%Paper: 9204217
%From: Frank Hoogeveen <HOOGEVEN%CERNVM.BITNET@pucc.princeton.edu>
%Date: Mon, 13 Apr 92 10:08:18 SET

%***  this is file PHYS as of 13.10.87  ***
%***  PHYS-TeX  Version 2.0  ***


% I make @ signs act like letters, temporarily, to avoid conflict
% with user-defined control sequences in certain cases.

\catcode`\@=11

\chardef\f@ur=4
\chardef\l@tter=11
\chardef\@ther=12
\dimendef\dimen@iv=4
\toksdef\toks@i=1 % global only
\toksdef\toks@ii=2
\newtoks\emptyt@ks % permanently empty token list register

\def\glet{\global\let}
\def\gz@#1{\global#1\z@}
\def\gm@ne#1{\global#1\m@ne}
\def\g@ne#1{\global\advance#1\@ne}

\def\@height{height}       % save token space
\def\@depth{depth}         %   (5 or 6 tokens at the cost
\def\@width{width}         %    of one macro expansion)

\def\@plus{plus}           % save
\def\@minus{minus}         %   token space


%\input physmain
%***  this is file PHYSMAIN as of 19.05.88  ***
%***  macros for text  ***

%***  allow the constructions:
%            `\loop ... \if... ... \else ... \repeat'
%      and   `\loop ... \ifcase ... \or ... \else ... \repeat'
\def\loop#1\repeat{\def\iter@te{#1\expandafter\iter@te \fi}\iter@te
  \let\iter@te\undefined}
\let\iterate=\undefined

\def\bpargroup{\bp@rgroup\ep@r} % group that ends with \par
\def\bgrafgroup{\bp@rgroup\ep@rgroup} % .... .... with \endgraf or \par
\def\bp@rgroup{\bgroup \let\par\ep@rgroup \let\endgraf}

\def\ep@r{\ifhmode \unpenalty\unskip \fi \p@r}
\def\ep@rgroup{\ep@r \egroup}

\let\p@r=\endgraf   % TeX's \par primitive
\let\par=\ep@r
\let\endgraf=\ep@r

\def\lb{\hfil\break}
\def\endpage{\par \vfil \eject}
\def\superendpage{\par \vfil \supereject}

%***  allow change of \catcodes's inside \leftline, ...

\let\plainleftline=\leftline       % rename \leftline from PLAIN
\let\plainrightline=\rightline     % rename \rightline from PLAIN
\let\plaincenterline=\centerline   % rename \centerline from PLAIN

\def\leftline{\@line\hsize\empty\hss}
\def\rightline{\@line\hsize\hss\empty}
\def\centerline{\@line\hsize\hss\hss}

\let\plainrlap=\rlap   % rename \rlap from PLAIN
\let\plainllap=\llap   % rename \llap from PLAIN

\def\rlap{\@line\z@\empty\hss}
\def\llap{\@line\z@\hss\empty}

\def\lftline{\@line\hsize\empty\hfil}
\def\rtline{\@line\hsize\hfil\empty}
\def\ctrline{\@line\hsize\hfil\hfil}

\def\@line#1#2#3{\hbox to#1\bgroup#2\let\n@xt#3%
  \afterassignment\@@line \setbox\z@\hbox}
\def\@@line{\aftergroup\@@@line}
\def\@@@line{\unhbox\z@ \n@xt\egroup}

\def\after@arg#1{\bgroup\aftergroup#1\afterassignment\after@@arg\@eat}
\def\after@@arg{\ifcat\bgroup\noexpand\n@xt\else \n@xt\egroup \fi}
\def\@eat{\let\n@xt= } % eat one token
\def\eat#1{}           % eat one argument
\def\@eat@#1{\@eat}    % eat one argument and one token

%***  ctrlines  ***

\let\nl=\space

\def\ctrlines#1#2{\par \bpargroup
  \bgroup \parskip\z@skip \noindent \egroup
  \let\ctr@style#1\let\nl\ctr@lines \hfil \ctr@style{#2}\strut
  \interlinepenalty\@M \par}
\def\ctr@lines{\strut \lb \strut \hfil \ctr@style}

%***  phantom and smash  ***

\def\begin@{\ifmmode \expandafter\mathpalette\expandafter\math@ \else
  \expandafter\make@ \fi}
\def\make@#1{\setbox\z@\hbox{#1}\fin@}
\def\math@#1#2{\setbox\z@\hbox{$\m@th#1{#2}$}\fin@}

\def\ph@nt{\let\fin@\finph@nt \begin@}
\let\makeph@nt=\undefined
\let\mathph@nt=\undefined

\newif\ift@ \newif\ifb@
\def\topsmash{\t@true\b@false\sm@sh}
\def\botsmash{\t@false\b@true\sm@sh}
\def\smash{\t@true\b@true\sm@sh}
\def\sm@sh{\let\fin@\finsm@sh \begin@}
\let\makesm@sh=\undefined
\let\mathsm@sh=\undefined
\def\finsm@sh{\ift@\ht\z@\z@\fi \ifb@\dp\z@\z@\fi \box\z@}

%***  boxit  ***

\newdimen\boxitsep   \boxitsep=5pt

\def\boxit{\fboxit{.4}}
\def\fboxit#1#2{\hbox{\vrule \@width#1\p@
    \vtop{\vbox{\hrule \@height#1\p@ \vskip\boxitsep
        \hbox{\hskip\boxitsep #2\hskip\boxitsep}}%
      \vskip\boxitsep \hrule \@height#1\p@}\vrule \@width#1\p@}}

