%&latex209
\documentstyle[12pt]{article}
\textheight=225mm
\begin{document}
\begin{flushright}
Preprint SSU-HEP-99/07\\
Samara State University
\end{flushright}

\vspace{30mm}
\immediate\write16{<<WARNING: LINEDRAW macros work with emTeX-dvivers
		    and other drivers supporting emTeX \special's
		    (dviscr, dvihplj, dvidot, dvips, dviwin, etc.) >>}
\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
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{\rotate(\k**)\mov(#1,0){\sm*}}\advance\N*-1\ifnum\N*>0\repeat
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\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**
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\d**=#2\Lengthunit \d*\d** \divide\d*57\multiply\d*\n*
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\m**\r* \divide\m**2 \l**\r* \advance\l**-\m** \N*\n* \divide\N*\r*
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\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}}}

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\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{center}
{\bf HADRONIC VACUUM POLARIZATION CONTRIBUTION\\
TO THE LAMB SHIFT IN MUONIC HYDROGEN}\footnote{Talk presented at the
XIVth International Workshop on High Energy Physics and Quantum Field
Theory, Moscow, Russia, 27 May - 2 June 1999}\\

\vspace{4mm}

R.N.~Faustov \\Scientific Council "Cybernetics" RAS\\
117333, Moscow, Vavilov, 40, Russia,\\
A.P.~Martynenko\\ Department of Theoretical Physics, Samara State University,\\
443011, Samara, Pavlov, 1, Russia
\end{center}

\begin{abstract}
The contribution of hadronic vacuum polarization to the Lamb shift in
muonic hydrogen is evaluated with the account of modern experimental data
on the cross section of $e^+e^-$ annihilation into hadrons. The numerical value
of this contribution to the (2P-2S) shift in muonic hydrogen is equal to
10.77 $\mu eV$.
\end{abstract}

\newpage

In recent years, experimental accuracy of the hydrogen - like atom (hydrogen
atom, positronium, muonium, et. al. ) energy level
measurements as well as of the anomalous magnetic moments of electron and muon
was essentially increased. In some cases this accuracy reached such a level,
which requires further theoretical investigations of the strong and even weak
interaction contributions. The hadronic vacuum polarization contribution
(HVP) to the muon anomalous magnetic moment, which was calculated by
different authors \cite{HK,KNO,EJ}, makes the value
\begin{equation}
a_\mu^{\rm HVP}=(6771\pm 96)\times 10^{-11}.
\end{equation}
The experimental precision in the measurement of $a_\mu$ is expected to be
improved by the E821 experiment at Brookhaven National Laboratory to the
level $(1\div 2) \times 10^{-10}$. The precision of the muonium hyperfine
splitting measurement
%will be
was raised up to the order of $10^{-2}$ kHz in the
new Los Alamos experiment \cite{L}. This suggests the calculation of the higher order
contributions to the hyperfine structure as well as the contribution of
hadronic vacuum polarization, which takes the value \cite{FKM}:
\begin{equation}
\Delta E^{\rm HVP}_{\rm hfs}(\mu e)=0.2397\pm 0.0070~~~kHz.
\end{equation}

At present time there are attempts to realize the measurement of the $2P_{1/2}
-2S_{1/2}$ Lamb shift in muonic hydrogen to about 10 ppm level.
Such high accuracy Lamb shift measurement in the muonic hydrogen
may become one more task, permitting the experimental
verification of the HVP contribution \cite{P}. The energy levels of system
$(\mu p)$ are defined in the same way as for electronic hydrogen. By virtue of the fact that
the electron to muon mass ratio $m_e/m_\mu = 4.836332\cdot 10^{-3}$,
some quantum electrodynamical corrections, connected with the vacuum polarization
(including hadronic vacuum polarization) increase substantially in the case of
muonic hydrogen. High precision measurement of the (2P-2S) Lamb shift in muonic
hydrogen makes it possible to determine with higher accuracy the proton
charge radius $R_{\rm p} = \sqrt{<\rm r^2>}$ \cite{K}.

