 d^4 x_3
\int\frac{d^4p_1 d^4p_2}{{(2\pi)}^8}
\Phi(x_1,\theta)
\Phi(x_2,\theta)\times\nonumber\\&\times&
\Phi(x_3,\theta)
%\times\nonumber\\&\times&
\exp[i(-p_1 x_1- p_2 x_2+(p_1+p_2)x_3)]
S(p_1, p_2)
\end{eqnarray}
Since superfields in the case under consideration are slowly varying in
space-time we can put
$
\Phi(x_1,\theta)
\Phi(x_2,\theta)
\Phi(x_3,\theta)
\simeq \Phi^3(x_1,\theta)
$.
As a result one gets
\begin{eqnarray}
I&=&\frac{\lambda^5}{12}\int d^2\theta \int d^4 x_1 d^4 x_2 d^4 x_3
\int\frac{d^4p_1 d^4p_2}{{(2\pi)}^8}
\Phi^3(x_1,\q)
\times\nonumber\\&\times&
\exp[i(-p_1 x_1- p_2 x_2+(p_1+p_2)x_3)]
S(p_1, p_2)
\end{eqnarray}
Integration over $d^4 x_2 d^4 x_3$ leads to delta-functions
$\delta(p_2)\delta(p_1+p_2)$. Hence the eq. (\ref{cont1}) takes the form
\begin{equation}
I=\frac{\lambda^5}{12}\int d^2\theta \int d^4 x_1 
\Phi^3(x_1,\q)
S(p_1,p_2)|_{p_1,p_2=0}
\end{equation}
Therefore final result for two-loop correction to chiral 
(often called holomorphic) effective
potential looks like
\begin{equation}
\label{l2c}
W^{(2)}=
\frac{6}{{(16\pi^2)}^2}\zeta(3)\Phi^3(z)
\end{equation}
We took into account that
$$
\int \frac{d^4k d^4l}{{(2\pi)}^8}
%\times\nonumber\\&\times&
\frac{k^2 p_1^2+ l^2 p_2^2 +2 (k_1 k_2)(p_1 p_2)}
{k^2 l^2 {(k+p_1)}^2{(l+p_2)}^2{(l+k)}^2{(l+k+p_1+p_2)}^2}|_{p_1=p_2=0}
=\frac{6}{{(4\pi)}^4}\zeta(3)
$$
We see that the correction (\ref{l2c}) is finite and does not require
renormalization. 

Chiral contributions to effective action arise also in other theories
describing dynamics of chiral superfields. F.e. in general chiral
superfield theory the leading chiral contribution is also chiral effective
potential (see next section), in dilaton supergravity leading chiral
contribution is of second order in space-time derivatives of chiral
superfield (see section 10), and these corrections are
finite. The Situation in $N=1$ super-Yang-Mills theory, however, possesses
some peculiarities. Really, in this model both finite (Fig. 7a) and divergent
(Fig.7b) two-loop chiral contributions are possible.

\hspace*{2cm}
\GRAPH(hsize=3){\ind(0,-15){Fig.7a}
\mov(.5,0){\halfcirc(2)[u]\lin(-1,0)\wavelin(1,0)
\mov(-.1,-1){\arcto(1,1)[-0.7]\wavearcto(-1,1)[0.7]}
\ind(-2,10){|}
\ind(-2,-3){\bar{D}^2}\ind(-2,0){|}\ind(-2,-10){|}\ind(-2,-13){\bar{D}^2}
\ind(-2,7){\bar{D}^2}
\ind(-9,2){D^2} \ind(-10,-2){D^2} \ind(8,2){D^2} \ind(8,-2){D^2}
\ind(-18,2){-} \ind(-18,-2){-} \ind(0,2){-} \ind(0,-2){-}
%\Linewidth{0.3pt}
\mov(-1,1){\lin(-.7,.7)}
\mov(-1,-.9){\lin(.7,-.7)}
\mov(-1,0){\lin(-.7,.7)}
}}
\hspace*{4cm}
\Lengthunit=1cm
\GRAPH(hsize=3){\ind(0,-20){Fig.7b}
\mov(.5,0){\halfcirc(2)[u]\mov(-1,0){\lin(-1,0)}
\mov(-.1,-1){\wavearcto(1,1)[-0.7]\arcto(-1,1)[0.7]\lin(0,-.7)}
\mov(1.8,0){\halfcirc(2)[u]\halfwavecirc(2)[d]}\mov(2.8,0){\lin(1,0)}
\ind(-15,2){-}\ind(-17,2){\bar{D}^2}\ind(4,2){-}\ind(2,2){D^2}
\ind(8,2){-}\ind(12,2){\bar{D}^2}\ind(26,2){-}\ind(29,2){D^2}
\ind(-8,-9){|}\ind(-7,-13){D^2}
}}

Finite contributions (there are more than 10 supergraphs of the form
similar to Fig. 7a \cite{dis}) all give contributions to effective potential
proportional to $\frac{\zeta(3)}{(4\pi)^4}\Phi^3$ (cf. (\ref{l2c})),
i.e. finite chiral contributions are analogous to the case of
Wess-Zumino model. 
As for divergent contribution given by Fig. 4b it is, after D-algebra
transformation, equal to
\bea
\label{kdiv}
& &\int d^2\q\int\frac{d^4p_1 d^4p_2}{(2\pi)^8}
(p_1+p_2)^2\int\frac{d^4k  d^4l}{(2\pi)^8}\frac{1}{k^2(k+p_1)^2(k+p_2)^2}
\times\nonumber\\&\times&
\frac{1}{l^2(l+p_1+p_2)^2}\Phi(-p_1,\q)
\Phi(-p_2,\q)\Phi(p_1+p_2,\q)
\eea  
To obtain the low-energy leading contribution we must consider the limit at 
$p_1,p_2\to 0$. It is known \cite{West4} that
$$
\lim_{p_1,p_2\to 0}
(p_1+p_2)^2\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2(k+p_1)^2(k+p_2)^2}=
\frac{1}{16\pi^2}\int_0^1 d\a \frac{\log [\a(1-\a)]}{1-\a(1-\a)}=
\frac{C_0}{16\pi^2}
$$
where $C_0$ is a some constant. The integral over $l$ is divergent, and
after dimensional regularization it is equal to
$$
\int\frac{d^{4-\epsilon}l}{(2\pi)^{4-\epsilon}}\frac{1}{l^2(l+p_1+p_2)^2}=
\frac{1}{16\pi^2}(\frac{2}{\epsilon}+\log\frac{(p_1+p_2)^2}{\mu^2})
$$
After cancellation of divergence via one-loop counterterm and
transforming
to coordinate representation we see that expression (\ref{kdiv}) for
slowly varying in space-time superfields takes the form
\bea
\int d^6 z \frac{1}{(16\pi^2)^2}\Phi^2(z)\log(-\frac{\Box}{\mu^2})\Phi(z)
\eea
Thus, the leading chiral correction in $N=1$ super-Yang-Mills theory
with chiral matter is nonlocal one (detailed discussion is given in 
\cite{dis}).
We note that nonlocal chiral corrections in this theory arise also in
the pure gauge sector \cite{KuzYar}.
Hence presence of quantum contributions to chiral effective Lagrangian is
quite characteristic for theories including chiral superfields. 

\section{General chiral superfield model}
From viewpoint of superstring theory low-energy models of elementary
particles are effective theories in which integration over massive
string modes is carried out, 10-dimensional background manifold has the
form $M^4\times K$ where $M^4$ is four-dimensional Minkovski space,
and $K$ is some six-dimensional compact manifold. Then reduction to $M^4$
is carried out. As a result we arrive at theory in $M^4$ with action 
\cite{GSH}
\bea
\label{genact}
S[\Phi,\bar{\Phi}]=\int d^8 z K(\Phi^i,\bar{\Phi}^i)+
(\int d^6 z W(\Phi^i)+h.c.)
\eea  
We can use matrix denotions via introduction of column vector
$\vec{\Phi}=\{\Phi\}$, after which the consideration in the 
case of several chiral superfield is analogous to the case of one
chiral superfield. We can consider this theory for arbitrary functions
$K$ and $W$. Note that there is no higher derivatives in the classical
action. Therefore the theory with the action (\ref{genact}) is the
most general theory without higher derivatives describing dynamics of
chiral superfield. There are a lot of phenomenological applications of
this model in string theory (see \cite{Cvet} and references therein).
In general case this theory is nonrenormalizable; however, it is
an effective theory aimed for studying of low-energy domain. Therefore
all integrals over momenta are effectively cut by condition $p\ll
M_{String}$ where $p$ is momentum, and $M_{String}=10^{17} GeV
\sim 10^{-2} M_{Pl}$ is a characteristic string mass.

Let $\Gamma [\bar{\Phi},\Phi]$ be effective action in the model (\ref{genact}).
It can be expressed with use of path integral representation \cite{BK0,BO}
\begin{eqnarray}
\label{Green1g}
 \exp(\frac{i}{\hbar}\Gamma[\bar{\Phi},\Phi]) &=&
 \int {\cal D} \phi {\cal D} \bar{\phi}
 \exp\big(
\frac{i}{\hbar}
S[\bar{\Phi}+\sqrt{\hbar}\bar{\phi},\Phi+\sqrt{\hbar}\phi]
-\nonumber\\&-&
(\int d^6 z
\frac{\delta\Gamma[\bar{\Phi},\Phi]}{\delta\Phi(z)}\phi(z)+h.c. )
\big)
\end{eqnarray}
Here $\Phi,\bar{\Phi}$ are the background superfields and $\phi,\bar{\phi}$
are the quantum ones.
The effective action can be written as
$\Gamma[\bar{\Phi},\Phi]=S[\bar{\Phi},\Phi]+\tilde{\Gamma}[\bar{\Phi},\Phi]$,
where $\tilde{\Gamma}[\bar{\Phi},\Phi]$ is a quantum correction.
Eq. (\ref{Green1g}) allows to obtain $\tilde{\Gamma}[\bar{\Phi},\Phi]$
in 
the form of
loop expansion
%\begin{equation}
$\label{Gamma}
 \tilde{\Gamma}[\bar{\Phi},\Phi] = \sum_{n=1}^{\infty}\hbar^n
\Gamma^{(n)} [\bar{\Phi},\Phi]$
%\end{equation}
and hence, to get loop expansion for the effective lagrangians
${\cal L}_{eff}$ and ${\cal L}^{(c)}_{eff}$.

The effective action can be presented as a series
in supercovariant derivatives
$D_A=(\partial_a,D_{\alpha},\bar{D}_{\dot{\alpha}})$ in the form
\begin{eqnarray}
\label{efexpa}
 \Gamma[\bar{\Phi},\Phi] &=& \int d^8z {\cal L}_{eff}
(\Phi,D_A\Phi,D_A D_B\Phi;\bar{\Phi},D_A\bar{\Phi},D_A D_B\bar{\Phi})
+\nonumber\\
&+&(\int d^6z {\cal L}^{(c)}_{eff}(\Phi) + h.c.) +\ldots
\end{eqnarray}
Here
${\cal L}_{eff}$ is called general effective lagrangian,
${\cal L}^{(c)}_{eff}$ is called chiral effective lagrangian. Both these
lagrangians are the series in supercovariant derivatives
of superfields and can be written in the form of loop expansion
\begin{eqnarray}
\label{ep}
{\cal L}_{eff}&=&K_{eff}(\bar{\Phi},\Phi)+\ldots
=K(\bar{\Phi},\Phi)+\sum_{n=1}^{\infty}K^{(n)}_{eff}(\bar{\Phi},\Phi)\nonumber\\
{\cal L}^{(c)}_{eff}&=&W_{eff}(\Phi)+\ldots=
W(\Phi)+\sum_{n=1}^{\infty}W^{(n)}_{eff}(\Phi)+\ldots
\end{eqnarray}
Here dots mean terms depending on covariant derivatives
of superfields
$\Phi,\bar{\Phi}$. Here $K_{eff}(\bar{\Phi},\Phi)$ is called
kahlerian effective potential, $W_{eff}(\Phi)$ is called chiral (or holomorphic)
effective potential,
$K^{(n)}_{eff}$ is a $n$-th correction to kahlerian potential
and $W^{(n)}_{eff}$ is a $n$-th correcton to chiral (holomorphic)
potential $W$.

To find loop corrections $\Gamma^{(n)} [\bar{\Phi},\Phi]$ in explicit
form we expand the right-hand side of eq. (\ref{Green1g}) 
in power series in quantum
superfields $\phi$, $\bar{\phi}$. As usual, the quadratic part of
expansion of
$\frac{1}{\hbar}S[\bar{\Phi}+\sqrt{\hbar}\bar{\phi},\Phi+\sqrt{\hbar}\phi] $
\begin{eqnarray}
\label{qua}
S_2=\frac{1}{2}\int d^8 z \left(\begin{array}{cc}\phi&\bar{\phi}
\end{array}\right)
\left(\begin{array}{cc}
K_{\Phi\Phi}&K_{\Phi\bar{\Phi}}\\
K_{\Phi\bar{\Phi}}&K_{\bar{\Phi}\bar{\Phi}}
\end{array}\right)
\left(\begin{array}{c}\phi\\
\bar{\phi}
\end{array}
\right)+[\int d^6 z \frac{1}{2}W^{''}\phi^2+h.c.]
\end{eqnarray}
defines the propagators and the the higher terms of 
expansion define the vertices.
Here
$K_{\Phi\bar{\Phi}}=\frac{\partial^2 K(\bar{\Phi},\Phi)}
{\partial\Phi\partial\bar{\Phi}}$,
$K_{\Phi\Phi}=\frac{\partial^2 K(\bar{\Phi},\Phi)}
{\partial\Phi^2}$ etc,  $W^{''}=\frac{d^2 W}{d\Phi^2}$.