%***  active characters `:,`; and `^^M  ***

\begingroup
  \catcode`\:=\active
  \lccode`\*=`\\ \lowercase{\gdef:{*}}   %   `: -> `\
  \catcode`\;=\active
  \lccode`\* `\% \lowercase{\gdef;{*}}   %   `; -> `%
  \catcode`\^^M=\active \glet^^M=\space  %   `^^M -> \space
\endgroup

%***  comment  ***

\begingroup
  \catcode`\:=\active
  \outer\gdef\comment{\begingroup
    \catcode`\\\@ther \catcode`\%\@ther \catcode`\^^M\@ther
    \catcode`\{\@ther \catcode`\}\@ther \catcode`\#\@ther
    \wlog{* input between `:comment' and `:endcomment' ignored *}%
    \c@mment}
\endgroup
{\lccode`\:=`\\ \lccode`\;=`\^^M
  \lowercase{\gdef\c@mment#1;{\c@@mment:endcomment*}}}
\def\c@@mment#1#2*#3{\if #1#3%
    \ifx @#2@\def\n@xt{\endgroup\ignorespaces}\else
      \def\n@xt{\c@@mment#2*}\fi \else
    \def\n@xt{\c@mment#3}\fi \n@xt}
\let\endcomment=\relax   % avoid errmessage for unmatched \endcomment

\message{date and time,}

%***  language  ***

\newcount\langu@ge

\begingroup \catcode`\"=\@ther \gdef\dq{"}
            \catcode`\"=\active
            \gdef"#1{{\accent\dq 7F #1}\penalty\@M \hskip\z@skip}
  \endgroup

\outer\def\english{\gm@ne\langu@ge
  \global\catcode`\"\@ther \glet\3\undefined}
\outer\def\german{\gz@\langu@ge
  \global\catcode`\"\active \glet\3\ss}

\def\case@language#1{\ifcase\expandafter\langu@ge #1\fi}
\def\case@abbr#1{{\let\nodot\n@dot\case@language{#1}.^^>}}
\def\n@dot{\expandafter\eat}
\let\nodot=\empty

%***  date  ***

\def\themonth{\xdef\themonth{\noexpand\case@language
  {\ifcase\month \or Januar\or Februar\or M\noexpand\"arz\or April\or
  Mai\or Juni\or Juli\or August\or September\or Oktober\or November\or
  Dezember\fi
  \noexpand\else
  \ifcase\month \or January\or February\or March\or April\or May\or
  June\or July\or August\or September\or October\or November\or
  December\fi}}\themonth}

\def\thedate{\case@language{\else\themonth\ }\number\day
  \case@language{.\ \themonth \else ,} \number\year}

\def\date{\number\day.\,\number\month.\,\number\year}

%***  \PhysTeX logo  ***

\def\PhysTeX{$\Phi\kern-.25em\raise.4ex\hbox{$\Upsilon$}\kern-.225em
  \Sigma$-\TeX}
%***  end of file PHYSMAIN  ***
%\input phystime
%***  this is file PHYSTIME as of 29.01.87  ***
%***  time  ***

%***  this file is read twice, once if the format is generated
%***                       and once if a dump file is loaded

\count255=\time \divide\count255 by 60 \edef\thetime{\the\count255 :}
\multiply\count255 by -60 \advance\count255 by\time
\edef\thetime{\thetime \ifnum10>\count255 0\fi \the\count255 }
%***  end of file PHYSTIME  ***
%\input physfont
%***  this is file PHYSFONT as of 28.04.88  ***
%***  spacing parameters  ***

\newskip\refbetweenskip   \newskip\chskiptamount
\newskip\chskiplamount   \newskip\secskipamount
\newskip\footnotebaselineskip   \newskip\interfootnoteskip

\newdimen\chapstretch   \chapstretch=2.5cm
\newcount\chappenalty   \chappenalty=-800
\newdimen\sectstretch   \sectstretch=2cm
\newcount\sectpenalty   \sectpenalty=-400

\def\chskipt{\chapbreak \vskip\chskiptamount}
\def\chskipl{\nobreak \vskip\chskiplamount}
\def\unchskip{\vskip-\chskiplamount}
\def\secskipt{\sectbreak \vskip\secskipamount}
\def\chapbreak{\par \vskip\z@\@plus\chapstretch \penalty\chappenalty
  \vskip\z@\@plus-\chapstretch}
\def\sectbreak{\par \vskip\z@\@plus\sectstretch \penalty\sectpenalty
  \vskip\z@\@plus-\sectstretch}

%***  fonts  ***

%***  load am... or cm... fonts according to PLAIN's selection  ***

\begingroup \lccode`\*=`\r
  \lowercase{\def\n@xt#1*#2@{#1}
    \xdef\font@sel{\expandafter\n@xt\fontname\tenrm*@}}\endgroup

\message{loading \font@sel\space fonts,}

\font\twelverm=\font@sel r12 % roman text
%\font\tenrm=\font@sel r10 % already defined in PLAIN
\font\ninerm=\font@sel r9
\font\eightrm=\font@sel r8
%\font\sevenrm=\font@sel r7 % already defined in PLAIN
\font\sixrm=\font@sel r6
%\font\fiverm=\font@sel r5 % already defined in PLAIN

\font\twelvei=\font@sel mi12 \skewchar\twelvei='177 % math italic
%\font\teni=\font@sel mi10 % already defined in PLAIN
\font\ninei=\font@sel mi9    \skewchar\ninei='177
\font\eighti=\font@sel mi8   \skewchar\eighti='177
%\font\seveni=\font@sel mi7 % already defined in PLAIN
\font\sixi=\font@sel mi6   \skewchar\sixi='177
%\font\fivei=\font@sel mi5 % already defined in PLAIN