\begin{figure}
\magnitude=2000
\GRAPH(hsize=15){
\mov(3,0){\lin(5,0)}%
\mov(3,3){\lin(5,0)}%
\mov(5.5,0){\wavelin(0,1)}%
\mov(5.5,3){\wavelin(0,-1)}%
\mov(5.5,1.5){\Circle*(1)}%
\mov(2.5,0){p}%
\mov(2.5,3){$\mu$}%
}
\vspace{3mm}
\caption{Contribution of HVP to the one-photon exchange interaction
in $(\mu p)$.}
\end{figure}

In the framework of the quasipotential method the interaction operator
in the system $(\mu p)$, accounting for $1\gamma$- and $2\gamma$- processes,
takes the form \cite{MF1}:
\begin{equation}
V=V_{1\gamma}+V_{2\gamma}=V^c+\Delta V,
\end{equation}
\begin{equation}
V_{1\gamma}=T_{1\gamma},~~~V_{2\gamma}=T_{2\gamma}-T_{1\gamma}\times
G^f\times T_{1\gamma},
\end{equation}
where $V^c$ is the Coulomb potential, $T_{1\gamma}$, $T_{2\gamma}$ are the
one- and two-photon scattering amplitudes off the energy shell,
$[G^f]^{-1}=(b^2-{\vec p}^2)/2\mu_R$ is the free two-particle Green function.
The dominant HVP contribution to the Lamb shift of muonic hydrogen is
determined by the diagram of Fig.1. Then the corresponding correction
to the muon-proton quasipotential in the configurational space can be
expressed in the following way \cite{FGK}:
\begin{equation}
\Delta V^{\rm HVP}_{\rm Ls}(\vec r)=-4\alpha(Z\alpha)\int_{4m_\pi^2}^\infty
\frac{\rho(s)ds}{s}\cdot \delta (\vec r),
\end{equation}
where the spectral function $\rho(s)$ is related to the known cross section
of $e^+e^-$ annihilation into hadrons $\sigma^h$:
\begin{equation}
\rho(s)=\frac{R(s)}{3s}=\frac{\sigma^h(e^+e^-\rightarrow hadrons)}{3s\sigma_
{\mu\mu}(e^+e^-\rightarrow \mu^+\mu^-)},
\end{equation}
and $\sigma_{\mu\mu}(e^+e^-\rightarrow \mu^+\mu^-)=4\pi\alpha^2/3s$ is
annihilation cross section to a muon pair. The S-level energy shift can be
obtained under averaging of (5) over Coulomb wave functions in the kind:
\begin{equation}
\Delta E^{\rm HVP}_{\rm Ls}=-\frac{4\alpha(Z\alpha)^4\mu^3}{\pi n^3}\int_{4m^2_\pi}^\infty
\frac{\rho(s)ds}{s}.
\end{equation}

In recent years the precision of $\sigma^h$ measurement for different
energy intervals was substantially increased \cite{A}. The main contribution to the cross section
$\sigma^h$ is determined by the process $e^++e^-\rightarrow \pi^++\pi^-$.
The cross section of this reaction is proportional to the modulus squared
of the pion form factor $F_\pi$. As a consequence of new experiments on CMD-2
detector the experimental data on form factor $F_\pi$ were obtained in the
region $0.61\leq\sqrt{s}\leq 0.96$. We have used them in the calculation
of HVP contribution in this work. To describe the experimental data the
Gounaris - Sakurai model, accounting for the $\rho-\omega$ interference, can be
used:
\begin{equation}
F_\pi(s)=\frac{BW^{GS}_{\rho(770)}(s)\frac{1+\delta~BW_\omega(s)}{1+\delta}+
\beta~BW^{GS}_{\rho(1450)}(s)}{1+\beta}.
\end{equation}
The parameters $\beta$, $\gamma$ and the values of the $\rho(770)$,
$\rho(1450)$, $\omega$ parameters were taken from \cite{A,PDG}. Substituting
(8) to the spectral function
\begin{equation}
\rho_{\pi\pi}(s)=\frac{(s-4m_\pi^2)^{3/2}}{12s^{5/2}}|F_\pi(s)|^2,
\end{equation}
we have made numerical integration in (7) for the energy interval
$4m_\pi^2\leq \sqrt{s}\leq 0.81$. The contributions of other
energy intervals to $\Delta E^{HVP}_{Ls}$
were obtained in the same way as in the papers \cite{EJ,FKM,MF2}. The results of numerical
integration in (7) are shown in the table. It is evident that the basic
contribution to $\Delta E^{\rm HVP}_{\rm Ls}$ is related to the form factor $F_\pi$.
So, it was useful to compare this contribution to that, which can be obtained
by means of the theoretical representation of the pion form factor, suggested
in \cite{GP}. As expected, the corresponding contributions to
$\Delta E^{\rm HVP}_{\rm Ls}$ are coincide.