The matrix superpropagator has the form
\bea
G(z_1,z_2)=\left(
\begin{array}{cc}
G_{++}(z_1,z_2)&G_{+-}(z_1,z_2)\\
G_{-+}(z_1,z_2)&G_{--}(z_1,z_2)
\end{array}
\right)
\eea
where $+$ denotes chirality with respect to corresponding argument,
and $-$ correspondingly -- antichirality.
This propagator satisfies the equation
\bea
& &\left(\begin{array}{cc}
W^{''}-\frac{1}{4}(\bar{D}^2K_{\Phi\Phi})&-\frac{1}{4}\vec{\bar{D}^2}
K_{\Phi\bar{\Phi}}\\
-\frac{1}{4}\vec{D^2}K_{\Phi\bar{\Phi}} & \bar{W}^{''}-\frac{1}{4}
(D^2 K_{\bar{\Phi}\bar{\Phi}})
\end{array}\right)
\left(\begin{array}{cc}
G_{++}(z_1,z_2)&G_{+-}(z_1,z_2)\\
G_{-+}(z_1,z_2)& G_{--}(z_1,z_2)
\end{array}
\right)=\nonumber\\&-&\left(\begin{array}{cc}
\delta_+&0\\
0&\delta_-
\end{array}
\right)
\eea
To consider k\"{a}hlerian effective potential we must omit all
derivatives of superfields $\Phi,\bar{\Phi}$, and the equation takes
the form
\bea
\left(\begin{array}{cc}
W^{''}&-\frac{1}{4}K_{\Phi\bar{\Phi}}\bar{D}^2\\
-\frac{1}{4}K_{\Phi\bar{\Phi}} D^2& \bar{W}^{''}
\end{array}\right)
\left(\begin{array}{cc}
G_{++}(z_1,z_2)&G_{+-}(z_1,z_2)\\
G_{-+}(z_1,z_2)& G_{--}(z_1,z_2)
\end{array}
\right)=-
\left(\begin{array}{cc}
\delta_+&0\\
0&\delta_-
\end{array}
\right)
\eea
The solution of this equation looks like
\bea
\label{grematg}
G=
\frac{1}{K^2_{\Phi\bar{\Phi}}\Box-W^{''}\bar{W}^{''}}
\left(\begin{array}{cc}
W^{''}&\frac{1}{4}K_{\Phi\bar{\Phi}}\bar{D}^2\\
\frac{1}{4}K_{\Phi\bar{\Phi}} D^2& \bar{W}^{''}
\end{array}\right)
\left(\begin{array}{cc}
\delta_+&0\\
0&\delta_-
\end{array}
\right)
\eea
Now we turn to studying of quantum contributions to k\"{a}hlerian effective
potential
%being the superfield analog of standard Coleman - Weinberg potential and
depending only on
superfields $\Phi$, $\bar{\Phi}$ but not on their derivatives.

The one-loop diagrams contributing to kahlerian effective potential are

\hspace{0.5cm}
\unitlength=.6mm
%\thicklines
%\GRAPH(hsize=3){%\ind(50,-30){.1.}
\begin{picture}(20,20)
\put(0,10){\circle{20}}\put(-10,10){\line(-1,0){5}}\put(-10,8.5){\line(-1,0){5}}
\put(10,10){\line(1,0){5}}\put(10,8.5){\line(1,0){5}}
%\ind(24,0){W''}\ind(-26,0){\bar{W''}}
%\put(4.8,0){\GRAPH(hsize=3){
\end{picture}
\hspace{2cm}
\begin{picture}(20,20)
\put(0,10){\circle{20}}\put(-10,10){\line(-1,0){5}}
\put(-10,8.5){\line(-1,0){5}}
\put(10,10){\line(1,0){5}}\put(10,8.5){\line(1,0){5}}
\put(0,20){\line(0,1){5}}\put(-1,20){\line(0,1){5}}
\put(0,0.5){\line(0,-1){5}}
\put(-1,0){\line(0,-1){5}}
%}}
%\ind(-10,20){W''}\ind(-10,-20){W''}
%\ind(-49,0){\bar{W}''}\ind(5,0){\bar{W''}}
%\put(30,0){\ldots}
\end{picture}
%}}}}
\hspace{2cm}
\begin{picture}(30,30)
\put(0,10){\circle{20}}
\put(-10,10){\line(-1,0){5}}\put(-10,8.5){\line(-1,0){5}}
\put(10,10){\line(1,0){5}}\put(10,8.5){\line(1,0){5}}
\put(-9,5){\line(-1,-1){6}}\put(-8,4){\line(-1,-1){6}}
\put(9,5){\line(1,-1){6}}\put(8,4){\line(1,-1){6}}
\put(-9,15){\line(-1,1){6}}\put(-8,16){\line(-1,1){6}}
\put(9,15){\line(1,1){6}}\put(8,16){\line(1,1){6}}
\put(30,10){\ldots}
\end{picture}

\vspace{2mm}

Double external lines correspond to alternating $W''$ and $\bar{W}''$.
Internal lines are $<\phi\bar{\phi}>$-propagators of
the form
$$
G_0\equiv<\phi\bar{\phi}>=-\frac{\bar{D}^2 D^2}{16 K_{\Phi\bar{\Phi}}\Box}
$$
Supergraph of such structure with $2n$
legs represents itself as a ring containing $n$ links of the following form

\hspace{2cm}
\unitlength=2mm
%\GRAPH(hsize=3){
\begin{picture}(40,15)
\put(10,4){\line(1,0){20}}\put(10,4){\line(0,1){10}}\put(10.5,4){\line(0,1){10}}
\put(20,4){\line(0,1){10}}\put(20.5,4){\line(0,1){10}}
%\ind(20,-10){.2}
\put(12,1){$\bar{D}^2$}\put(12,4){$|$}\put(22,1){$D^2$}\put(22,4){$|$}
\put(11,14){$W''$}\put(21,14){$\bar{W}''$}
\end{picture}
%\vspace*{-2mm}

The total contribution of all these diagrams
after $D$-algebra transformations, summation, integration over momenta
and subtraction of divergences is equal to
\begin{equation}
\label{k1}
K^{(1)}=-\int d^4 \theta {\rm tr}
\frac{1}{32\pi^2}W''\frac{1}{K^2_{\Phi\bar{\Phi}}}\bar{W}''%\Big\{
\ln\Big(W''\frac{1}{\mu^2 K^2_{\Phi\bar{\Phi}}}\bar{W}''\Big)
%\Big\}
\end{equation}
This form is more convenient for analysis of many-field model than
that one given in \cite{my2,my5}, and ${\rm tr}$ denotes trace of
product of the given matrices.
It is easy to show that the present result corresponds to the known
expression for the Wess-Zumino model where $W''=\frac{1}{2}\lambda\Phi$.

%\section{General properties of chiral effective potential}

Let us consider the holomorphic effective potential
$W_{eff}(\Phi)$. The mechanism of its arising is just the same than in
Wess-Zumino model.
We note again that the chiral contributions to effective action can be
generated by supergraphs containing massless propagators only.
To find chiral corrections to effective action we put
$\bar{\Phi}=0$ in eqs. (\ref{Green1g},\ref{qua}).
{\bf Therefore here and further all derivatives of $K$, $W$ and $\bar{W}$ will
be taken at $\bar{\Phi}=0$.} Under this condition
the action of quantum superfields $\phi,\bar{\phi}$ in external superfield
$\Phi$ looks like
\begin{eqnarray}
\label{qch}
S[\bar{\phi},\phi,\Phi]=\frac{1}{2}\int d^8 z
\left(\begin{array}{cc}\phi&\bar{\phi} \end{array}\right)
\left(\begin{array}{cc}
K_{\Phi\Phi}&K_{\Phi\bar{\Phi}}\\
K_{\Phi\bar{\Phi}}&K_{\bar{\Phi}\bar{\Phi}}
\end{array}\right)
\left(\begin{array}{c}\phi\\
\bar{\phi}
\end{array}
\right)+\int d^6 z \frac{1}{2}W^{''}\phi^2+\ldots
\end{eqnarray}
The dots here denote the terms of third, fourth and higher orders in quantum
superfields.
We  call the theory massless if $W^{''}|_{\Phi=0}=0$.
Further we consider only massless theory.

To calculate the corrections to $W(\Phi)$ we use supergraph technique
(see f.e. \cite{BK0}).
For this purpose one splits the action (\ref{qch}) into sum of free
part and vertices of interaction.
As a free part we take the action $S_0=\int d^8 z \phi\bar{\phi}$.
The corresponding superpropagator is
%\begin{equation}
%\label{pr1}
$G(z_1,z_2) = -\frac{D^2_1\bar{D}^2_2}{16\Box}\delta^8(z_1-z_2)$.
%\end{equation}
And the term $S[\bar{\phi},\phi,\Phi]-S_0$
will be treated as vertices where $S[\bar{\phi},\phi,\Phi]$
is given by eq. (\ref{qch}). Our purpose is to find the 
first leading contribution to
$W_{eff}(\Phi)$. As we will show, chiral loop contributions begin with two
loops. Therefore we keep in eq. (\ref{qch})  only the terms of second, 
third and fourth
orders in quantum fields.

Non-trivial corrections to chiral potential can arise only
if $2L+1-n_{W^{''}}-n_{V_c}=0$
where $L$ is a number of loops, $n_{W^{''}}$ is a number of vertices
proportional to $W^{''}$, $n_{V_c}$ is that one of vertices of third and
higher orders in quantum superfields,
otherwise corresponding contribution
will either vanish or lead to singularity in the infrared linit.
In one-loop
approximation this equation leads to $n_{W^{''}}+n_{V_c}=3$.
However, all supergraphs satisfying this condition have zero contribution.
Therefore first correction to chiral effective potential is two-loop one.
In two-loop approximation this equation has the form $n_{W^{''}}+n_{V_c}=5$.
Since the number of purely chiral (antichiral)
vertices independent of $W^{''}$ in two-loop supergraphs can be equal
to 0, 1 or 2, the number of external vertices $W^{''}$ takes values from 3
to 5.

We note that non-trivial contribution to holomorphic
effective potential from any diagram can arise only if the number of
$D^2$-factors is more by one than the number of $\bar{D}^2$-factors
(see details in \cite{Buch5}).
The only Green function in the theory is the propagator
$<\phi\bar{\phi}>$. Therefore total number of quantum chiral superfields
$\phi$ corresponding to all vertices
must be equal to that one of antichiral ones $\bar{\phi}$.
As a result we find that the only two-loop supergraph contributing to
chiral effective potential looks like

\vspace*{1mm}

\hspace{4.5cm}
\Lengthunit=1.5cm
\Linewidth{1.2pt}
\GRAPH(hsize=3){\ind(0,-16){Fig.8}%\thicklines
\mov(.5,0){\Circle(2)\mov(-1,0){\lin(2,0)}
\ind(-2,10){|}
\ind(-2,-3){\bar{D}^2}\ind(-2,0){|}\ind(-2,-10){|}\ind(-2,-13){\bar{D}^2}
\ind(-2,7){\bar{D}^2}
\ind(-9,2){D^2} \ind(-10,-2){D^2} \ind(8,2){D^2} \ind(8,-2){D^2}
\ind(-18,2){-} \ind(-18,-2){-} \ind(0,2){-} \ind(0,-2){-}
\Linewidth{0.3pt}
\mov(-1,1){\lin(-.7,.7)}\mov(-1.1,1){\lin(-.7,.7)}
\mov(-1,-1){\lin(.7,-.7)}\mov(-1.1,-1){\lin(.7,-.7)}
\mov(-1,0){\lin(-.7,.7)}\mov(-1.1,0){\lin(-.7,.7)}}
}

\vspace*{3mm}

Here internal lines are propagators $<\phi\bar{\phi}>$ depending on background
chiral superfields which have the form
\begin{equation}
\label{propc}
<\phi\bar{\phi}>=-\bar{D}^2_1 D^2_2
\frac{\delta^8(z_1-z_2)}{16 K_{\Phi\bar{\Phi}}(z_1)\Box}.
\end{equation}
We note that the superfield $K_{\Phi\bar{\Phi}}$ is not constant here.
Double external lines are $W''$.

After $D$-algebra transformations and loop integrations
we find that two-loop contribution to holomorphic effective
potential in this model looks like
\begin{equation}
\label{l2cg}
W^{(2)}=
\frac{6}{{(16\pi^2)}^2}\zeta(3) \bar{W}^{'''2}
{\Big\{\frac{W''(z)}{K^2_{\Phi\bar{\Phi}}(z)}\Big\}}^3
\end{equation}
One reminds that $\bar{W}^{'''}=\bar{W}^{'''}(\bar{\Phi})|_{\bar{\Phi}=0}$ and
$K_{\Phi\bar{\Phi}}(z)=\frac{\partial ^2 K(\bar{\Phi},\Phi)}
{\partial\Phi\partial\bar{\Phi}}
|_{\bar{\Phi}=0}$ here.
We see that the correction (\ref{l2cg}) is finite and does not require
renormalization in any case despite the theory is non-renormalizable
in general case. 

We note that the calculation of two-loop k\"{a}hlerian effective
potential can be carried out with help of matrix superpropagator 
(\ref{grematg}). The results are given in (\cite{my2,my5}).

Now let us consider some phenomenological applications of the theory
characterized by the action (\ref{genact}). Let us suppose that the column
vector $\vec{\Phi}$ describes two superfield: light (massless)
$\phi$ and heavy $\Phi$, $\vec{\Phi}=\left(\begin{array}{c}\phi\\ \Phi
\end{array}\right)$. We calculate for this case one-loop effective
action and eliminate heavy superfields with use of effective equations
of motion. As a result we arrive at the effective action of light
superfields. There is a decoupling theorem \cite{Collins,Syma}
according to which this effective action after redefining of
parameters
(fields, masses, coupling)
can be expressed in the form of a sum of effective action
of the theory obtained from initial one by putting heavy fields to zero
and terms proportional to different powers of $\frac{1}{M}$ where $M$
is mass of heavy superfield (which in the case under consideration is
put, by fenomenological reasons,
to be equal to $M_{String}$ \cite{Cvet,my5}). 