\font\twelvesy=\font@sel sy10 scaled \magstep1
  \skewchar\twelvesy='60 % math symbols
%\font\tensy=\font@sel sy10 % already defined in PLAIN
\font\ninesy=\font@sel sy9    \skewchar\ninesy='60
\font\eightsy=\font@sel sy8   \skewchar\eightsy='60
%\font\sevensy=\font@sel sy7 % already defined in PLAIN
\font\sixsy=\font@sel sy6   \skewchar\sixsy='60
%\font\fivesy=\font@sel sy5 % already defined in PLAIN

%\font\tenex=\font@sel ex10 % math extension already defined in PLAIN

%\font\preloaded=\font@sel ss10 % sans serif
%\font\preloaded=\font@sel ssq8

%\font\preloaded=\font@sel ssi10 % sans serif italic
%\font\preloaded=\font@sel ssqi8

\font\twelvebf=\font@sel bx12 % boldface extended
%\font\tenbf=\font@sel bx10 % already defined in PLAIN
\font\ninebf=\font@sel bx9
\font\eightbf=\font@sel bx8
%\font\sevenbf=\font@sel bx7 % already defined in PLAIN
\font\sixbf=\font@sel bx6
%\font\fivebf=\font@sel bx5 % already defined in PLAIN

\font\twelvett=\font@sel tt12 % typewriter
%\font\tentt=\font@sel tt10 % already defined in PLAIN
%\font\preloaded=\font@sel tt9
\font\eighttt=\font@sel tt8

%\font\preloaded=\font@sel sltt10 % slanted typewriter

\font\twelvesl=\font@sel sl12 % slanted roman
%\font\tensl=\font@sel sl10 % already defined in PLAIN
\font\ninesl=\font@sel sl9
\font\eightsl=\font@sel sl8
\let\sevensl=\eightsl % there is no sl7 font
\let\sixsl=\sevensl % there is no sl6 font
\let\fivesl=\sixsl % there is no sl5 font

\font\twelveit=\font@sel ti12 % text italic
%\font\tenit=\font@sel ti10 % already defined in PLAIN
\font\nineit=\font@sel ti9
\font\eightit=\font@sel ti8
\font\sevenit=\font@sel ti7
\let\sixit=\sevenit % there is no ti6 font
\let\fiveit=\sixit % there is no ti5 font

%\font\preloaded=\font@sel u10 % unslanted text italic

%\font\preloaded=\font@sel mib10 % bold math italic
%\font\preloaded=\font@sel bsy10 % bold math symbols

\font\twelvecsc=\font@sel csc10 scaled \magstep1 % caps and small caps
\font\tencsc=\font@sel csc10
\let\eightcsc=\tencsc % there is no csc8 font

%\font\preloaded=\font@sel ssbx10 % sans serif bold extended

%\font\preloaded=\font@sel dunh10 % Dunhill style

%\font\preloaded=\font@sel r7 scaled \magstep4 % for titles
%\font\preloaded=\font@sel tt10 scaled \magstep2
%\font\preloaded=\font@sel ssbx10 scaled \magstep2

%\font\preloaded=manfnt % METAFONT logo and dragon curve
                        % and special symbols

%***  sizes  ***

\def\twelvepoint{\twelve@point
  \let\normal@spacing\twelve@spacing \set@spacing}
\def\tenpoint{\ten@point
  \let\normal@spacing\ten@spacing \set@spacing}
\def\eightpoint{\eight@point
  \let\normal@spacing\eight@spacing \set@spacing}

\def\rm{\fam\z@ \@fam}
\def\mit{\fam\@ne}
\def\oldstyle{\mit \@fam}
\def\cal{\fam\tw@}
\def\it{\fam\itfam \@fam}
\def\sl{\fam\slfam \@fam}
\def\bf{\fam\bffam \@fam}
\def\tt{\fam\ttfam \@fam}
\def\caps{\@caps}
\def\@fam{\the\textfont\fam}

%***  fonts for sizes  ***

\def\twelve@point{\set@fonts twelve ten eight }
\def\ten@point{\set@fonts ten eight six }
\def\eight@point{\set@fonts eight six five }

\def\set@fonts#1 #2 #3 {\textfont\ttfam\csname#1tt\endcsname
    \expandafter\let\expandafter\@caps\csname#1csc\endcsname
  \def\n@xt##1##2{\textfont##1\csname#1##2\endcsname
    \scriptfont##1\csname#2##2\endcsname
    \scriptscriptfont##1\csname#3##2\endcsname}%
  \set@@fonts}
\def\set@@fonts{\n@xt0{rm}\n@xt1i\n@xt2{sy}%
  \textfont3\tenex \scriptfont3\tenex \scriptscriptfont3\tenex
  \n@xt\itfam{it}\n@xt\slfam{sl}\n@xt\bffam{bf}\rm}

%***  spacing for sizes  ***

\def\singlespace{\chardef\@spacing\z@ \set@spacing}
\def\doublespace{\chardef\@spacing\@ne \set@spacing}
\def\triplespace{\chardef\@spacing\tw@ \set@spacing}
\chardef\@spacing=1   %  \doublespace

\def\set@spacing{\expandafter\expandafter\expandafter\set@@spacing
  \expandafter\spacing@names\expandafter\@@\normal@spacing
  \normalbaselines}
\def\set@@spacing#1#2\@@#3+#4*{#1#4\multiply#1\@spacing \advance#1#3%
  \ifx @#2@\let\n@xt\empty \else
    \def\n@xt{\set@@spacing#2\@@}\fi \n@xt}