\begin{figure}
\magnitude=2000
\GRAPH(hsize=15){
\mov(0,0){\lin(4,0)}%
\mov(0,5){\lin(4,0)}%
\mov(0,2){\lin(4,0)}%
\mov(7,1){\Circle*(1)}%
\mov(5,0){\lin(4,0)}%
\mov(5,2){\lin(4,0)}%
\mov(2,0){\wavelin(0,2)}%
\mov(7,0){\wavelin(0,0.5)}%
\mov(7,2){\wavelin(0,-0.5)}%
\mov(2,-0.4){c}%
\mov(7,-0.4){d}%
\mov(7,4.6){b}%
\mov(2,4.6){a}%
\mov(7,2){\halfwavecirc(2)[+]}%
\mov(2,3){\Circle*(1)}%
\mov(0.5,2){\wavelin(1,1)}%
\mov(3.5,2){\wavelin(-1,1)}%
\mov(0,8){\lin(4,0)}%
\mov(5,5){\lin(4,0)}%
\mov(5,8){\lin(4,0)}%
\mov(3.,6.5){\Circle*(1)}%
\mov(6.,6.5){\Circle*(1)}%
\mov(4.5,6.5){+}%
\mov(8.5,6.5){+}%
\mov(9.,7.5){diagrams}%
\mov(9.,7.){with crossed}%
\mov(9.,6.5){photon lines}%
\mov(9.,6.){and subtracted}%
\mov(9.,5.5){diagrams}%
\mov(1,5){\wavelin(0,3)}%
\mov(1,5){\Circle**(0.3)}%
\mov(3,5){\wavelin(0,1)}%
\mov(3,5){\Circle**(0.3)}%
\mov(3,7){\wavelin(0,1)}%
\mov(6,5){\Circle**(0.3)}%
\mov(6,5){\wavelin(0,1)}%
\mov(6,7){\wavelin(0,1)}%
\mov(8,5){\wavelin(0,3)}%
\mov(8,5){\Circle**(0.3)}%
}

\vspace{5mm}

\caption{Diagrams of two-photon interaction, giving HVP contribution
to the Lamb shift in ($\mu p$)}
\end{figure}