We study such a theory in one-loop approximation.
The low-energy leading one-loop contribution to effective action is
given by (\ref{k1}). The matrices $K_{\Phi\bar{\Phi}}=
\frac{\pa^2 K}{\pa\Phi^i\pa\bar{\Phi}^j}$ and $W''=
\frac{\pa^2 W}{\pa\Phi^i\pa\Phi^j}$ can
be diagonalized simultaneously, and the trace is a sum of proper values. 
We consider, as an example, the minimal theory \cite{my6} with
\bea
K&=&\phi\bar{\phi}+\Phi\bar{\Phi};\ 
W=\frac{M}{2}\Phi^2+\frac{\lambda}{2}\Phi\phi^2+\frac{g}{3!}\phi^3
\eea
After calculations described in \cite{my6} and analogous ones to those of
Wess-Zumino model we get
\bea
K^{(1)}&=&-\frac{1}{32\pi^2}\big(R^2_1\log\frac{R^2_1}{\mu^2}+
R^2_2\log\frac{R^2_2}{\mu^2}\big)
\eea
where
\bea
R_{1,2}&=&|\l\Phi+g\phi|^2+2\l^2|\phi|^2+M^2
\pm\nonumber\\&\pm&
\sqrt{(|\l\Phi+g\phi|^2-M^2)^2+4|\l^2\Phi\bar{\phi}+
\l M\phi+\l g|\phi|^2|^2}
\eea
The low-energy effective action is given by
\bea
\label{leea}
\Gamma^{(1)}&=&\int d^8 z (\phi\bar{\phi}+\Phi\bar{\Phi}+\hbar
K^{(1)})
+[\int d^6 z (\frac{M}{2}\Phi^2+\frac{\lambda}{2}\Phi\phi^2+
\frac{g}{3!}\phi^3)+h.c.]
\eea
The effective equations of motion for heavy superfield $\Phi$ looks like
\bea
\label{eem}
-\frac{1}{4}\bar{D}^2(\bar{\Phi}+\hbar\frac{\pa K^{(1)}}{\pa\Phi})
+M\Phi+\frac{\l}{2}\phi^2=0
\eea
We can solve this equation by iterative method, i.e. we suppose that 
\bea
\label{iter}
\Phi=\Phi_0+\Phi_1+\ldots+\Phi_n+\ldots,
\eea 
here $\Phi_k$ is $k$-th 
approximation. Since mass $M$ is very large we suppose that
$|D^2\Phi|\ll M\Phi$.
Zero approximation is obtained from condition
$M\Phi_0+\frac{\l}{2}\phi^2=0$,
i.e. $\Phi_0=-\frac{\l\phi^2}{2M}$. And
$k$-th approximation is proportional to $M^{-k}$.
Substituting the expansion of $\Phi$ (\ref{iter}) into (\ref{eem}) we
get the following recurrent relation for $\Phi_{n+1}$:
\bea
\Phi_{n+1}=\frac{\bar{D}^2}{4M}[\bar{\Phi}_n+\hbar
\frac{\pa K^{(1)}}{\pa\Phi}|_{\Phi=\Phi_0+\ldots+\Phi_n}
-\hbar\frac{\pa K^{(1)}}{\pa\Phi}|_{\Phi=\Phi_0+\ldots+\Phi_{n-1}}]
\eea
As a result, we get the following solution for $\Phi$ in leading order
\bea
\Phi=-\frac{\lambda\phi^2}{2M}-\frac{\hbar \bar{D}^2}{64\pi^2 M}
[\l g\bar{\phi}(1+\log\frac{2g^2|\phi|^2}{\mu^2})]+O(\frac{1}{M^2})
\eea
This solution can be substituted into the low-energy effective action
(\ref{leea}).
The effective action of light superfields is defined as
\bea
S_{eff}[\phi,\bar{\phi}]=\Gamma^{(1)}[\phi,\bar{\phi};
\Phi(\phi,\bar{\phi}),\bar{\Phi}(\phi,\bar{\phi})]
\eea
where $\Phi(\phi,\bar{\phi})$ and the same notation for $\bar{\Phi}$
mean that heavy superfields are expressed in terms of light ones via
effective equations of motion.
In the case under consideration, effective action of light superfields
looks like
\bea
S_{eff}=S^{(0)}_{eff}+S^{(1)}_{eff}+O(\frac{1}{M^2})
\eea
where $S^{(0)}_{eff}$ and $S^{(1)}_{eff}$ are of zeroth and first
orders in $M^{-1}$ respectively. These expressions look like
\bea
S^{0}_{eff}&=&\int d^8 z\Big[\phi\bar{\phi}-\frac{\hbar}{32\pi^2}
[4\l^2|\phi|^2(1+\log\frac{2M^2}{\mu^2})+
2g^2|\phi|^2\log\frac{2g^2|\phi|^2}{\mu^2}]+\nonumber\\&+&
(\int d^6 z\frac{g}{3!}\phi^3+h.c.)
\Big]\nonumber\\
S^{(1)}_{eff}&=&-\frac{\hbar}{32\pi^2 M}\int d^8 z \Big[
-2\l^2g(\phi+\bar{\phi})|\phi|^2\log\frac{g^2|\phi|^2}{M^2}
+2\l^2g[\bar{\phi}(\phi^2+O(\hbar^2))+h.c.]
\Big]+\nonumber\\
&+&\frac{1}{M}\Big[\int d^6 z\Big(\frac{1}{8}\l^2\phi^4-\l^2g^2
(\frac{\hbar\bar{D}^2}{64\pi^2}(\bar{\phi}(1+\log\frac{2g^2|\phi|^2}{\mu^2}))^2
\Big)+h.c.
\Big]
\eea
We see that this effective action contains term of the
form
$$
-\frac{\hbar}{32\pi^2}4\l^2|\phi|^2(1+\log\frac{2M^2}{\mu^2})
$$
which increases with growth of $M$. If we put the renormalization
condition $\log\frac{2M^2}{\mu^2}=-1$ this term vanishes. The
effective action, however, in this case takes the form
\bea
S^{0}_{eff}&=&\int d^8 z\Big[\phi\bar{\phi}-\frac{\hbar}{32\pi^2}
2g^2|\phi|^2\log\frac{2g^2|\phi|^2}{eM^2}+\nonumber\\&+&
(\int d^6 z\frac{g}{3!}\phi^3+h.c.)
\Big]
\eea
This expression contains the term
$2g^2|\phi|^2\log\frac{2g^2|\phi|^2}{eM^2}$ 
which also increases as $M$ grows. Therefore we see that modifications
of light superfield effective action caused by presence of heavy
superfields are significant \cite{my6}.

\section{Dilaton supergravity as an example of superfield theory with
higher derivatives}
\setcounter{equation}{0}

The starting point of our consideration is the supertrace anomaly of
matter superfield in curved superspace. The action generating this
anomaly is obtained in \cite{anom}. In conformally flat superspace
(in which vector supergravity prepotential $H^m$ is equal to zero)
this action looks like
\begin{eqnarray}
\label{actan}
\Gamma_A&=&\frac{1}{16\pi^2}\int d^8 z
(8c\partial_a\bar{\sigma}\partial^a\sigma +
\frac{1}{32}(32c-b)\bar{D}^{\dot{\alpha}}\bar{\sigma}D^{\alpha}\sigma
(\partial_{\alpha\dot{\alpha}}(\sigma+\bar{\sigma})-\nonumber\\&-&
\frac{1}{2}\bar{D}_{\dot{\alpha}}\bar{\sigma} D_{\alpha}\sigma))
\end{eqnarray}
Here we have used the "flat" supercovariant derivatives
$D_{\alpha}$, $\bar{D}_{\dot{\alpha}}$, $\partial_{\alpha\dot{\alpha}}$;
and $\sigma=\ln\Phi$, $\bar{\sigma}=\ln\bar{\Phi}$.

Let us also consider the superfield action of N=1 supergravity, in conformally
flat superspace we obtain

\begin {equation}
\label{actsg} 
S_{SG}=-\frac{m^2}{2}\int d^8 z \Phi\bar{\Phi}
+[\Lambda\int d^6 z \Phi^3 +h.c.]
\end {equation}
where $m^2=\frac{6}{\kappa^2}$, $\Lambda$ is the cosmological constant and
$\kappa$ is the gravitational constant. We will investigate the theory action
of which is a sum of the actions (\ref{actan}) and (\ref{actsg})
   Denoting
\begin{eqnarray}
\frac{Q^2}{2}=8c\nonumber,\
\xi_1=\frac{1}{32{(4\pi)}^2}(32c-b),\ 
\xi_2=-\frac{1}{64{(4\pi)}^2}(32c-b)
\end{eqnarray}
we get a superfield theory in flat superspace with the action of the form
\begin {eqnarray}
\label{actiondsg}
S&=&\int d^8z(-\frac{Q^2}{2{(4\pi)}^2}\bar{\sigma}\Box\sigma+
\bar{D}^{\dot\alpha}\bar{\sigma}
D^{\alpha}\sigma\times\nonumber\\
&\times&({\xi_1\partial_{\alpha}}_{\dot{\alpha}}(\sigma+\bar{\sigma})
+\xi_2\bar{D}_{\dot{\alpha}}\bar{\sigma} D_{\alpha}\sigma)-
\frac{m^2}{2} e^{\sigma+\bar{\sigma}})+(\Lambda\int d^6z e^{3\sigma}+h.c.)
\end {eqnarray}
The $Q^2, \xi_1, \xi_2, m^2, \Lambda$ will be considered as the arbitrary and
independent parameters of the model. We will call the model with action
(\ref{actiondsg}) the four-dimensional dilaton supergravity model.

   In order to calculate the counterterms and to find the divergences we should
study the structure of supergraphs of the theory. The strucure of supergraphs
is defined by a form of propagators and vertices. 

The theory under consideration is characterized by a matrix propagator
\begin {equation}
G(z,z')=\left(
\begin{array}{cc}
   G_{++}(z,z') & G_{+-}(z,z')\\
   G_{-+}(z,z') & G_{--}(z,z')
\end{array}
\right)
\end {equation}
  satisfying the equation
\begin {equation}
\label{eq}
\left(
\begin{array}{cc}
  9\Lambda & (\frac{Q^2}{{(4\pi)}^2}\Box+m^2)\frac{\bar{D}^2}{4}\\
  (\frac{Q^2}{{(4\pi)}^2}\Box+m^2)\frac{D^2}{4} & 9\bar{\Lambda}
  \end{array}
\right)
\left(
\begin{array}{cc}
    G_{++} & G_{+-}\\
    G_{-+} & G_{--}
\end{array}
\right)
=
\left(
\begin{array}{cc}
   \delta_+ & 0\\
     0 & \delta_-
\end{array}
\right)
\end{equation}
where $\delta_+=-\frac{1}{4} \bar{D}^2\delta^8(z_1-z_2)$,
$\delta_-=-\frac{1}{4} D^2\delta^8(z_1-z_2)$.
The solution of this equation is written in the form
\begin{equation}
G=\frac{1}{81\Lambda\bar{\Lambda}-\Box{(\frac{Q^2}{16\pi^2}\Box+m^2)}^2}
\left[
\begin {array} {cc}
  9\bar{\Lambda} & -(\frac{Q^2}{16\pi^2}\Box+m^2)\frac{\bar{D}^2}{4}\\
  -(\frac{Q^2}{16\pi^2}\Box+m^2)\frac{D^2}{4} & 9\Lambda
  \end {array}
\right]
\end{equation}
   This propagator acts on columns
$\left(
\begin{array}{c}
   \phi\\
   \bar{\phi}
\end{array}
\right)$
where $\phi$ is a chiral superfield and $\bar{\phi}$ is an antichiral
superfield.

  The propagator in momentum representation looks like follows
\begin{eqnarray}
\label{Greensg}
G_{++}(k)&=&\bar{G}_1(k)\frac{-\bar{D}^2}{4}\delta_{12}\,
G_{+-}(k)=G_2(k)\frac{\bar{D}^2 D^2}{16}\delta_{12}\\
G_{-+}(k)&=&G_2(k)\frac{D^2\bar{D}^2}{16}\delta_{12}\,
G_{--}(k)=G_1(k)\frac{-D^2}{4}\delta_{12}\nonumber
\end{eqnarray}
 where
\begin{eqnarray}
\label{Gr2}
%\delta_{12}&=&\delta^4(\theta_1-\theta_2)\nonumber\\
G_1(k)&=&\frac{9\Lambda}{k^2{(\frac{Q^2}{16\pi^2}k^2-m^2)}^2+
81\Lambda\bar{\Lambda}}\,
\bar{G}_1(k)=\frac{9\bar{\Lambda}}{k^2{(\frac{Q^2}{16\pi^2}k^2-m^2)}^2+
81\Lambda\bar{\Lambda}}\nonumber\\
G_2(k)&=&\frac{-\frac{Q^2}{16\pi^2}k^2+m^2}{k^2{(\frac{Q^2}{16\pi^2}k^2-m^2)}^2
+81\Lambda\bar{\Lambda}}\nonumber
\end{eqnarray}

A structure of vertices is taken from action (\ref{actiondsg}).
There are four vertices:
\begin{eqnarray}
\label{vertsg}
V_1&=&\xi_1 
\bar{D}^{\dot{\alpha}}\bar{\sigma}
D^{\alpha}\sigma\partial_{\alpha\dot{\alpha}}(\sigma+\bar{\sigma})\nonumber\\
V_2&=&\xi_2\bar{D}_{\dot{\alpha}}\bar{\sigma}\bar{D}^{\dot{\alpha}}\bar{\sigma}
D^{\alpha}\sigma D_{\alpha}\sigma\nonumber\\
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
V_3&=&-\frac{m^2}{2}(e^{\sigma+\bar{\sigma}}-1-(\sigma+\bar{\sigma})-\frac{1}{2}
{(\sigma+\bar{\sigma})}^2)\\
V_4&=&\Lambda (-\frac{D^2}{4\Box})(e^{3\sigma}-1-3\sigma-\frac{9}{2}\sigma^2)+
\bar{\Lambda} (-\frac{\bar{D}^2}{4\Box})(e^{3\bar{\sigma}}-1-3\bar{\sigma}
-\frac{9}{2}\bar{\sigma}^2)\nonumber
\end{eqnarray}
The factors $(-\frac{D^2}{4\Box})$ and $(-\frac{D^2}{4\Box})$ arise in $V_4$
since the all vertices correspond to the action written as the integrals over
whole superspace.

The eqs. (\ref{Greensg}--\ref{vertsg}) are the basis of supergraph
technique 
allowing to develop a perturbative treatment for the model 
(\ref{actiondsg}).

Let us consider the superficial degree of divergence (SDD) from this
model. Since space-time derivatives give contribution to SDD equal to
1, and spinor ones -- to 1/2 we see that any $V_1,V_2$-type vertex
contributes to SDD with 2, and all such vertices -- with $V_{1,2}$
(here $V_{1,2}$ is a number of such vertices). Any loop as usual
contributes with 2 (4, because any loop includes integration over
$d^4k$, and $-2$, since contraction of any loop into a point requires
four $D$-factors and reduces possible contribution to SDD by 2), hence
all loops -- with 2L. Contribution of all propagators $G_{++},G_{--}$
is equal to $-5P_1$ where $P_1$ is a number of such propagators, and
contribution of all propagators $G_{+-},G_{-+}$ -- to
$-2P_2$. The $V_3$-vertices do not contribute at all, and 
$V_4$-type totalize a contribution of $-V_4$ 
\cite{my1}. Therefore total SDD is equal to
$$
\omega=2L+2V_{1,2}-5P_1-2P_2-V_4
$$
Since $V_{1,2}+V_3+V_4=V$ and $L+V-P=1$ we get
\bea
\label{index}
\omega=2-3P_1-2V_3-3V_4
\eea
Note that this is only the upper limit for SDD. Really some $D$-factors are
transported to external lines, and $\omega$ in general case is less by
$N_D/2$, where $N_D$ is a number of $D$-factors acting on external lines.
However, this equation allows one to make some conclusions. First of all,
divergent diagrams cannot contain $V_4$-vertices (hence all chiral
contributions to effective action are finite). Second, divergent
diagrams cannot contain 
$<\sigma\sigma>,<\bar{\sigma}\bar{\sigma}>$-propagators and can
contain no more than one vertex proportional to $m^2$. However, they
can contain arbitrary number of $V_{1,2}$-vertices, hence there is an
infinite number of divergent structures 
(f.e. divergent corrections of the form
$\xi_1\xi_2^{n+1}\sigma^n\bar{\sigma}^n\bar{D}^{\dot{\alpha}}\bar{\sigma}
D^{\alpha}\sigma\partial_{\alpha\dot{\alpha}}(\sigma+\bar{\sigma})$
can arise for any $n$). Hence the theory is non-renormalizable.
However, if we put $\xi_1=\xi_2=0$ the theory is super-renormalizable.