\def\normalbaselines{\lineskip\normallineskip
  \setbaselineskip\normalbaselineskip
  \lineskiplimit\normallineskiplimit}

\def\setbaselineskip{\afterassignment\set@strut \baselineskip}
\def\set@strut{\setbox\strutbox\spacer\z@\baselineskip}

\def\spacer{\hbox\bgroup \afterassignment\x@spacer \dimen@}
\def\x@spacer{\ifdim\dimen@=\z@\else \hskip\dimen@ \fi
  \afterassignment\y@spacer \dimen@}
\def\y@spacer{\setbox\z@\hbox{$\vcenter{\vskip\dimen@}$}%
  \vrule \@height\ht\z@ \@depth\dp\z@ \@width\z@ \egroup}

\def\spacing@names{%  list of all variable spacing parameters
  \normalbaselineskip
  \normallineskip
  \normallineskiplimit
  \footnotebaselineskip
  \interfootnoteskip
  \parskip
  \refbetweenskip
  \abovedisplayskip
  \belowdisplayskip
  \abovedisplayshortskip
  \belowdisplayshortskip
  \chskiptamount
  \chskiplamount
  \secskipamount
  }

\def\twelve@spacing{%  parameters and increments for twelvepoint
  14\p@              +5\p@        *%\normalbaselineskip
  \p@                +\z@         *%\normallineskip
  \z@                +\z@         *%\normallineskiplimit
  14\p@              +\p@         *%\footnotebaselineskip
  20\p@              +\z@         *%\interfootnoteskip
  5\p@\@plus\p@      +-2\p@       *%\parskip
  \z@                +6\p@        *%\refbetweenskip
  8\p@\@plus2\p@\@minus3\p@  +%
    4\p@\@plus3\p@\@minus5\p@     *%\abovedisplayskip
  8\p@\@plus2\p@\@minus3\p@  +%
    4\p@\@plus3\p@\@minus5\p@     *%\belowdisplayskip
  \p@\@plus2\p@\@minus\p@    +%
    4\p@\@plus3\p@\@minus2\p@     *%\abovedisplayshortskip
  8\p@\@plus2\p@\@minus3\p@  +%
    \p@\@plus2\p@\@minus2\p@      *%\belowdisplayshortskip
  20\p@\@plus5\p@    +\z@         *%\chskiptamount
  5.5\p@             +\z@         *%\chskiplamount
  6\p@\@plus2\p@     +\z@         *%\secskipamount
  }

\def\ten@spacing{%  parameters and increments for tenpoint
  11\p@              +4.5\p@      *%\normalbaselineskip
  \p@                +\z@         *%\normallineskip
  \z@                +\z@         *%\normallineskiplimit
  12\p@              +\p@         *%\footnotebaselineskip
  16\p@              +\z@         *%\interfootnoteskip
  5\p@\@plus\p@      +-2\p@       *%\parskip
  \z@                +5\p@        *%\refbetweenskip
  8\p@\@plus2\p@\@minus3\p@  +%
    4\p@\@plus3\p@\@minus5\p@     *%\abovedisplayskip
  8\p@\@plus2\p@\@minus3\p@  +%
    4\p@\@plus3\p@\@minus5\p@     *%\belowdisplayskip
  \p@\@plus2\p@\@minus\p@    +%
    4\p@\@plus3\p@\@minus2\p@     *%\abovedisplayshortskip
  8\p@\@plus2\p@\@minus3\p@  +%
    \p@\@plus2\p@\@minus2\p@      *%\belowdisplayshortskip
  20\p@\@plus5\p@    +\z@         *%\chskiptamount
  5.5\p@             +\z@         *%\chskiplamount
  6\p@\@plus2\p@     +\z@         *%\secskipamount
  }

\def\eight@spacing{%  parameters and increments for eightpoint
  9\p@               +3.5\p@      *%\normalbaselineskip
  \p@                +\z@         *%\normallineskip
  \z@                +\z@         *%\normallineskiplimit
  10\p@              +\p@         *%\footnotebaselineskip
  14\p@              +\z@         *%\interfootnoteskip
  5\p@\@plus\p@      +-2\p@       *%\parskip
  \z@                +5\p@        *%\refbetweenskip
  8\p@\@plus2\p@\@minus3\p@  +%
    4\p@\@plus3\p@\@minus5\p@     *%\abovedisplayskip
  8\p@\@plus2\p@\@minus3\p@  +%
    4\p@\@plus3\p@\@minus5\p@     *%\belowdisplayskip
  \p@\@plus2\p@\@minus\p@    +%
    4\p@\@plus3\p@\@minus2\p@     *%\abovedisplayshortskip
  8\p@\@plus2\p@\@minus3\p@  +%
    \p@\@plus2\p@\@minus2\p@      *%\belowdisplayshortskip
  20\p@\@plus5\p@    +\z@         *%\chskiptamount
  5.5\p@             +\z@         *%\chskiplamount
  6\p@\@plus2\p@     +\z@         *%\secskipamount
  }

\twelvepoint

%***  macros for sizes  ***

\def\large{\par \bgroup \twelvepoint \after@arg\@size}
\def\medium{\par \bgroup \tenpoint \after@arg\@size}
\def\small{\par \bgroup \eightpoint \after@arg\@size}
\def\@size{\par \egroup}

\def\LARGE#1{{\twelve@point #1}}
\def\MEDIUM#1{{\ten@point #1}}
\def\SMALL#1{{\eight@point #1}}
%***  end of file PHYSFONT  ***
%\input phystext
%***  this is file PHYSTEXT as of 02.02.87  ***
%***  texts and headings  ***