Consider one-loop HVP contributions to the Lamb shift in $(\mu p)$,
shown in Fig.2. The necessary HVP correction, coming from these two-photon
amplitudes $T_{2\gamma}$ can be obtained after the  following substitution
in the photon propagator:
\begin{equation}
\frac{1}{k^2+i\epsilon}\rightarrow\left(\frac{\alpha}{\pi}\right)
\int_{s_{th}}^\infty\frac{\rho(s)ds}{k^2-s+i\epsilon}.
\end{equation}
Then the displacement of S-energy levels can be written in the following way:
\cite{FM}:
\begin{equation}
\Delta E^{\rm HVP}_{\rm Ls,~2\gamma}=-\frac{2\mu^3}{\pi^2 n^3}\delta_{l0}\alpha(Z\alpha)^5
\int_0^\infty V(k)dk\int\frac{\rho(s)ds}{k^2+s},
\end{equation}
\begin{equation}
V(k)=\frac{2kF_1^2}{m_1m_2}+\frac{k^3}{2m_1^3m_2^3}\left[2F_1^2(m_1^2+m_2^2)+
4m_1^2F_1F_2+3m_1^2F_2^2\right]+
\end{equation}
\begin{displaymath}
+\frac{\sqrt{k^2+4m_1^2}}{2m_1^3m_2(m_1^2-m_2^2)}\left[k^2(2m_2^2F_1^2+
4m_1^2F_1F_2+3m_1^2F_2^2)+8m_1^4F_1F_2+\frac{16m_1^4m_2^2F_1^2}{k^2}\right]
\end{displaymath}
\begin{displaymath}
-\frac{\sqrt{k^2+4m_2^2}m_1}{2m_2^3(m_1^2-m_2^2)}\left[k^2(2F_1^2+
4F_1F_2+3F_2^2)-8m_2^2F_1F_2+\frac{16m_2^4F_1^2}{k^2}\right]+
\end{displaymath}
\begin{displaymath}
+\frac{8m_1[F_2(0)+4m_2^2F_1'(0)-\frac{2m_2^2}{k^2}]}{m_2(m_1+m_2)},
\end{displaymath}
where the iteration part of the quasipotential (4) was taken into account.
There are no infrared divergences in expression (12). For the calculation
of the corresponding correction to the (2P-2S) transition energy we use
the dipole parameterization of the proton form factors $F_1$ and $F_2$
\cite{BY}:
\begin{equation}
F_1(k^2)=\frac{G_E-\frac{k^2}{4m_2^2}G_M}{1-\frac{k^2}{4m_2^2}},~~
F_2(k^2)=\frac{G_M-G_E}{1-\frac{k^2}{4m_2^2}},
\end{equation}
\begin{displaymath}
G_M=\frac{1+\kappa}{\left(1-\frac{k^2}{\Lambda^2}\right)^2},~~
G_E=\frac{1}{\left(1-\frac{k^2}{\Lambda^2}\right)^2},
\end{displaymath}
where the proton structure parameter $\Lambda=0.898 m_2$ \cite{BY},
$\kappa$ = 1.792847 is the proton anomalous magnetic moment.
The numerical value of the contribution (11), accounting for (9), is equal to
\begin{equation}
\Delta E^{\rm HVP}_{\rm Ls,~2\gamma(a+b)}=-0.047~\mu eV.
\end{equation}

The contribution of the same order in $\alpha$ to the energy spectrum due to
hadronic vacuum polarization is determined by the diagram (c) of Fig.2.
Using on-shell approximation for the external particle legs, we can
express it as follows:
\begin{equation}
\Delta E_{2\gamma~(c)}=\frac{\mu^3(Z\alpha)^4}{m_1^2n^3}\left[4m_1^2\rho'_{HVP}(0)
\delta_{l0}+f_{HVP}(0)\frac{C_{jl}}{2l+1}\right],
\end{equation}
\begin{equation}
C_{jl}=\delta_{l0}+(1-\delta_{l0})\frac{j(j+1)-l(l+1)-\frac{3}{4}}{l(l+1)},
\end{equation}
where the quantities $\rho'_{\rm HVP}(0)$ and $f_{\rm HVP}(0)$ denote the HVP
contribution to the slope of the charge form factor and to the muon anomalous
magnetic moment. The following integral representation for $f_{\rm HVP}(0)$
is commonly used:
\begin{equation}
f_{\rm HVP}(0)=\frac{1}{3}\left(\frac{\alpha}{\pi}\right)^2\int_{s_{th}}^\infty
\frac{R(s)ds}{s}\int_0^1\frac{y^2(1-y)dy}{(y^2+\frac{s}{m_1^2}(1-y))}.
\end{equation}
{\bf Table. HVP contributions to the Lamb shift (2P-2S) \\in muonic
hydrogen for the different energy intervals.}\\[3mm]
\begin{tabular}{|c|c|c|}  \hline
final state &energy range $\sqrt{s}$, GeV & $\Delta E_{Ls}^{HVP}(\mu p),~~\mu eV$\\ \hline
$\rho,\omega\rightarrow 2\pi$, $\omega\rightarrow 3\pi$
&(0.28, 0.95)&$ 7.035\pm 0.193$ \\
$\phi  $ & &$0.625\pm 0.023$  \\
$J/\Psi$ & & $0.115\pm 0.010$  \\
$\Upsilon$  & & $0.001$ \\
hadrons &(0.95, 1.4)   &$1.766\pm 0.073$  \\
hadrons &(1.4, 2.2)   &$0.602\pm 0.039$  \\
hadrons &(2.2, 3.1)   &$0.279\pm 0.024$ \\
hadrons &(3.1, 5.0)   &$0.181\pm 0.012$  \\
hadrons &(5.0, 10.0)  &$0.099\pm 0.002$   \\
hadrons &(10.0, 40.0)&$0.034\pm 0.001$	  \\
hadrons &$\sqrt{s}\geq 40.0$ &	0.003  \\  \hline
   &Total contribution	 & $10.772\pm 0.377$   \\
   &accounting for (14),(19)	    &		      \\   \hline
\end{tabular}
\vspace{5mm}