Consider one-loop counterterms leading to renormalization of
$\xi_1,\xi_2$.
\unitlength=.4mm
\begin{center}
\begin{picture}(100,100)
\put(50,50){\circle{40}}
\put(30,50){\line(-1,-1){20}}
\put(15,40){\line(0,-1){10}}
\put(10,20){$D^{\alpha}$}
\put(35,70){\line(0,-1){10}}
\put(30,70){$\bar{D}_{\dot{\alpha}}$}
\put(30,50){\line(-1,1){20}}
\put(15,70){\line(0,-1){10}}
\put(35,40){\line(0,-1){10}}
\put(30,20){$D_{\alpha}$}
\put(10,75){$\bar{D}^{\dot{\alpha}}$}
\put(70,50){\line(1,0){20}}
\put(80,40){$\partial_{\beta\dot{\beta}}$}
\put(80,55){\line(0,-1){10}}
\put(65,40){\line(0,-1){10}}
\put(65,70){\line(0,-1){10}}
\put(70,75){$D^{\beta}$}
\put(70,20){$\bar{D}^{\dot{\beta}}$}
\put(45,80){$G_{+-}$}
\put(45,20){$G_{-+}$}
\put(40,0){Fig.9}
\end{picture}
%\end{center}
%\begin{center}
\begin{picture}(100,100)
\put(50,50){\circle{40}}
\put(30,50){\line(-1,-1){20}}
\put(30,50){\line(-1,1){20}}
\put(70,50){\line(1,-1){20}}
\put(70,50){\line(1,1){20}}
\put(15,40){\line(0,-1){10}}
\put(10,20){$D^{\alpha}$}
\put(15,70){\line(0,-1){10}}
\put(10,75){$\bar{D}^{\dot{\alpha}}$}
\put(85,70){\line(0,-1){10}}
\put(85,40){\line(0,-1){10}}
\put(85,20){$D^{\beta}$}
\put(85,75){$\bar{D}^{\dot{\beta}}$}
\put(35,40){\line(0,-1){10}}
\put(30,20){$D_{\alpha}$}
\put(35,70){\line(0,-1){10}}
\put(30,70){$\bar{D}_{\dot{\alpha}}$}
\put(65,40){\line(0,-1){10}}
\put(65,70){\line(0,-1){10}}
\put(70,75){$D_{\beta}$}
\put(70,20){$\bar{D}_{\dot{\beta}}$}
\put(45,80){$G_{+-}$}
\put(45,20){$G_{-+}$}
\put(40,0){Fig.10}
\end{picture}
\end{center}
The supergraphs associated with Fig.9 and Fig.10 lead to the following
contributions respectively
\begin{eqnarray}
{S_1}^{(1)}&=&72\mu^{-\epsilon}\xi_1\xi_2\int d^4\theta_1 d^4\theta_2
\frac{d^d p_1 d^d p_2}{{(2\pi)}^{2d}}\times\nonumber\\
&\times&\bar{D}^{\dot{\alpha}}\bar{\sigma}(-p_1,\theta_1)
D^{\alpha}\sigma(-p_2,\theta_1)
\partial_{\beta\dot{\beta}}(\sigma(p,\theta_2)+ \bar{\sigma}(p,\theta_2))
\times\nonumber\\
&\times&\int\frac{d^d k}{{(2\pi)}^d} D_{\alpha}\bar{D}^{\dot{\beta}}G_{+-}(k)
\bar{D}_{\dot{\alpha}}D^{\beta} G_{-+}(k+p)\\
{S_2}^{(1)}&=&72\mu^{-\epsilon}\xi_2^2\int d^4\theta_1 d^4\theta_2
\frac{d^d p_1 d^d p_2 d^d p_3}{{(2\pi)}^{3d}}\times\nonumber\\
&\times&\bar{D}^{\dot{\alpha}}\bar{\sigma}(-p_1,\theta_1)
D^{\alpha}\sigma(-p_2,\theta_1)
\bar{D}_{\dot{\beta}}\bar{\sigma}(p_3,\theta_2)
 D_{\beta}\sigma(p-p_3,\theta_2)\times\nonumber\\
&\times&\int\frac{d^d k}{{(2\pi)}^d}
D_{\alpha}\bar{D}^{\dot{\beta}}G_{+-}(k) \bar{D}_{\dot{\alpha}} D^{\beta}
G_{-+}(k+p)\nonumber
\end{eqnarray}
Here ${S_1}^{(1)}$ and ${S_2}^{(1)}$ are one-loop divergent corrections to
vertices $V_1$ and $V_2$ correspondingly, $p_1$, $p_2$ and $p_3$ are external
momenta, $p=p_1+p_2$. $\mu$ is a standard arbitrary parameter of mass dimension
introduced in dimensional regularization and $\epsilon=4-d$. 
After calculations described in \cite{my1} we get
one-loop quantum corrections from this supergraphs
in the form
\begin{eqnarray}
\label{correct}
S_1^{(1)}&=&-576\mu^{-\epsilon}\frac{\xi_1\xi_2(16\pi^2)^2}{Q^4}
\int d^8z
\bar{D}^{\dot{\alpha}}\bar{\sigma}D^{\alpha}\sigma
\partial_{\alpha\dot{\alpha}}(\sigma+ \bar{\sigma})
(\frac{2}{16\pi^2\epsilon}+fin)\equiv\\
&\equiv& S^{(1)}_{1_{div}}+S^{(1)}_{1_{fin}}\nonumber\\
S_2^{(1)}&=&-576\mu^{-\epsilon}\frac{\xi^2_2(16\pi^2)^2}{Q^4}
\int d^8 z
\bar{D}^{\dot{\alpha}}\bar{\sigma}
D^{\alpha}\sigma
\bar{D}_{\dot{\alpha}}\bar{\sigma}
 D_{\alpha}\sigma
(\frac{2}{16\pi^2\epsilon}+fin)\equiv\nonumber\\
&\equiv& S^{(1)}_{2_{div}}+S^{(1)}_{2_{fin}}\nonumber
\end{eqnarray}

In order to renormalize the theory we introduce the one-loop counterterms\\
$-{S_1^{(1)}}_{div}$, $-{S_2^{(1)}}_{div}$. It corresponds to the following
renormalization transformation

\begin{eqnarray}
\label{ren0}
Q^2_{(0)}=\mu^{-\epsilon}Z_Q Q^2\nonumber\\
\xi_{1(0)}=\mu^{-\epsilon}Z_1\xi_1\\
\xi_{2(0)}=\mu^{-\epsilon}Z_2\xi_2\nonumber
\end{eqnarray}
where
$Q^2_{(0)}, \xi_{1(0)}, \xi_{2(0)}$ are the bare parameters and
$Q^2, \xi_1, \xi_2 $- are the renormalized ones. As a result one obtains

\begin{equation}
\label{ren12}
Z_1=Z_2=(1+\frac{72\xi_2(16\pi)^2}{Q^4\epsilon})
\end{equation}
We see that in one-loop approximation there is the same independent
renormalization constant both for $\xi_1$ and $\xi_2$. It means in particular
that if we put $\xi_2^{(0)}=c\xi_1^{(0)}$, where $c$ is a constant, then the
renormalized parameters $\xi_1$ and $\xi_2$ will satisfy the same relation
$\xi_2=c\xi_1$. One-loop renormalization does not destroy the relationship
between the parameters in lower order.

Next step is a calculation of $Z_Q$. Let us consider the supergraph given on
Fig.12 (note that we considered such a supergraph in Section 3)
\begin{center}
\begin{picture}(100,100)
\put(50,50){\circle{40}}
\put(30,50){\line(-1,0){20}}
\put(70,50){\line(1,0){20}}
\put(15,55){\line(0,-1){10}}
\put(80,55){\line(0,-1){10}}
\put(80,40){$\partial_{\alpha\dot{\alpha}}$}
\put(10,60){$\partial_{\beta\dot{\beta}}$}
\put(65,40){\line(0,-1){10}}
\put(65,70){\line(0,-1){10}}
\put(70,75){$D^{\alpha}$}
\put(70,20){$\bar{D}^{\dot{\alpha}}$}
\put(35,40){\line(0,-1){10}}
\put(30,20){$D^{\beta}$}
\put(35,70){\line(0,-1){10}}
\put(30,70){$\bar{D}^{\dot{\beta}}$}
\put(45,80){$G_{+-}$}
\put(45,20){$G_{-+}$}
\put(40,0){Fig.12}
\end{picture}
\end{center}
The corresponding contribution looks like this
\begin{eqnarray}
S_Q^{(1)}&=&-18\mu^{-\epsilon}\xi^2_1\int d^4\theta_1 d^4\theta_2\int \frac
{d^d p}{{(2\pi)}^d}
\partial_{\beta\dot{\beta}}\bar{\sigma}(-p,\theta_1)
\partial_{\alpha\dot{\alpha}}\sigma(p,\theta_2)\times\\
&\times&\int\frac{d^d k}{{(2\pi)}^d}
\frac{\bar{D}^{\dot{\beta}} D^{\alpha}\bar{D}^2 D^2}{16}\delta_{12}
\frac{D^{\beta}\bar{D}^{\dot{\alpha}} D^2 \bar{D}^2}{16}\delta_{12}
G_2(k)G_2(k+p)\nonumber
\end{eqnarray}
 Carrying out the transformations analogous to those used above we obtain
\begin{equation}
S_Q^{(1)}=-\mu^{-\epsilon}\int d^4\theta d^d x\partial_{\alpha\dot{\alpha}}
\sigma\partial^{\alpha\dot{\alpha}}\bar{\sigma}
(\frac{18\xi_1^2{(4\pi)}^4}{Q^4\epsilon}+fin)\equiv S^{(1)}_{Q_{div}}+
S^{(1)}_{Q_{fin}}
\end{equation}
After introducing the one-loop counterterm $-{S_Q^{(1)}}_{div}$ one will obtain
using
(\ref{ren0})
\begin {equation}
\label{Z}
Z_Q=(1+\frac{32\pi^2}{Q^2}\frac{18\xi_1^2{(4\pi)}^4}{Q^4\epsilon})
\end {equation}

   So we have studied renormalization of $\xi_1$, $\xi_2$ and $Q^2$. As for
$\Lambda$, it was already noted that all diagrams containing vertex of type
$V_4$ are finite, it means that the coupling $\Lambda$ is not renormalized.

   Now it remains to investigate renormalization of $m^2$. It follows from
(\ref{index}), that divergent diagrams can contain no more than one vertex 
of $V_3$-type corresponding to coupling constant $m^2$. All other
possible vertices should be of  $V_1$- or $V_2$-types.

We will study the divergent corrections to $m^2$ in the case when $\xi_1$=
$\xi_2=0$. It means that the vertices $V_1$ and $V_2$ are absent at
all.  It will
be proved further that this case corresponds to infrared limit of the theory.
It means that only $V_3$-type vertex can be presented in the diagrams giving
contribution to divergent correction to $m^2$. All these diagrams contain only
one vertex $V_3$-type, one internal line $G_{+-}$-type and an arbitrary
number of external lines corresponding to $\sigma$, $\bar{\sigma}$
\begin{center}
\begin{picture}(100,100)
\put(50,50){\circle{40}}
\put(30,50){\line(-1,-1){20}}
\put(30,50){\line(-1,1){20}}
\put(20,45){$\vdots$}
\put(75,50){$G_{+-}$}
\put(40,0){Fig.13}
\end{picture}
\end{center}

   Let us consider such a diagram with a given number $N$ of external lines,
$l$ from those are chiral and other are antichiral. Contribution of
this diagram has the form
\begin{equation}
-\frac{m^2}{2}\int d^8 z\frac{\sigma^l(z)\bar{\sigma}^{N-l}(z)}{l!(N-l)!}
 G_{+-}(z,z)
\end{equation}
  Sum of all these contributions is equal to
\begin{equation}
S_3=-\frac{m^2}{2}\int d^8 z \sum_{N=0}^{\infty}\sum_{l=0}^{N}
\frac{\sigma^l(z)\bar{\sigma}^{N-l}(z)}{l!(N-l)!}G_{+-}(z,z)=
\frac{m^2}{2}\int d^8 z e^{\sigma+\bar{\sigma}} G_{+-}(z,z)
\end{equation}
In momentum representation the $S_3$ can be written as follows
\begin{equation}
S_3=-\frac{m^2}{2}\int d^4\theta d^d x e^{\sigma+\bar{\sigma}}\int
\frac{d^d k}{{(2\pi)}^d}\frac{-A^2 k^2+m^2}
{k^2{(-A^2 k^2+m^2)}^2- 81\Lambda\bar{\Lambda}}
\frac{\bar{D}^2_1 D^2_1}{16}\delta_{11}
\end{equation}
Taking into account
$\frac{\bar{D}^2_1 D^2_1}{16}\delta_{11}=\frac{\bar{D}^2_1 D^2_1}{16}
 \delta_{12}\big|_{\theta_1=\theta_2}=1 $ we get
 \begin{equation}
S_3=-\frac{m^2}{2}\int d^4\theta  d^d x e^{\sigma+\bar{\sigma}}
\int\frac{d^d k}{{(2\pi)}^d}
\frac{-A^2 k^2+m^2}{k^2{(-A^2 k^2+ m^2)}^2-81\Lambda\bar{\Lambda}}
\end{equation}
After integration over momentum we get
$$ S_3=\frac{m^2}{2}\int d^d x d^4\theta e^{\sigma+\bar{\sigma}}
(\frac{2}{16\pi^2 A^2\epsilon}+fin)$$
We note that despite these diagrams are tadpole-type, their
contribution is not equal to zero unlike  Wess-Zumino model.