\def\submittextone{Zur Ver\"offentlichung in\else Submitted to}
\def\submittexttwo{ eingereicht\else}
\def\abstracthead{Zusammenfassung\else Abstract}
\def\ackhead{Danksagung\else Acknowledgements}
\def\appendixhead{Anhang\else Appendix}
\def\eqabbr{Gl\else eq}
\def\eqsabbr{Gln\else eqs}
\def\figpref{Abb\else Fig}
\def\fighead{Abbildungen\else Figure captions}
\def\figabbr{Bild\nodot\else Fig}
\def\tabpref{Tab\else Tab}
\def\tabhead{Tabellen\else Table captions}
\let\tababbr=\tabpref
\def\refpref{Lit\else Ref}
\def\refhead{Literaturverzeichnis\else References}
\def\refabbr{???\else Ref}
\def\refsabbr{????\else Refs}
\def\tocpref{Inh\else Toc}
\def\tochead{Inhaltsverzeichnis\else Table of contents}
\def\footpref{Anm\else Foot}
\def\foothead{Anmerkungen\else Footnotes}
\def\prfhead{Beweis\else Proof}

%***  styles  ***

\def\UPPERCASE#1{\edef\n@xt{#1}\uppercase\expandafter{\n@xt}}

\let\headlinestyle=\twelverm
\let\footlinestyle=\twelverm
\let\pagestyle=\twelverm       % page numbers
\let\titlestyle=\bf            % title of paper, chapters, appendices
\let\authorstyle=\caps
\let\addressstyle=\sl
\let\sectstyle=\caps           % section titles
% \def\secsstyle{\sectstyle}   %   will be defined by \newsect
\def\subsecstyle#1{\hbox{$\underline{\hbox{#1}}$\enskip}\ignorespaces}
\let\headstyle=\UPPERCASE      % headlines for list of figure capts. etc
\let\captionstyle=\it          % figure and table captions
\let\journalstyle=\sl
\let\volumestyle=\bf
\let\figstyle=\empty           % list of figure captions
\def\figindent{to4em}
\def\figbreak{\goodbreak \vskip\refbetweenskip}
\let\tabstyle=\empty           % list of table captions
\let\tabindent=\figindent
\let\tabbreak=\figbreak
\let\refstyle=\empty           % list of references
\let\refindent=\figindent
\def\namrefindent{2em}
\def\refbreak{\filbreak \vskip\refbetweenskip}
\let\tocstyle=\empty           % table of contents
\let\tocindent=\empty
\let\tocbreak=\refbreak
\let\footstyle=\empty          % (list of) footnotes
\let\footindent=\figindent
\let\footbreak=\figbreak
\let\stmttitlestyle=\bf        % mathematical statements
\let\stmtstyle=\sl
\let\prftitlestyle=\caps
\let\prfstyle=\sl

%***  user exits  ***                    % called before

\def\skipuserexit{\setbox\z@\box\@cclv}  % page is not printed
\def\shipuserexit{\unvbox\@cclv}         % page is printed

\def\chapuserexit{\sectuserexit}         % chapter title
\def\appuserexit{\chapuserexit}          % appendix title
\def\sectuserexit{\secsuserexit}         % section title
\let\secsuserexit=\relax                 % subsection title
%***  end of file PHYSTEXT  ***
%\input physpage
%***  this is file PHYSPAGE as of 09.12.87  ***
%***  page numbers and pagesel  ***

\newcount\firstp@ge   \firstp@ge=-10000
\newcount\lastp@ge   \lastp@ge=10000
\outer\def\pagesel#1#2{\global\firstp@ge#1 \global\lastp@ge#2
  \wlog{**************************}\wlog{*}%
  \wlog{* don't use \string\p agesel}%
  \wlog{* this will not be supported in future}%
  \wlog{*}\wlog{**************************}%
  \wlog{(* pages #1-#2 selected for printing, others will be skipped *)}}

\newbox\pageb@x

\outer\def\toppagenum{\glet\page@tbn T%
  \glet\headb@x\pageb@x \glet\footb@x\voidb@x}
\outer\def\botpagenum{\glet\page@tbn B%
  \glet\headb@x\voidb@x \glet\footb@x\pageb@x}
\outer\def\nopagenum{\glet\page@tbn N%
  \glet\headb@x\voidb@x \glet\footb@x\voidb@x}

\outer\def\lefthead{\glet\head@lrac L}
\outer\def\righthead{\glet\head@lrac R}
\outer\def\althead{\glet\head@lrac A}
\outer\def\centhead{\glet\head@lrac C}
\outer\def\leftfoot{\glet\foot@lrac L}
\outer\def\rightfoot{\glet\foot@lrac R}
\outer\def\altfoot{\glet\foot@lrac A}
\outer\def\centfoot{\glet\foot@lrac C}

\newtoks\lheadtext   \newtoks\cheadtext   \newtoks\rheadtext
\newtoks\lfoottext   \newtoks\cfoottext   \newtoks\rfoottext

\headline={\headlinestyle \head@foot\skip@head\head@lrac
  \lheadtext\cheadtext\rheadtext\headb@x}
\footline={\footlinestyle \head@foot\skip@foot\foot@lrac
  \lfoottext\cfoottext\rfoottext\footb@x}
\lheadtext={}   \cheadtext={}   \rheadtext={}
\lfoottext={}   \cfoottext={}   \rfoottext={}

\newbox\page@strut
\setbox\page@strut\hbox{\vrule \@height 15mm\@depth 10mm\@width \z@}