The numerical value for $f_{\rm HVP}(0)$ has been improved by the new calculations
in recent years \cite{HK}. We have used for it the following value:
$f_{\rm HVP}(0)$ = 695.1$\times 10^{-10}$. The contribution of HVP
to the slope of charge form factor can be expressed also in
the integral form similar to (17). To obtain it we consider Feynman's
parameterization for loop momentum integration and a once subtracted
dispersion relation for the polarization operator \cite{LPR,SGK}. The
corresponding expression reads as:
\begin{equation}
\rho'_{\rm HVP}(0)=\frac{1}{12}\left(\frac{\alpha}{\pi}\right)^2\frac{1}{m_1^2}
\int_{s_{th}}^\infty\frac{R(s)ds}{s}\int_0^1ydy\times
\end{equation}
\begin{displaymath}
\left[\frac{1}{30}\frac{y^2(36y-y^2-40)}{D^2(y,s)}+\frac{1}{6}\frac{(22-14y-y^2)}
{D(y,s)}+\frac{m_1^2}{s}\ln\left(\frac{y^2}{D(y,s)}\right)\right],
\end{displaymath}
\begin{displaymath}
D(y,s)=y^2+\frac{s}{m_1^2}(1-y).
\end{displaymath}
The integration over y in (18) can be done exactly. The numerical value of the
correction (15) to the Lamb shift (2P-2S) of muonic hydrogen, obtained
by means of (17), (18) and (8), is
\begin{equation}
\Delta E^{\rm HVP}_{\rm Ls,~2\gamma(c)}=-0.015~\mu eV.
\end{equation}
The contribution of diagram (d) in Fig.2 to the energy spectrum is of the order of
$O(\alpha^8)$ and can be neglected. Total value of hadronic vacuum
polarization contribution to the Lamb shift (2P-2S) is shown in the table.
It is in good agreement with the results of papers \cite{FMS,KN}. This
contribution has the same order as a Lamb shift correction due to the proton
structure and proton polarizability $(Z\alpha)^5$ \cite{SPH,KS}. This result
allows to increase the theoretical accuracy of the Lamb shift calculation
in $(\mu p)$. Indeed, taking into account our result for HVP contribution, obtained
in this work, and the proton polarizability contribution, found in \cite{FM},
and calculations of papers \cite{P,KN}, we can write total
expression for the Lamb shift in muonic hydrogen in the form:
\begin{equation}
\Delta E_{\rm Ls}(\mu p)=(206.085 (2)-5.1975 R_p^2)~meV.
\end{equation}
The uncertainty in the first term of (20) is determined by experimental errors
in the measurement of structure functions of deep inelastic e-p scattering
\cite{FM} and cross section $\sigma^h$.
The result (20) should be used when extracting the proton
charge radius $R_p$ from the future experiment of muonic hydrogen
Lamb shift.

We are grateful to S.G. Karshenboim, I.B. Khriplovich,
R.A. Sen'kov, for useful discussions.
The work was performed under the financial
support of the Russian Foundation for Fundamental Research
(grant 98-02-16185) and the Program "Universities of Russia - Fundamental
Researches" (grant 2759).

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\end{document}