To cancel the divergence we should introduce a counterterm $-{S_3}_{div}$. It
corresponds to mass renormalization
\begin{eqnarray}
\label{rmass}
m^2_0&=&\mu^{-\epsilon} Z_m m^2\nonumber\\
Z_{m^2}&=&1+\frac{2}{16\pi^2 A^2\epsilon}=1+\frac{2}{Q^2\epsilon}
\end{eqnarray}
Here $m^2_0$ is a bare mass and $m^2$ is a renormalized one.  

Next step is consideration of beta functions. As usual, beta function
for any renormalized parameter $g(\mu)$ is defined as
$$
\b_g=\mu\frac{dg}{d\mu}
$$
where the renormalized parameter is expressed in term of the 
bare one $g_0$ which
does not depend on $\mu$, $\frac{dg_0}{d\mu}=0$.

Using (\ref{ren12},\ref{ren0},\ref{rmass}) we obtain the following
beta functions
\begin{eqnarray}
\label{rgr}
\beta_{\xi_1}=\frac{72{(16\pi)}^2}{Q^4}\xi_1\xi_2;\ 
\beta_{\xi_2}=\frac{72{(16\pi)}^2}{Q^4}\xi_2^2;\
\beta_{Q^2}=32\pi^2\frac{18{(16\pi)}^2 \xi^2_1}{Q^4}
\end{eqnarray}

As a result the equations for running couplings have the form
\begin{eqnarray}
\label{eqr}
\frac{d\xi_1}{dt}=a\frac{\xi_1\xi_2}{Q^4};\
\frac{d\xi_2}{dt}=a\frac{\xi_2^2}{Q^4};\
\frac{d Q^2}{dt}=b\frac{\xi_1^2}{Q^4}
\end{eqnarray}
where $a=2^{11}3^2\pi^2$, $b=3^2 2^{14}\pi^4$.
The solutions of these equations are
\begin{eqnarray}
\xi_1(t)&=&\frac{\xi_1}{\xi_2}\xi_2(t)\nonumber\\
Q^2(t)&=& Q^2+8\pi^2\frac{\xi_1^2}{\xi_2^2}(\xi_2(t)-\xi_2)\nonumber\\
t&=&\frac{1}{2^{11} 3^2\pi^2}\Big\{-[Q^2-8\pi^2\frac{\xi^2_1}{\xi^2_2}]
(\frac{1}{\xi_2(t)}-\frac{1}{\xi_2})-\nonumber\\
&-&16\pi^2{(\frac{\xi_1}{\xi_2})}^2 [Q^2-8\pi^2\frac{\xi^2_1}{\xi^2_2}]
\ln\frac{\xi_2(t)}{\xi_2}+\\
&+&64\pi^4{(\frac{\xi_1}{\xi_2})}^4(\xi_2(t)-\xi_2)\Big\}\nonumber
\end{eqnarray}

Let us investigate the behaviour of running couplings $\xi_1(t)$, $\xi_2(t)$
and $Q^2(t)$ in infrared domain when $t\rightarrow-\infty$. It is easy to see
that in this case $\xi_2(t)\rightarrow 0$ and hence $\xi_1(t)\rightarrow 0$.
It means that $\xi_1^{(0)}$= $\xi_2^{(0)}$= 0 is an infrared fixed point. For
$Q^2(t)$ we obtain $Q^2(t)\rightarrow Q^2-8\pi^2\frac{\xi^2_1}{\xi_2}$.
If we take quantities of initial $\xi_1$ and $\xi_2$ so that they correspond to
infrared fixed point $\xi_1=\xi_2=0$ one gets $Q^2(t)\rightarrow Q^2$.
In particular, only the diagrams given on Fig.4 can contribute to mass
renormalization in infrared limit.

To investigate a behaviour of running mass we should use a notion of scaling
dimension of superfields. We note that the action of the theory 
(\ref{actiondsg}) at $\xi_1=\xi_2=0$
is invariant under the transformations

\begin{eqnarray}
\label{trans}
\delta\sigma=(x^a \partial_a +\frac{1}{2}\theta^{\alpha} D_{\alpha})\sigma+1\\
\delta\bar{\sigma}=(x^a
\partial_a+\frac{1}{2}\bar{\theta}_{\dot{\alpha}}\bar{D}
^{\dot{\alpha}})\bar{\sigma}+1\nonumber
\end{eqnarray}
   Let $V$ be is some function depending on superfields 
$\sigma$, $\bar{\sigma}$
and their derivatives $\partial_a\sigma$, $\partial_a\bar{\sigma}$,
$D_{\alpha}\sigma$, $\bar{D}_{\dot{\alpha}}\bar{\sigma}$,\ldots. We call that
$V$
has the scaling dimension $\Delta$ if the transformation law of $V$ under
transformations (\ref{trans}) looks like this
\begin{equation}
\delta V[\sigma,\bar{\sigma}]=(x^a\partial_a+
\frac{1}{2}\theta^{\alpha} D_{\alpha}+
\frac{1}{2}\bar{\theta}_{\dot{\alpha}} \bar{D}^{\dot{\alpha}}+\Delta) V
\end{equation}

 It is easy to see that the superfields $\sigma$, $\bar{\sigma}$ have no
definite scaling dimension, the derivatives $\partial_a\sigma$,
$\partial_a\bar{\sigma}$ have scaling dimension equal to 1, the spinor
derivatives $D_{\alpha}\sigma$,$\bar{D}_{\dot{\alpha}}\bar{\sigma}$ have no
definite scaling dimensions. However, the functions $e^{\sigma}$,
$e^{\bar{\sigma}}$ have definite scaling dimensions $\Delta$=1.

  Let us fulfil the transformations $\sigma\rightarrow\alpha\sigma$,
$\bar{\sigma}\rightarrow\alpha\bar{\sigma}$ and
$S\rightarrow\frac{1}{\alpha^2}S$
in the action (\ref{actiondsg}) at $\xi_1=\xi_2=0$. It leads to the following
action depending on arbitrary real parameter $\alpha$
\begin{eqnarray}
\label{iract}
S=\int d^8 z(-\frac{1}{2}\frac{Q^2}{16\pi^2}\bar{\sigma}\Box\sigma-
\frac{m^2}{2\alpha^2}e^{\alpha(\sigma+\bar{\sigma})})
+(\frac{\Lambda}{\alpha^2}\int d^6 z e^{3\alpha\sigma}+ h.c.)\nonumber
\end{eqnarray}

  We consider the calculation of the 
renormalization constant $Z_m$ in the theory
(\ref{iract}). The only modification in comparison with eq. (\ref{rmass}) is
that we should use the propagator $\alpha^2G_{+-}$ in supergraph given by
Fig.4. The parameter $\alpha$ is resulted here because of expansion of
$e^{\alpha(\sigma+\bar{\sigma})}$. It leads immediately to
$$ Z_{m^2}=1+\frac{2\alpha^2}{Q^2\epsilon}$$
Therefore the equation for running mass will be
\begin{equation}
\label{eqren}
\frac{d m^2(t)}{dt}=\frac{2\alpha^2 m^2(t)}{Q^2}+\Delta_{m^2}m^2(t)
\end{equation}
where $\Delta_{m^2}$ is a scaling dimension of  $m^2(t)$.

To find $\Delta_{m^2}$ we consider the term
$\frac{m^2}{2\alpha^2}e^{\alpha(\sigma+\bar{\sigma})}$ in the action
(\ref{iract}).
The scaling dimension of this term is $-2$, $\alpha$ is dimensionless
and scaling
dimension of $e^{\alpha(\sigma+\bar{\sigma})}$ is $2\alpha$. Hence
$\Delta_{m^2}=2-2\alpha$.
Therefore the equation (\ref{eqren}) looks like this
\begin{eqnarray}
\frac{d m^2(t)}{dt}&=&(2-2\alpha+\frac{2\alpha^2}{Q^2})m^2(t)\\
m^2(0)&=&m^2\nonumber
\end{eqnarray}
where we took into account that $Q^2(t)= Q^2$ in infrared limit. A solution
of this equation can be written in the form
\begin{equation}
\label{efm}
m^2(t)=m^2\exp((2-2\alpha+\frac{2\alpha^2}{Q^2})t)
\end{equation}
It is evident that at $2-2\alpha+\frac{2\alpha^2}{Q^2}>0$
we get $m^2(t)\rightarrow 0$ in infrared limit (note that this
condition is satisfied at $\a=1$, i.e. when there is no rescaling). 
It corresponds to
$\kappa^2(t)\rightarrow \infty$ where $\kappa^2(t)$ is the running 
gravitational constant.

  As for coupling constant $\Lambda$, its beta-function is equal to zero since
the vertex of $V_4$-type is always finite (see above) and the fields
$\sigma$, $\bar{\sigma}$ are not renormalized in this approach. 

Therefore in infrared limit we stay with the following action
\bea
%\begin{eqnarray}
\label{iract1}
S=\int d^8 z(-\frac{1}{2}\frac{Q^2}{16\pi^2}\bar{\sigma}\Box\sigma+
(\Lambda\int d^6 z e^{3\sigma}+ h.c.)
\end{eqnarray}
%\eea
Our aim consists of calculation of low-energy leading contributions to
one-loop effective action. As usual, the first step is background-quantum
splitting
\bea
\sigma\to\sigma+\chi,\bar{\sigma}\to\bar{\sigma}+\bar{\chi}
\eea
Here $\sigma,\bar{\sigma}$ are background superfields, and
$\chi,\bar{\chi}$ are quantum ones. It is known that to find one-loop
contribution to effective action it is enough to consider the quadratic
action of quantum superfields which has the form
\begin{eqnarray}
\label{iractq}
S^{(2)}_q[\sigma,\bar{\sigma};\chi,\bar{\chi}]=
\int d^8 z(-\frac{1}{2}\frac{Q^2}{16\pi^2}\bar{\chi}\Box\chi
+(\frac{9}{2}\Lambda\int d^6 z e^{3\sigma}\chi^2+ h.c.)\nonumber
\end{eqnarray}
The one-loop effective action $\Gamma^{(1)}[\sigma,\bar{\sigma}]$ can
  be read from expression
\bea
\label{efq}
\exp(i\Gamma^{1}[\sigma,\bar{\sigma}])=\int D\chi D\bar{\chi}
\exp(iS^{(2)}_q[\sigma,\bar{\sigma};\chi,\bar{\chi}])
\eea
We suggest that the effective action, as usual, has the structure described by
(\ref{efexpa},\ref{ep}).
The one-loop effective action in this theory can be expressed in the form of
effective action of some real scalar superfield just as we done in
Wess-Zumino model:
we consider the theory of a real superfield $v$ with action
\bea
S_v=\frac{Q^2}{16\pi^2}\frac{1}{16}\int d^8 z v D^{\a}\bar{D}^2 D_{\a}
\Box v
\eea
One-loop effective action $W_v$ for this theory in the framework of
  Faddeev-Popov approach is given by
\bea
\label{wv}
e^{iW_v}=\int Dv \exp (iS_v)\delta(\frac{1}{4}D^2v-\bar{\chi})
\delta(\frac{1}{4}\bar{D}^2v-\chi)\Delta
\eea
where $\Delta$ is Faddeev-Popov determinant which is a constant as in 
Wess-Zumino model. We note that $W_v$ is also a constant.
After multiplying of left-hand side and right-hand side of equations 
(\ref{efq}) and (\ref{wv}) respectively and integration over
$\chi,\bar{\chi}$
we get one-loop contribution to effective action $\bar{\Gamma}^{(1)}$ 
in the form
\bea
\Gamma^{(1)}=-\frac{i}{2}{\rm Tr}\log
(\frac{Q^2}{16\pi^2}\Box^2-9\Lambda e^{3\sigma}\frac{\bar{D}^2}{4}-
9\Lambda e^{3\bar{\sigma}}\frac{D^2}{4}) 
\eea
In Schwinger representation the one-loop contribution to effective
action looks like
\bea
\Gamma^{(1)}=-\frac{i}{2}\int_0^{\infty}\frac{ds}{s}{\rm Tr}
\exp[is(\frac{Q^2}{16\pi^2}\Box^2-9\Lambda e^{3\sigma}\frac{\bar{D}^2}{4}-
9\Lambda e^{3\bar{\sigma}}\frac{D^2}{4})]
\eea
After change $s\frac{Q^2}{16\pi^2}\to s,\Lambda\frac{16\pi^2}{Q^2}\to
\Lambda$ we can express this effective action as
\bea
\label{gdsg}
\Gamma^{(1)}=-\frac{i}{2}\int_0^{\infty}\frac{ds}{s}{\rm Tr}e^{is\Box^2}
\exp[is
(-9\Lambda e^{3\sigma}\frac{\bar{D}^2}{4}-
9\Lambda e^{3\bar{\sigma}}\frac{D^2}{4})
]
\eea 
Really, commutators of $\Box^2$ with background superfields can lead
only to terms depending on space-time derivatives of background
superfields which lead only to higher orders in $\pa_a\sigma$.
We can calculate exponent of
$\tilde{\Delta}=-9\Lambda e^{3\sigma}\frac{\bar{D}^2}{4}-
9\Lambda e^{3\bar{\sigma}}\frac{D^2}{4}
$
by the same way as in Wess-Zumino model. The
necessary expressions are
\bea
\label{neces}
e^{is\Box^2}\delta^4(x_1-x_2)|_{x_1=x_2}&=&\int\frac{d^4k}{(2\pi)^4}e^{isk^4}=
\frac{1}{32is\pi^2}\\
\Box
e^{is\Box^2}\delta^4(x_1-x_2)|_{x_1=x_2}&=&\int\frac{d^4k}{(2\pi)^4}(-k^2)
e^{isk^4}=-\frac{\sqrt{\pi}}{32\pi^2(is)^{3/2}}
\eea
The expression (\ref{gdsg}) can be exactly found in two special
cases:

(i) k\"{a}hlerian effective potential, in this case all derivatives of
$\sigma,\bar{\sigma}$ are equal to zero. The expression (\ref{gdsg})
is analogous to the expression (\ref{gl0}) after redefinitions
$9\Lambda e^{3\sigma}\to\psi,9\Lambda e^{3\bar{\sigma}}\to\bar{\psi}$.
As a result we can easily restore expression for Schwinger
coefficients $A(s),\tilde{A}(s)$ (\ref{tu}):
\bea
A(s)=\tilde{A}(s)=\frac{1}{\Box}
[\cosh(9is\Lambda\sqrt{e^{3(\sigma+\bar{\sigma})}\Box})-1]
\eea
The one-loop k\"{a}hlerian effective potential is given by
\bea
\label{kd}
K^{(1)}=-\frac{i}{2}\int_0^{\infty}[A(s)+\tilde{A}(s)]U(x,x';s)|_{x=x'}
\eea
where $U(x,x';s)=e^{is\Box^2}\delta^4(x-x')$.
We can write
\bea
K^{(1)}=-\frac{i}{2}\int_0^{\infty}\frac{d\tilde{s}}{\tilde{s}}
\sum_{n=0}^{\infty}
\Big[\frac{({9\Lambda\tilde{s}e^{\frac{3}{2}(\sigma+\bar{\sigma})}})^{2n+2}}
{(2n+2)!}\Box^n\Big] e^{is\Box^2}\delta^4(x-x')|_{x=x'}
\eea
We can separate sum over $n$ into sum over odd $n$ and sum over even
$n$. We use expressions (\ref{neces}) and take into account
that
\bea
\Box^{2n} e^{is\Box^2}\delta^4(x_1-x_2)&=&(\frac{\pa}{\pa is})^n
e^{is\Box^2}\delta^4(x_1-x_2)\\
\Box^{2n+1} e^{is\Box^2}\delta^4(x_1-x_2)&=&(\frac{\pa}{\pa is})^n
\Box e^{is\Box^2}\delta^4(x_1-x_2)
\eea
As a result we find one-loop k\"{a}hlerian effective potential in the
form
\bea
\label{kdsg}
K^{(1)}=c\Lambda^{2/3}(\frac{16\pi^2}{Q^2})^{2/3}e^{\sigma+\bar{\sigma}}
\eea
where $c$ is a constant given in \cite{my7}.