\def\head@foot#1#2#3#4#5#6{\unhcopy\page@strut
  \if#1T\hfil \else
    \if#2C\head@@foot{\the#3}{\copy#6}{\the#5}\else
      \if#2A\ifodd\pageno \let#2R\else \let#2L\fi \fi
      \if#2R\head@@foot{\the#4}{\the#5}{\copy#6}\else
        \head@@foot{\copy#6}{\the#3}{\the#4}\fi \fi \fi}
\def\head@@foot#1#2#3{\plainrlap{#1}\hfil#2\hfil\plainllap{#3}}

\let\startpage=\relax  % allow input to be read twice

\outer\def\pageall{\glet\page@ac A%
  \global\countdef\pageno\z@ \global\pageno\@ne
  \global\countdef\pageno@pref\@ne \gz@\pageno@pref
  \glet\page@pref\empty \glet\page@reset\count@
  \glet\chap@break\chskipt \outer\gdef\startpage{\global\pageno}}
\outer\def\pagechap{\glet\page@ac C%
  \global\countdef\pageno\@ne \gz@\pageno
  \global\countdef\pageno@pref\z@ \gz@\pageno@pref
  \gdef\page@pref{\dash@pref}%
  \gdef\page@reset{\global\pageno\@ne \global\pageno@pref}%
  \glet\chap@break\superendpage
  \outer\gdef\startpage##1.{\global\pageno@pref##1\global\pageno}}

%***  output  ***

\hsize=15 cm   \hoffset=0 mm
\vsize=22 cm   \voffset=0 mm

\newdimen\hoffset@corr@p   \newdimen\voffset@corr@p
\newdimen\hoffset@corrm@p   \newdimen\voffset@corrm@p
\newdimen\hoffset@corr@l   \newdimen\voffset@corr@l
\newdimen\hoffset@corrm@l   \newdimen\voffset@corrm@l

\outer\def\portrait{\switch@pl P%
  \glet\hoffset@corr\hoffset@corr@p
  \glet\voffset@corr\voffset@corr@p
  \glet\hoffset@corrm\hoffset@corrm@p
  \glet\voffset@corrm\voffset@corrm@p}
\outer\def\landscape{\switch@pl L%
  \glet\hoffset@corr\hoffset@corr@l
  \glet\voffset@corr\voffset@corr@l
  \glet\hoffset@corrm\hoffset@corrm@l
  \glet\voffset@corrm\voffset@corrm@l}
\def\switch@pl#1{\if #1\ori@pl \else \superendpage \glet\ori@pl#1%
  \dimen@\ht\page@strut \advance\dimen@\dp\page@strut
  \advance\vsize\dimen@ \dimen@ii\hsize \global\hsize\vsize
  \advance\dimen@ii-\dimen@ \global\vsize\dimen@ii \fi}
\let\ori@pl=P

\def\m@g{\dimen@\ht\page@strut \advance\dimen@\dp\page@strut
  \advance\vsize\dimen@ \divide\vsize\count@
  \multiply\vsize\mag \advance\vsize-\dimen@
  \divide\hsize\count@ \multiply\hsize\mag
  \divide\dimen\footins\count@ \multiply\dimen\footins\mag
  \mag\count@}

\output={\physoutput}

\def\physoutput{\make@lbl
  \ifnum \pageno<\firstp@ge \skipp@ge \else
  \ifnum \pageno>\lastp@ge \skipp@ge \else \shipp@ge \fi \fi
  \advancepageno \skippagenum F\skipheadline F\skipfootline F%
  \ifnum\outputpenalty>-\@MM \else \dosupereject \fi}

\def\skippagenum{\glet\skip@page}
\def\skipheadline{\glet\skip@head}
\def\skipfootline{\glet\skip@foot}

\def\skipp@ge{{\skipuserexit \setbox\z@\box\topins
  \setbox\z@\box\footins}\deadcycles\z@}
\def\shipp@ge{\setbox\pageb@x\hbox{%
    \if F\skip@page \pagestyle{\page@pref \folio}\fi}%
  \dimen@-.5\hsize \advance\dimen@\hoffset@corrm
  \divide\dimen@\@m \multiply\dimen@\mag
  \advance\hoffset\dimen@ \advance\hoffset\hoffset@corr
  \dimen@\ht\page@strut \advance\dimen@\dp\page@strut
  \advance\dimen@\vsize \dimen@-.5\dimen@
  \advance\dimen@\voffset@corrm
  \divide\dimen@\@m \multiply\dimen@\mag
  \advance\voffset\dimen@ \advance\voffset\voffset@corr
  \shipout\vbox{\makeheadline \vbadness\@M \setbox\z@\pagebody
    \dimen@\dp\z@ \box\z@ \kern-\dimen@ \makefootline}}

\def\pagecontents{\ifvbox\topins\unvbox\topins\fi
  \dimen@\dp\@cclv \shipuserexit % open up \box255
  \ifvbox\footins % footnote info is present
    \vskip\skip\footins \footnoterule \unvbox\footins\fi
  \ifr@ggedbottom \kern-\dimen@ \vfil \fi}

\def\folio{\ifnum\pageno<\z@ \ifcase\langu@ge \MEDIUM{\uppercase
  \expandafter{\romannumeral-\pageno}}\else \romannumeral-\pageno \fi
  \else \number\pageno \fi}

\def\makeheadline{\line{\the\headline}\nointerlineskip}
\def\makefootline{\nointerlineskip \line{\the\footline}}

\skippagenum=F   \skipheadline=F   \skipfootline=F
%***  end of file PHYSPAGE  ***
%\input physchap
%***  this is file PHYSCHAP as of 31.05.89  ***
%***  title page macros  ***

\outer\def\titlepage{\glet\titl@fill\vfil}
\outer\def\notitlepage{\gdef\titl@fill{\vskip20\p@}}

\newbox\t@pleft   \newbox\t@pright
\def\t@pinit{%
  \global\setbox\t@pleft\vbox{\hrule \@height\z@ \@width.26\hsize}%
  \global\setbox\t@pright\copy\t@pleft}
\t@pinit

\def\topleft{\t@p\t@pleft}
\def\topright{\t@p\t@pright}
\def\t@p#1#2{\global\setbox#1\vtop{\unvbox#1\hbox{\strut #2}}}

\outer\def\submit#1{\topleft{\case@language\submittextone}%
  \topleft{{#1}\case@language\submittexttwo}}

\let\pubdate=\topright

\outer\def\title{\vbox{\line{\box\t@pleft \hss \box\t@pright}}%
  \skippagenum T\skipheadline T\skipfootline T%
  \t@pinit \titl@fill \vskip\chskiptamount \@title}
\let\titcon=\relax  % allow input to be read twice
\outer\def\titcon{\errmessage{please use \noexpand\nl in the title
  instead of \noexpand\titcon}\unchskip \@title}
\def\@title#1{\ctrlines\titlestyle{#1}\chskipl}
\def\titl@#1{\edef\n@xt{\noexpand\@title{#1}}\n@xt}

\def\author{\aut@add\authorstyle}
\def\autcon{\and@con \author}
\def\address{\aut@add\addressstyle}
\def\addcon{\and@con \address}
\def\and@con{\titl@fill \ctrline{\case@language{und\else and}}}

\def\aut@add#1{\titl@fill \ctrlines{#1\use@nl}}
\def\use@nl{\let\\\use@@nl}
\def\use@@nl{\errmessage{please use \noexpand\nl in addresses and
  (lists of) authors instead of \string\\}\nl}

\def\abstract{\titl@fill \he@d{\case@language\abstracthead}%
  \after@arg\titl@fill}

\def\ack{\chskipt \he@d{\case@language\ackhead}}

\def\he@d#1{\ctrline{\headstyle{#1}}\chskipl}

\message{chapters, sections and appendices,}

%***  label and pagelabel  ***

\newtoks\l@names   \l@names={\\\the@label}
\let\the@label=\empty

\def\label{\num@lett\@label}
\def\@label#1{\def@name\l@names#1{\the@label}}
\def\quote{\num@lett\empty}

\newinsert\lbl@ins
\count\lbl@ins=0   \dimen\lbl@ins=\maxdimen   \skip\lbl@ins=0pt
\newcount\lbln@m   \lbln@m=0
\let\lbl@saved\empty

\def\pagelabel{\num@lett\@pagelabel}
\def\@pagelabel#1{\ifx#1\undefined \let#1\empty \fi
  \toks@\expandafter{#1}\expandafter\testcr@ss\the\toks@\cr@ss\@@
  \ifcr@ss\else
    \toks@{\cr@ss\lbl@undef}\def@name\l@names#1{\the\toks@}\fi
  \g@ne\lbln@m \insert\lbl@ins{\vbox{\vskip\the\lbln@m sp}}%
  \count@\lbln@m \do@label\store@label#1}
\begingroup \let\save=\relax  % allow input to be read twice
  \gdef\lbl@undef{\message{unresolved \string\pagelabel, use
    \string\save\space and \string\crossrestore}??}
\endgroup
\def\do@label#1{\expandafter#1\csname\the\count@\endcsname}
\def\store@label#1#2{\expandafter\gdef\expandafter\lbl@saved
  \expandafter{\lbl@saved#1#2}}

\def\make@lbl{\setbox\z@\vbox{\let\MEDIUM\relax
  \unvbox\lbl@ins \loop \setbox\z@\lastbox \ifvbox\z@
    \count@\ht\z@ \do@label\make@label \repeat}}
\def\make@label#1{\def\make@@label##1#1##2##3#1##4\@@{%
    \gdef\lbl@saved{##1##3}%
    \ifx @##4@\errmessage{This can't happen}\else
    \def@name\l@names##2{\page@pref\folio}\fi}%
  \expandafter\make@@label\lbl@saved#1#1#1\@@}

\outer\def\lblrestore{\all@restore\l@names}

%***  initialization  ***

\let\sect=\relax  \let\s@ct=\relax  % allow input to be read twice

\def\chapinit{\chap@init{\chap@pref}\glet\sect@@eq\@chap@sect@eq
  \sectinit}
\def\appinit{\chap@init{\char\the\appn@m}\glet\sect@@eq\@chap@eq
  \glet\sect@dot@pref\empty \glet\sect@pref\dot@pref
  \glet\sect\undefined}
\def\sectinit{\xdef\sect@dot@pref{\the\sectn@m.}%
  \xdef\sect@pref{\dot@pref\sect@dot@pref}\glet\sect\s@ct}
\def\chap@init#1{\xdef\the@label{#1}\xdef\dot@pref{\the@label.}%
  \xdef\dash@pref{\the@label--}\glet\chap@@eq\@chap@eq}

%***  chap  ***

\newcount\chapn@m   \chapn@m=0

\outer\def\chappage{\glet\chap@page T}
\outer\def\nochappage{\glet\chap@page F}

\outer\def\arabicchapnum{\glet\chap@ar A\gdef\chap@pref{\the\chapn@m}}
\outer\def\romanchapnum{\glet\chap@ar R%
  \gdef\chap@pref{\uppercase{\romannumeral\chapn@m}}}