(ii) chiral effective Lagrangian. To calculate it we put $\bar{\sigma}=0$
which leads to calculation of
\bea
\label{tudsg}
\tilde{U}(\sigma|s)=\exp[is(-9\Lambda e^{3\sigma})\frac{\bar{D}^2}{4}-
9\Lambda\frac{D^2}{4})] 
\eea
up to the second (leading) order is spinor derivatives of $\sigma$. 
We note that as a result one-loop effective action in leading order
takes the form
\bea
\label{gled}
\Gamma^{(1)}&=&\frac{i}{2}\int d^8 z \int_0^{\infty}\frac{ds}{s}
[A_1(\sigma|\Box)D^{\a}\sigma D_{\a}\sigma+A_2(\sigma|\Box) D^2\sigma+
A_3(\sigma|\Box)]\times\nonumber\\&\times&
U_0(x,x';s)|_{x=x'}
\eea 
After transforming of this expression to the form of integral over
$d^6z$
it is at least of second order in space-time derivatives of $\sigma$. 
Hence we can put all coefficients $A_1,A_2,A_3$ to depend only on
$\sigma$ but not on its derivatives. It means that
at the step of calculating
$\tilde{U}(\sigma|s)$ we can put all space-time derivatives of 
$\sigma,\bar{\sigma}$ to zero.
After calculation of $\tilde{U}(\sigma|s)$ (\ref{tudsg}) we get the
coefficients of Schwinger expansion $A(s),\tilde{A}(s)$
(note again that only they contribute to one-loop effective action, see
(\ref{kd})):
\bea
\tilde{A}(s)&=&\frac{1}{\Box}[\cosh(iWs)-1]\\
A(s)&=&-16h(-8h\phi(is)^2+64\frac{h^2}{W^2}(\frac{\sinh(iWs)}{W}-is)D^2\phi
+\nonumber\\&+&
8192\frac{h^4}{W^5}[iWs\cosh(iWs)-3\sinh(iWs)+2iWs]
D^{\a}\phi D_{\a}\phi
\eea
Here $\phi=e^{3\sigma}$, $h=-\frac{9}{4}\Lambda\frac{16\pi^2}{Q^2}$,
$W=16he^{3/2\sigma}\sqrt{\Box}$.
This expression for $A$ and $\tilde{A}$ can be substituted into
(\ref{gled}). After expansion of $A$ and $\tilde{A}$ into power
series,
transformation of the contribution to the form of integral over
$d^6z$ and integration over $s$ (see details in \cite{my7}),
one-loop leading chiral contribution to effective action takes the
form
\bea
\label{chdsg}
{\cal L}^{(1)}_c=\Lambda^{1/3}\Big[\{(c_1+3c_3)e^{-\sigma}+c_2
e^{2\sigma}+3c_4 e^{-4\sigma}\}\pa^m\sigma\pa_m\sigma+
(c_3e^{-\sigma}+c_4e^{-4\sigma})\Box\sigma\Big]
\eea
Here $c_1,c_2,c_3,c_4$ are finite constants given in \cite{my7}.
We find that the leaing chiral contribution to effective action
is of second order in space-time derivatives of chiral superfield
$\sigma$, therefore one-loop chiral effective potential is absent.

Then, if we sum classical action (\ref{actiondsg}) and leading
quantum corrections \\ (\ref{kdsg},\ref{chdsg}) we get one-loop corrected
effective action. If we put in it all derivatives of superfields to
zero we get the following low-energy leading effective action:
\bea
\Gamma=c(\frac{\Lambda}{Q^2})^{2/3}\int d^8 z e^{\sigma+\bar{\sigma}}
+(\Lambda\int d^6 z e^{3\sigma}+h.c.)
\eea
We remind that $\sigma=\log\Phi$, $\bar{\sigma}=\log\bar{\Phi}$ where 
$\Phi,\bar{\Phi}$ are chiral and antichiral supergravity prepotentials
(so called chiral compensators). Expression of this effective action
in terms of $\Phi,\bar{\Phi}$ gives
\bea
\Gamma=c(\frac{\Lambda}{Q^2})^{2/3}\int d^8 z \Phi\bar{\Phi}+
(\Lambda\int d^6 z \Phi^3 +h.c.)
\eea
This action has the structure similar to the classical action of
Wess-Zumino model. Hence we see that Wess-Zumino model is generated at
infrared limit of four-dimensional dilaton supergravity.

\section{Supergauge theories}
\setcounter{equation}{0}

This section is a brief review of results on supergauge theories. 
Unfortunately, restricted volume of this section does not allow to
discuss all essential results of last years in this sphere hence we
only give here main ones.

The starting point of our consideration is an action of ${\cal N}=1$
super-Yang-Mills theory:
\bea
\label{symact}
S_{SYM}=\frac{1}{4g^2}\int d^6 z {\rm tr} W^{\a}W_{\a}
\eea
where
\bea
W_{\a}=-\frac{1}{8}\bar{D}^2(e^{-2gV}D_{\alpha}e^{2gV}); V(z)=V^I(z) T^I
\eea
The $V(z)=V^I(z)T^I$ is a real scalar Lie-algebra-valued superfield.
We can expand the action (\ref{symact}) into power series in coupling
$g$. As a result we get
\bea
\label{gauge1}
S=\frac{1}{16}\int d^8 z {\rm tr} (V D^{\a}\bar{D}^2 D_{\a}V+\ldots)
\eea
Here dots denote higher orders in $g$.
The action (\ref{symact}) is invariant under gauge transformations
\bea
\label{gtr}
e^{2gV}\to e^{-2ig\bar{\L}}e^{2gV}e^{2ig\L}
\eea
where $\bar{D}_{\ad}\L=0$. The equivalent form of this transformation
\cite{BK0}
is
\bea
\label{gauge2}
\delta (gV)=L_{gV}(2ig\L-2ig\bar{\L}+{\rm coth}_{gV}(2ig\L+2ig\bar{\L}))
\eea
Here $L_{gV}A=[gV,A]$ is a Lie derivative.
It is easy to see that strengths $W_{\a},\bar{W}_{\ad}$ are invariant
under such transformations.
The leading order in (\ref{gauge2}) is
\bea
\delta(gV)=2ig(\Lambda-\bar{\Lambda})
\eea

Since the theory is gauge invariant we must introduce gauge-fixing
functions for quantization. The most natural form of them is (cf. section 8
where these gauge-fixing functions were used for calculation of
one-loop effective action in Wess-Zumino model)
\bea
\chi(V)&=&-\frac{1}{4}\bar{D}^2V+f(z)\\
\bar{\chi}(V)&=&-\frac{1}{4}D^2 V+\bar{f}(z)\nonumber
\eea
Here $f(z)$ is arbitrary chiral superfield.
Variation of these gauge fixing functions under transformations
(\ref{gauge2}) is
\bea
\label{vargau}
\delta\left(
\begin{array}{c}
\chi(V)\\
\bar{\chi}(V)
\end{array}
\right)
=
\left(
\begin{array}{cc}
0&-\frac{1}{4}\bar{D}^2\\
-\frac{1}{4}D^2& 0
\end{array}
\right)
\left(
\begin{array}{c}
g\L\\
g\bar{\L}
\end{array}
\right)
\eea
According to Faddeev-Popov approach we can introduce the ghost action
\bea
S_{GH}={\bf c}'\delta\chi|_{g\Lambda=c,g\bar{\Lambda}=\bar{c}}
\eea
i.e. parameters of transformation $g\L,g\bar{\L}$ are ghosts.
Here $\delta\chi\equiv\delta\left(
\begin{array}{c}
\chi(V)\\
\bar{\chi}(V)
\end{array}
\right)
$ from (\ref{vargau}), ${\bf c}'$ is a line $(c' \bar{c}')$ and since 
$\L$ is chiral $c,c'$ are also chiral ones. Here $c,c'$ are chiral
ghosts and $\bar{c},\bar{c}'$ are antichiral ones. As usual, ghosts
are fermions.

Therefore
\bea
S_{GH}=\int d^6 z {\rm tr} c'\frac{\delta\chi}{\delta V}\delta V+
\int d^6 \bar{z} {\rm tr} \bar{c}'\frac{\delta\bar{\chi}}{\delta
  V}\delta V
\eea
where 
\bea
\delta V=L_{gV}(c-\bar{c}+{\rm coth}_{gV}(c+\bar{c}))
\eea
Therefore the action of ghosts looks like
\bea
S_{GH}=\int d^8 z {\rm tr} (c'+\bar{c}')L_{gV}(c-\bar{c}+{\rm
  coth}_{gV} (c+\bar{c}))
\eea
Then, the generating functional for this theory at zero sources  
according to Faddeev-Popov approach looks like
\bea
\label{gf0}
Z[J]|_{J=0}=\int Dv D\{c\}e^{i(S_{SYM}+S_{GH})}
\delta_+(\frac{1}{4}\bar{D}^2V-f)\delta_-(\frac{1}{4}D^2V-\bar{f})
\eea
Here $D\{c\}\equiv Dc Dc' D\bar{c} D\bar{c}'$
We can average over functions $f$ and $\bar{f}$ with weight
\bea
\exp(\frac{i}{\xi}\int d^8 z (f\bar{f}+b\bar{b}))
\eea
where $\xi$ is a some number (gauge parameter). The $b,\bar{b}$ are
Nielsen-Kallosh ghosts (in this case their contribution to effective
action is a constant, but in background-covariant formulation it is
non-trivial). 
As a result (\ref{gf0}) takes the form
\bea
Z[J]_{J=0}=\int Dv D\{c\}e^{i(S_{SYM}+S_{GH}+S_{GF})}
\eea 
where
\bea
S_{GF}=\frac{1}{16\xi}\int d^8 z {\rm tr} (\bar{D}^2 V) (D^2 V)
\eea
is a gauge-fixing action \cite{BK0}.

We introduce total action
\bea
\label{total}
S_{total}=S_{SYM}+S_{GF}+S_{GH}
\eea
and generating functional
\bea
Z[J,\{\eta\}]&=&\int DV D\{c\}\exp(i(S_{total}+\int d^8 z {\rm tr} JV
+\int d^6 z {\rm tr}(\eta'c'+ \eta c)+\nonumber\\&+&\int d^6\bar {z}
(\bar{\eta}'\bar{c}'+\bar{\eta}\bar{c}) ))
\eea
Here $\{\eta\}$ is the set of all sources: $\eta,\eta',\bar{\eta},\bar{\eta}'$.
To develop diagram technique we must split action $S_{total}$ into
free (quadratic) part and vertices. It is easy to see (\cite{BK0,Kovacz})
that
\bea
e^{-2gV}D^{\a}e^{2gV}=2g D_{\a}V-2g^2 [V,D_{\a}V]+\frac{4}{3}
[V,[V,D_{\a}V]]+\ldots
\eea
Therefore (\ref{symact}) looks like
\bea
\label{ssym}
S_{SYM}&=&\int d^8 z {\rm tr}\Big(\frac{1}{16}V D^{\a} \bar{D}^2 D_{\a}
V+\frac{1}{8}g (\bar{D}^2 D^{\a} V)[V, D_{\a}V]-\nonumber\\&-&
\frac{1}{16}g^2
[V,D^{\a}V]\bar{D}^2[V,D_{\a}V]-\frac{1}{12}g^2 (\bar{D}^2D^{\a}V)
[V,[V,D_{\a}V]]+\ldots
\Big)
\eea
And ghost action is
\bea
\label{sgh}
S_{GH}=\int d^8 z {\rm tr}\Big(\bar{c}'c-\bar{c}c'+
g(c'+\bar{c}')[V,c-\bar{c}]+\frac{g^2}{3}(c'+\bar{c}')[V,[V,c+\bar{c}]]
\Big)
\eea
This expansion is enough in one- and two-loop calculations.

The quadratic action is
\bea
S_0=\frac{1}{2}\int d^8 z {\rm tr}V\Big(-\Box
+\frac{1}{16}(1+\frac{1}{\chi}\{D^2,\bar{D}^2\})
\Big)V+\int d^8 z {\rm tr}(\bar{c}'c-\bar{c}c')
\eea
Vertices can be read off from (\ref{ssym},\ref{sgh}).
Propagators look like
\bea
<V(z_1)V(z_2)>&=&-\frac{1}{\Box}\Big(-\frac{1}{8\Box}D^{\a}\bar{D}^2
D_{\a}+\xi\frac{\{D^2,\bar{D}^2\} }{16\Box}
\Big)\delta^8(z_1-z_2)\\
<\bar{c}'(z_1)c(z_2)>&=&<c'(z_1)\bar{c}(z_2)>=\frac{1}{\Box}
\delta^8(z_1-z_2)\nonumber
\eea
We note that ghosts are fermions hence any ghost loop corresponds to
minus sign. Then, $D$-factors are associated with vertices containing
ghosts just by the same rule as with vertices containing any chiral 
superfields.
We note that if we choose $\xi=-1$ (Feynman gauge) the propagator of
gauge superfield takes the simplest form
\bea
\label{pfe}
<v(z_1)v(z_2)>=\frac{1}{\Box}\delta^8(z_1-z_2)
\eea
Note that its sign is opposite to the the sign of propagator of chiral 
superfield.

If we want to introduce interaction of a chiral superfield with the gauge
one the quadratic part in $\Phi,\bar{\Phi}$ looks like
\bea
\label{chint}
S=\int d^8 z \bar{\Phi}_i (e^{gV})^i_j\Phi^j
\eea
if chiral superfield $\Phi_i$ is transformed under some representation
of the gauge group (i.e. it is an isospinor) or
\bea
S=\int d^8 z {\rm tr} (\bar{\Phi}e^{-gV}\Phi e^{gV})
\eea
if chiral superfield $\Phi=\Phi^a T^a$ is Lie-algebra-valued. Note
that under gauge transformations (\ref{gtr}) the chiral superfield is
transformed as
\bea
\Phi\to e^{-2ig\L}\Phi
\eea
for isospinor chiral superfield and as
\bea
\Phi\to e^{-2ig\L}\Phi e^{2ig\L}
\eea
for Lie-algebra-valued chiral superfield. Note that $\L,\bar{\L}$ are 
Lie-algebra-valued in both cases.
The vertices can be easily obtained by expanding into power series
expressions corresponding to interaction: in first case
\bea 
\int d^8 z [\bar{\Phi}_i (e^{gV})^i_j\Phi^j-\Phi_i\bar{\Phi}^i]=
\int d^8 z \sum_{n=1}^{\infty}\frac{1}{n!}\bar{\Phi}_i {(gV^n)}^i_j\Phi^j
\eea
and in second one --
\bea
\int d^8 z ({\rm tr} (\bar{\Phi}e^{-gV}\Phi e^{gV})-\bar{\Phi}\Phi)=
\int d^8 z {\rm tr}(\Phi[V,\bar{\Phi}]+\bar{\Phi}[V,\Phi]+\frac{1}{2}
[\Phi,[V,[V,\bar{\Phi}]]+\ldots)
\eea

The diagram technique derived now is very suitable for calculations in
sector of background $\Phi,\bar{\Phi}$ only and for calculation of 
divergences.

Example:

consider the theory with action (see f.e. \cite{Gio})
\bea
S&=&{\rm tr}\frac{1}{g^2}\int d^6 z W^{\a} W_{\a} +{\rm tr} \int d^8 z
\bar{\Phi}e^{-gV}\Phi e^{gV}+\nonumber\\&+&
\sum_{i=1}^n\Big[ig(\int d^6 z Q_i \Phi
\tilde{Q}_i+h.c.)+\int d^8 z \bar{\tilde{Q}}_i e^{-gV}\tilde{Q}_i
+\int d^8 z \bar{Q}_i e^{gV} Q_i
\Big]
\eea
Here $\Phi$ is Lie-algebra-valued chiral superfield, and $Q^i,\tilde{Q}_i$
are chiral superfields transformed under mutually conjugated
representations of Lie algebra. They are often called matter hypermultiplets.
Let us consider the structure of one-loop divergences in this theory. For
simplicity we choose Feynman gauge $\xi=-1$ in which the 
propagator has the most
simple structure (\ref{pfe}), therefore all tadpole diagrams given in
\cite{Kovacz} evidently vanish.

First we consider contributions to wave function renormalization of
$\Phi$ field

\vspace*{2mm}

\Lengthunit=1cm
\hspace*{3cm}\GRAPH(hsize=3)
{\Linewidth{1.2pt}\Circle(2)\Linewidth{.5pt}\mov(-1,0){\lin(-.5,0)}
\mov(.9,0){\lin(.5,0)}}
\hspace*{6cm}\GRAPH(hsize=3)
{\halfcirc(2)[u]\halfwavecirc(2)[d]\mov(-1,0){\lin(-.5,0)}
\mov(.9,0){\lin(.5,0)}
}

\vspace*{1mm}

Here thin line is propagator of $\Phi$, thick -- of hypermultiplets 
$Q,\tilde{Q}$, wavy -- of real superfield $v$, dashed -- from ghosts.

One-loop divergent contributions from these supergraphs are
respectively
\bea
2\sum_M\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2(k+p)^2}\Phi^a\bar{\Phi}^b {\rm
  tr}_M
(T^a T^b)
\eea
and
\bea
-\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2(k+p)^2}\Phi^a\bar{\Phi}^b
{\rm tr}_{ad} (T^a T^c T^d){\rm tr}_{ad} (T^b T^c T^d)
\eea
Here $tr_M$ denotes trace in representation under which
hypermultiplets are transformed. Coefficient 2 is caused by presence
of
two chiral hypermultiplets $Q$ and $\tilde{Q}$.
We see that if $\sum_M{\rm tr}_M(T^a T^b)=
{\rm tr}_{ad} (T^a T^c T^d){\rm tr}_{ad} (T^b T^c T^d)$
there is no divergent contributions to wave function renormalization.

Corrections to hypermultiplet wave function look like

\vspace*{1mm}

\Lengthunit=1cm
\hspace*{3cm}\GRAPH(hsize=3)
{\Linewidth{.5pt}\halfcirc(2)[u]\Linewidth{1.2pt}\halfcirc(2)[d]
\mov(-1.1,0){\lin(-.5,0)}\mov(.9,0){\lin(.5,0)}\Linewidth{.5pt}
}
\hspace*{6cm}\GRAPH(hsize=3)
{\halfwavecirc(2)[u]\Linewidth{1.2pt}\halfcirc(2)[d]
\mov(-1.1,0){\lin(-.5,0)}\mov(.9,0){\lin(.5,0)}\Linewidth{.5pt}
}

\vspace*{1mm}

One-loop divergent contributions from these supergraphs are
respectively
\bea
-\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2(k+p)^2}\bar{Q}_i Q_l
(T^a)^{ij} (T^a)^{jl}
\eea
and
\bea
\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2(k+p)^2}\bar{Q}_i Q_l
(T^a)^{ij} (T^a)^{jl}
\eea
These corrections evidently cancel each other, hence there is no
renormalization of hypermultiplet wave function.
In both these cases cancellation is caused by the difference in signs of
propagators of gauge superfield and chiral superfields. 

One-loop contributions to wave function renormalization for gauge
superfields are 

\vspace*{2mm}

\Lengthunit=1cm
\hspace*{3cm}\Linewidth{.5pt}
\GRAPH(hsize=3)
{\Circle(2)\mov(-1,0){\wavelin(-.5,0)}
\mov(1,0){\lin(.5,0)}}
\hspace*{6cm}\GRAPH(hsize=3)
{\wavecirc(2)\mov(-1,0){\wavelin(-.5,0)}
\mov(1,0){\wavelin(.5,0)}
}

\vspace*{3mm}

\Lengthunit=1cm
\hspace*{3cm}\GRAPH(hsize=3)
{\Linewidth{1.2pt}\Circle(2)\Linewidth{.5pt}\mov(-1,0){\wavelin(-.5,0)}
\mov(1,0){\wavelin(.5,0)}}
\hspace*{6cm}\GRAPH(hsize=3)
{\dashcirc(2)\mov(-1,0){\wavelin(-.5,0)}\mov(1,0){\wavelin(.5,0)}}

\vspace*{2mm}

Contribution of these four supergraphs are respectively
\bea
& &\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2(k+p)^2} V^a V^b
{\rm tr}_{ad} (T^a T^c T^d) {\rm tr}_{ad} (T^b T^c T^d)\nonumber\\
& &\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2(k+p)^2} V^a V^b{\rm tr}_{ad} 
(T^a T^c T^d) {\rm tr}_{ad} (T^b T^c T^d)\nonumber\\
2\sum_M& &\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2(k+p)^2} V^a V^b{\rm tr}_M 
(T^a T^b)\nonumber\\
-&4&\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2(k+p)^2} V^a V^b{\rm tr}_M 
(T^a T^b)\nonumber\\
\eea
Hence if $2\sum_M{\rm tr}_M(T^a T^b)+
2{\rm tr}_{ad} (T^a T^c T^d) {\rm tr}_{ad} (T^b T^c T^d)=4{\rm
  tr}_{ad} (T^a T^b)$ 
(or, as is the same with taking into account
condition for $<\phi\bar{\phi}>$-propagator, 
${\rm tr}_{ad} (T^a T^c T^d) {\rm tr}_{ad} (T^b T^c T^d)={\rm
  tr}_{ad} (T^a T^b)$)
these contributions are cancelled and there is
no divergent correction to $<VV>$-propagator, and the theory is
finite. This mechanism of vanishing divergences is discussed, f.e. in
\cite{SGRS}, \cite{HST}, where it is shown to be connected with $N=2$ 
superconformal symmetry.
The most important example of such theories is $N=4$ super-Yang-Mills
theory where we have one pair of hypermultiplets $Q,\tilde{Q}$ and
they are transformed under the adjoint representation of Lie algebra. 

However, the approach developed until this place is very useful for
consideration of divergences and corrections in the sector of chiral
superfields $\Phi,Q,\tilde{Q}$ only. To study contributions depending
on gauge superfields we must develop a method allowing to preserve
manifest gauge invariance at any step as earlier we obtained
contributions in terms of superfield $V$ which are in general case not
gauge invariant. Therefore we must introduce an approach in which
external lines are background strengths $W_{\a},\bar{W}_{\ad}$ and
their {\bf gauge covariant} derivatives.
This method was developed in \cite{GRS} (see also \cite{GZ} and
references therein), here we give its short description.

Starting point is background-quantum splitting in (\ref{symact}). 
To preserve gauge
invariance we carry out it in the following fashion.
First, $e^{gV}$ can be represented as $e^{g\Omega}e^{g\bar{\Omega}}$
where $\Omega$ is a some superfield. Then we can carry out splitting
$e^{g\Omega}\to e^{g\Omega}e^{g\omega}$ with
$e^{g\omega}e^{g\bar{\omega}}=e^{gv}$ where $v$ is a quantum
superfield.
It leads to the following expression for background-quantum splitting
for $V$:
\bea
e^{gV}\to e^{g\Omega}e^{gv}e^{g\bar{\Omega}}
\eea
Here and further $\Omega,\bar{\Omega}$ are background superfields.
We can introduce covariant derivatives as
$\nabla_{\a}=e^{-g\Omega}D_{\a}e^{g\Omega}$ (derivative
$D_{\a}$ here acts on all on the right). We note that
$\nabla_{\a}$ can be represented as $\nabla_{\a}=D_{\a}+i\Gamma_{\a}$.
Here $\Gamma_{\a}=-ie^{-g\Omega}(D_{\a}e^{g\Omega})$ is a connection.
And $\Gamma_{\a\ad}=D_{\a}\bar{\Gamma}_{\ad}-\bar{D}_{\ad}\Gamma_{\a}$.
Then we can introduce the covariantly chiral superfield $\tilde{\Phi}=
e^{g\bar{\Omega}}\Phi$.
As a result
(\ref{symact}) with chiral matter coupled to gauge superfield in a
fashion given by (\ref{chint}) takes the form:
\bea
\label{ymquant}
S&=&-\frac{1}{16g^2}{\rm tr}\int d^8 z (e^{-gv}\nabla^{\a}e^{gv})
\bar{\nabla}^2(e^{-gv}\nabla_{\a}e^{gv})+\int d^8 z \bar{\tilde{\Phi}}
e^{gv}\tilde{\Phi}+\nonumber\\&+&[\frac{1}{2}\int d^6 z m\Phi^2+h.c.]
\eea
%Here $W(\Phi)$ is a self-interaction of chiral superfield.
Background-quantum splitting for $\tilde{\Phi}$ can be carried out by
a standard way $\tilde{\Phi}\to\Phi_0+\phi$ where now both 
$\Phi_0$ and $\phi$ are covariantly chiral. Further in this section we
denote $\tilde{\Phi}$ as $\Phi$.

The action (\ref{ymquant}) is invariant under transformations \cite{GRS}
\bea
e^{gv}\to e^{ig\bar{\L}}e^{gv}e^{-ig\L};\ \Phi\to e^{ig\L}\Phi.
\eea
Therefore we must introduce gauge-fixing action 
\bea
S_{GF}=-\frac{1}{16}\int d^8 z (\nabla^2 v)(\bar{\nabla}^2 v)
\eea
The action of arisen Faddeev-Popov ghosts can be obtained in analogy
with formulation of super-Yang-Mills theory described above. It is
\bea
S_{FP}=
{\rm tr}\int d^8 z (c'-\bar{c}')L_{gV}[c+\bar{c}+{\rm coth} L_{gV}(c-\bar{c})]
\eea
Averaging over gauges leads to appearing of Nielsen-Kallosh ghosts with
action
\bea
S_{NK}={\rm tr}\int d^8 z b\bar{b}
\eea
The quadratic action in quantum superfields as a result takes the form
\bea
S_2&=&\int d^8 z
\Big[\frac{1}{2}v(\nabla^a\nabla_a-W^{\a}\nabla_{\a}-\bar{W}_{\ad}
\bar{\nabla}^{\ad}+g^2\Phi\bar{\Phi})v+b\bar{b}+\bar{c}'c+c'\bar{c}+b\bar{b}+
\phi\bar{\phi}\Big]+\nonumber\\&+&[\int d^6 z
\frac{1}{2}m\phi^2+
h.c.]
\eea
Here we took into account that
$$\frac{1}{8}\nabla^{\a}\bar{\nabla}^2\nabla_{\a}-\frac{1}{16}\{\nabla^2,
\bar{\nabla}^2\}=
\nabla^a\nabla_a-W^{\a}\nabla_{\a}-\bar{W}_{\ad}\bar{\nabla}^{\ad}
$$
Propagators look like
\bea
\label{prb}
<v(z)v(z')>&=&
[\nabla^a\nabla_a-W^{\a}\nabla_{\a}-
\bar{W}_{\ad}\bar{\nabla}^{\ad}]^{-1}\delta^8(z-z')\nonumber\\
<\phi(z)\bar{\phi}(z')>&=&-(\Box_++m^2)^{-1}\delta^8(z-z')\nonumber\\
<\phi(z)\phi(z')>&=&-m\nabla^2(\Box_+(\Box_+-m^2))^{-1}\nonumber\\
<\bar{\phi}(z)\bar{\phi}(z')>&=&-m\bar{\nabla}^2(\Box_-(\Box_--m^2))^{-1}
\eea
In analogy with standard supergraph technique at any vertex
$\bar{\phi}$ and $\phi$ are associated
with factors $\nabla^2,\bar{\nabla}^2$ respectively. 
Here 
\bea
\Box_+=\nabla^a\nabla_a-W^{\a}\nabla_{\a}-\frac{1}{2}(\nabla^{\a}W_{\a});\
\Box_-=\nabla^a\nabla_a-\bar{W}^{\ad}\bar{\nabla}_{\ad}-
\frac{1}{2}(\bar{\nabla}^{\ad}\bar{W}_{\ad})
\eea
We can introduce commutation relations \cite{GZ}
\bea
\{\nabla_{\a},\bar{\nabla}_{\ad}\}&=&i\nabla_{\a\ad};\
[\nabla_{\a},\nabla_{\b\bd}]=i\epsilon_{\a\b}\bar{W}_{\bd}\nonumber\\
\nabla^{\a}W_{\a}&=&\bar{\nabla}_{\ad}\bar{W}^{\ad};\
\nabla^2\Box_+\bar{\nabla}^2=\Box_-\nabla^2\bar{\nabla}^2;\
\bar{\nabla}^2\nabla^2\bar{\nabla}^2=\Box_+\bar{\nabla}^2
\eea
Some examples of application of background field method to calculation of
correction in pure super-Yang-Mills sector are given in
\cite{GZ}. 
This method was used also for calculating of one-loop effective action
in $N=4$ super-Yang-Mills theory \cite{BKT}. 
Unfortunately, application of this method for calculating
higher loop corrections depending both on background gauge superfields and 
background chiral ones being based in principle on action (\ref{ymquant})
and propagators (\ref{prb}) leads to very cumbersome expressions. 

\section{Conclusion}
We considered superfield method in supersymmetric field theory. This
method allows to preserve manifest supersymmetry at any step of
calculations, and it is much more compact that component approach.

We studied several examples of superfield theories and described
quantum calculations for them. These examples were Wess-Zumino model,
general chiral superfield model, dilaton supergravity and $N=1$
super-Yang-Mills theory with chiral matter. In these theories we
developed supergraph technique, studied general form of superfield
effective action and calculated low-energy leading contributions
to effective action. It is natural to expect that development of
superfield quantum calculations in other  superfield models formulated in
terms of $N=1$ superfields including different supergravity models is
in principle no more difficult.

Let us briefly discuss further prospects of superfield quantum theory.
The most important ways of development of superfield supersymmetry at present
time are:

1. {\bf Studying of theories with extended supersymmetry}. It is known that
theories with extended supersymmetry possess better renormalization
properties, f.e. as $N=1$ super-Yang-Mills theory is renormalizable,
the $N=4$ super-Yang-Mills theory is finite. 
The most important examples of theories with extended supersymmetry
are $N=2$ and $N=4$ super-Yang-Mills theories. During last years
numerous results in studying of these theories were obtained (see f.e.
\cite{BKT,bug1,bug2} and references therein).

It is natural to expect that the most adequate method for
consideration of such theories must possess manifest $N=2$
supersymmetry. Such a method is a harmonic superspace approach
developed in \cite{GIKOS1},\cite{GIKOS2}. 
This method is based on consideration of superfields being the
functions of bosonic space-time coordinates $x^a$, two sets of
Grassmann coordinates $\theta^{i\a},\bar{\theta}^{i\ad}$ with 
$i=1,2$ and spherical harmonics $u^{\pm i}$. Introducing of analytic
superfield \cite{GIKOS1} allows to develop formulation in terms of
unconstrained $N=2$ superfield and to avoid arising of component
fields with higher spins. The formulations of $N=2$ and
$N=4$ super-Yang-Mills theories in harmonic superspace is given
in \cite{GIKOS1,GIKOS2}, background field method for these theories is
developed in \cite{bbiko,bbko,bko}, and examples of quantum
calculations are given in \cite{bug1,bug2,bs,km,kt}. The most important results
are: calculation of holomorphic action of $N=2$ matter hypermultiplets
in external $N=2$ gauge superfield, calculation of one-loop
nonholomorphic effective potential in $N=4$ super-Yang-Mills theory
and proof of its absence in higher loops, calculation of one-loop
effective action for $N=4$ super-Yang-Mills theory for constant
strength tensor $F_{ab}$, calculation of superconformal anomaly of
$N=2$ matter interacting with $N=2$ supergravity.   

Further development of this approach is, first, in calculation of
contributions depending on derivatives of $N=2$ super-Yang-Mills
strength ${\cal W}$, second, in calculation of contributions depending
on background matter hypermultiplet fields, third, in further
development of harmonic superspace approach for $N=2$ supergravity and
$N=3$ super-Yang-Mills theory (formulation of harmonic superspace
approach for $N=3$ supersymmetric theory was given in paper \cite{Del}
but it did not get further development).

2. {\bf Noncommutative supersymmetric theories}. Noncommutative
theories have been intensively studied during last years. Concept of
space-time noncommutativity was introduced to quantum field theory due
to some consequences of D-branes theory \cite{SW2} and to
consideration of quantum theories on very small distances where
quantum fluctuations of geometry are essential. Consideration of
supersymmetric noncommutative theories is quite natural. During last
years some interesting results in studying of noncommutative
supersymmetric theories were obtained but they were mostly based on
component approach. Superfield results until this time are only
calculation of leading ($\sim F^4$) correction to one-loop effective
action for $N=4$ super-Yang-Mills theory \cite{Zan} and introduction of
supergraph technique for noncommutative Wess-Zumino model \cite{Popp}.
Therefore one sees that there are a lot of open questions in
superfield noncommutative theories, f.e. development of quantum
formulation for noncommutative analogs of all theories considered
here. Superfield noncommutative theory, thus, seems to be a very
promising sphere.

Then, there are a lot of applications of superfields approach to
problems of supersymmetric quantum field theory (f.e. to studying of
AdS/CFT correspondence which was carried out mostly on base of
component approach), and of course consideration of many problems
originated from superstrings and branes theory.

It allows us to suppose that superfield approach in quantum field
theory is a very perspective one, and there are a lot of ways for its
development and more applications. 



{\large\bf Acknowledgements.}
Author is grateful to Prof. V.O. Rivelles and Prof. M.O.C. Gomes for
discussions, to Prof. I.L. Buchbinder, Prof. M. Cvetic,
Dr. S.M. Kuzenko  
for collaboration, and to Instituto de F'{i}sica, Universidade de S\~{a}o Paulo
for hospitality. The work was supported by FAPESP, project No. 00/12671-7. 
 

\begin{thebibliography}{100}
\bibitem{GSH} M. Green, J. Schwarz, E. Witten. Theory of
  superstrings. NY., 1991.
\bibitem{VA} D.V. Volkov, V.P. Akulov. JETP Lett., 16, 621 (1972);
  Phys. Lett. B46, 109 (1973).
\bibitem{GL} Y.A. Golfand, E.S. Lichtman. JETP Lett., 13, 323 (1971).
\bibitem{WZ} J. Wess, B. Zumino. Nucl. Phys. B70, 139 (1974);
  Nucl. Phys. B78, 1 (1974).
\bibitem{WZ1} J. Wess, B. Zumino. Phys. Lett. B49, 52 (1974).
\bibitem{SS} A. Salam, J. Strathdee. Nucl. Phys. B76, 477 (1974).
\bibitem{FWZ} S. Ferrara, J. Wess, B. Zumino. Phys. Lett. B51, 239 (1974).
\bibitem{WB} J. Wess, J. Bagger. Supersymmetry and supergravity.
Princeton University Press, Princeton, 1983.
\bibitem{BK0} I.L. Buchbinder, S.M. Kuzenko. Ideas and Methods of
Supersymmetry and Supergravity. IOP Publishing, Bristol and
  Philadelphia, 1998.
\bibitem{OM} V. Ogievetsky, L. Mezinchesku. Uspehi Fiz. Nauk
(Phys. Sci. Achievments) 117, 637 (1975) (in Russian).
\bibitem{my2} I.L. Buchbinder, A.Yu. Petrov. Yad. Fiz. (Phys. Atom. Nucl.)
63, 9, 2557 (2000).
\bibitem{my1} I.L. Buchbinder, A.Yu. Petrov. Class. Quant. Grav.
13, 2081 (1996).
\bibitem{SW} N. Seiberg, E. Witten. Nucl. Phys. B430, 19 (1994).
\bibitem{West} P. West. Introduction to supersymmetry and
  supergravity. World Scientific, 1989.
\bibitem{West2} P. West. Phys. Lett. B258, 375 (1991).
\bibitem{Jack} I. Jack, D.R.T. Jones. Problems of supersymmetric
  regularization. {\it Preprint} hep-ph/9707207.
\bibitem{Collins} J. Collins. Renormalization. N.Y., 1982. 
\bibitem{West3} I. Jack, D.R.T. Jones, P. West. Phys. Lett. B258, 382
  (1991).
\bibitem{West4} P. West. Phys. Lett. B261, 396 (1991).
\bibitem{my3} I.L. Buchbinder, S.M. Kuzenko,
  A.Yu. Petrov. Phys. Lett. B321, 372 (1994).
\bibitem{my4} I.L. Buchbinder, S.M. Kuzenko, A.Yu. Petrov. Yad. Fiz.
(Phys. Atom. Nucl.), 59, 1, 157 (1996).
\bibitem{BO} I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro. Effective
  action in quantum gravity. IOP Publishing, Bristol and Philadelphia,
  1992.
\bibitem{BSD} B.S. De Witt. Dynamical theory of groups and fields.
Princeton Univ. Press,, Princeton, 1987.
\bibitem{Buch1} I.L. Buchbinder, S.M. Kuzenko,
  J.V. Yarevskaya. Nucl. Phys. B411, 665 (1994);
  Yad. Fiz. (Phys. Atom. Nucl), 56, 5, 202 (1993).
\bibitem{McA} I..N. McArthur, T.D. Gargett. Nucl. Phys. B497, 525 (1997).
\bibitem{CW} S. Coleman, S. Weinberg. Phys. Rev. D7, 1888 (1973).
\bibitem{GRS} M.T. Grisaru, M. Rocek, W. Siegel. Nucl. Phys. B159, 429
  (1979).
\bibitem{PW} A. Pickering, P. West. Phys. Lett. B353, 54 (1997).
\bibitem{dis} A.Yu. Petrov. Superfield effective action in
  supersymmetric field theories (PhD thesis), Tomsk, 1997.
\bibitem{KuzYar} S.M. Kuzenko,
  J.V. Yarevskaya. Yad. Fiz. (Phys. Atom. Nucl.) 56, 5, 195 (1993).
\bibitem{Cvet}
G. Cleaver, M. Cveti\v{c}, J.R.
Espinosa, L. Everett, and P. Langacker,  Nucl. Phys. B525, 3, 1998;
Phys. Rev, D59, 115003, 1999;
M. Cveti\v{c}, L. Everett, and J. Wang. Nucl. Phys. B538, 52, 1999.
\bibitem{my5} A.Yu. Petrov. {\it Preprint} hep-th/0002013.
\bibitem{Buch5} I.L. Buchbinder, A.Yu. Petrov. Phys.Lett. B461, 209 (1999).
\bibitem{Syma} K. Symanzik. Commun. Math. Phys, 34, 7 (1973);
T. Appelquist and J. Carrazone, Phys. Rev. D11, 2856 (1975).
\bibitem{my6} I.L. Buchbinder, M. Cvetic,
  A.Yu. Petrov. Mod.Phys. Lett. A15, 783 (2000);
Nucl. Phys. B571, 358 (2000).
\bibitem{anom} I.L. Buchbinder, S.M. Kuzenko. Phys. Lett. B202, 233 (1988).
\bibitem{my7} I.L. Buchbinder, A.Yu. Petrov. Class. Quant. Grav. 14,
21 (1997).
\bibitem{Kovacz} S. Kovacs. {\it Preprint} hep-th/9902047. 
\bibitem{Gio} A. De Giovanni, N.T. Grisaru, D. Zanon. Phys. Lett. B409,
251  (1997).
\bibitem{SGRS} S.J. Gates, M.T. Grisaru, M. Rocek,
  W. Siegel. Superspace or One Thousand and One Lectures in
  Supersymmetry, Benjamin/Cummings, 1983.
\bibitem{HST} P.S. Howe, K. Stelle, P. West. Phys. Lett. B124,
  55 (1983). 
\bibitem{GZ} M.T. Grisaru, D. Zanon. Nucl. Phys. B252, 578 (1985). 
\bibitem{BKT} I.L. Buchbinder, S.M. Kuzenko,
  A.A. Tseytlin. Phys. Rev. D62, 045001 (2000).
\bibitem{bug1} I.L. Buchbinder, S.M. Kuzenko. Phys. Lett. B446, 216
(1999); Mod. Phys. Lett. A13, 1623 (1998). 
\bibitem{bug2} I.L. Buchbinder, A.Yu. Petrov. Phys. Lett. B469, 482
(2000).
\bibitem{GIKOS1} A. Galperin, E.Ivanov, S. Kalitzin, V. Ogievetsky,
E. Sokachev. Class. Quant. Grav. 1, 469 (1984).
\bibitem{GIKOS2} A. Galperin, E. Ivanov, V. Ogievetsky, E. Sokachev.
Class. Quant. Grav. 2, 601 (1985); 2, 617 (1985).
\bibitem{bbiko} 
E.I. Buchbinder, I.L. Buchbinder, E.A. Ivanov,
S.M. Kuzenko, B.A. Ovrut. Phys. Lett. B412, 309 (1997). 
\bibitem{bbko} E.I. Buchbinder, I.L. Buchbinder,
S.M. Kuzenko, B.A. Ovrut. Phys. Lett. B417, 61 (1997).
\bibitem{bko} I.L. Buchbinder, S.M. Kuzenko, B.A. Ovrut.
Phys. Lett. B433, 335 (1997).
\bibitem{bs} I.L. Buchbinder, I.B. Samsonov. Mod. Phys. Lett. A14,
2537 (1999).
\bibitem{km} S.M. Kuzenko, I.N. McArthur. Phys. Lett. B506, 140 (2001).
\bibitem{kt} S.M. Kuzenko, S. Theisen. Class. Quant. Grav. 17, 665 (2000).
\bibitem{Del} F. Delduc, I. McCabe. Class. Quant. Grav. 6, 233 (1989).
\bibitem{SW2} N. Seiberg, E. Witten. {\it Preprint} hep-th/9908142.
\bibitem{Zan} D. Zanon. {\it Preprint} hep-th/0009196, 0001140, 0010275.
\bibitem{Popp} A.A. Bichl et al. {\it Preprint} hep-th/0007050.
\end{thebibliography}
\end{document}