\let\chap=\relax  \let\ch@p=\relax  % allow input to be read twice

\outer\def\chapters{\glet\chap@yn Y\glet\chap\ch@p
  \chap@init{0}\glet\sect@@eq\@chap@sect@eq \sectinit}
\outer\def\nochapters{\glet\chap@yn N\glet\chap\undefined
  \glet\dot@pref\empty \glet\dash@pref\empty
  \glet\chap@@eq\@eq \glet\sect@@eq\@sect@eq \sectinit}

\outer\def\ch@p#1{\if T\chap@page \superendpage \else \chap@break \fi
  \g@ne\chapn@m \sect@reset \chapinit \page@reset\chapn@m
  \eq@reset \fig@reset \tab@reset
  \toks@{\dot@pref}\toks@ii{#1}\chapuserexit
  \titl@{\the\toks@\ \the\toks@ii}%
  \ifnum\auto@toc>\m@ne \toks@store{#1}\@toc\dot@pref \fi}

%***  sect  ***

%  use \read@store to read (sub-)section titles in case they are used
%  in table of contents

\def\sec@title#1#2#3#4#5#6#{\ifx @#6@\g@ne#1\else\global#1#6\fi
  #2\secskipt \xdef\the@label{#3\the#1}\xdef#4{\the@label.}%
  \read@store{\sec@@title#4#5}}
\def\sec@@title#1#2#3#4{\toks@{\the@label.}\toks@ii\toks@store
  #4\bpargroup #2\varitem{\the\toks@}\interlinepenalty\@M
    \let\nl\lb \the\toks@ii \par\nobreak
  \ifnum#3<\auto@toc \@toc{#1}\fi}

\newcount\sectn@m   \sectn@m=0
\def\sect@reset{\gz@\sectn@m}
\outer\def\s@ct{\sec@title\sectn@m
  {\secs@reset \sectinit \eq@@reset \fig@@reset \tab@@reset}%
  \dot@pref\sect@pref{\sectstyle\z@\sectuserexit}}

\let\secs@reset=\relax

\outer\def\subsec#1{\par \noindent \subsecstyle{#1}}

%***  define new subsection levels

\def\sect@lev{\@ne}         % lowest subsection level
\def\sect@id{sect}          %   defined so far
\def\secs@id{secs}          % next lower level

\outer\def\newsect{\begingroup \count@\sect@lev
  \let\@\endcsname \let\or\relax
  \edef\n@xt{\new@sect}\advance\count@\@ne
  \xdef\sect@lev{\the\count@\space}\n@xt
  \glet\sect@id\secs@id \xdef\secs@id{\secs@id s}\endgroup}
\begingroup \let\newcount=\relax
  \gdef\new@sect{\wlog{\noexpand\string\secs@nm\@= subsection
      level \noexpand\sect@lev}\noexpand\newcount\secs@nm\n@m@
    \gdef\secs@nm\@reset@{\noexpand\gz@\secs@nm\n@m@}%
    \outer\gdef\secs@nm\@{\noexpand\sec@title\secs@nm\n@m@
      \secs@nm s\@reset@ \csn@me\sect@id\@pref@ \secs@nm\@pref@
      {\secs@nm\style@{\the\count@}\secs@nm\userexit@}}%
    \glet\secs@nm s\@reset@ \relax
    \gdef\secs@nm\style@{\csn@me\sect@id\style@}%
    \gdef\secs@nm\userexit@{\secs@nm s\userexit@}%
    \glet\secs@nm s\userexit@\relax
    \outer\xdef\csn@me toc\secs@id\@
      {\global\auto@toc\noexpand\the\count@\space}%
    \xdef\noexpand\save@@toc{\save@@toc\or\secs@id}}
\endgroup

\def\n@m@{n@m\@}
\def\@pref@{@pref\@}
\def\@reset@{@reset\@}
\def\style@{style\@}
\def\userexit@{userexit\@}

\def\secs@nm{\csn@me\secs@id}
\def\csn@me{\expandafter\noexpand\csname}

%***  appendix  ***

\newcount\appn@m   \appn@m=64

\outer\def\appendix{\if T\chap@page \superendpage \else\chap@break \fi
  \g@ne\appn@m \secs@reset \appinit \page@reset\appn@m
  \eq@reset \fig@reset \tab@reset \futurelet\n@xt \app@ndix}

\def\app@ndix{\ifcat\bgroup\noexpand\n@xt \expandafter\@ppendix \else
  \expandafter\@ppendix\expandafter\unskip \fi}

\def\@ppendix#1{\toks@{\case@language\appendixhead^^>\dot@pref}
  \toks@ii{#1}\appuserexit
  \titl@{\the\toks@\ \the\toks@ii}%
  \ifnum\auto@toc>\m@ne \toks@store{#1}%
  \@toc{\case@language\appendixhead\ \dot@pref}\fi}

\def\app#1{\chskipt \he@d{\case@language\appendixhead\ #1}}
%***  end of file PHYSCHAP  ***
%\input physequ
%***  this is file PHYSEQU as of 02.07.87  ***
%***  quotation of equations, figures, tables and references ***

\def\num@lett{\cat@lett \num@@lett}
\def\num@@lett#1#2{\egroup #1{#2}}

\def\num@l@tt{\cat@lett \num@@l@tt}
\def\num@@l@tt#1#{\egroup #1}

\def\cat@lett{\bgroup
  \catcode`\0\l@tter \catcode`\1\l@tter \catcode`\2\l@tter
  \catcode`\3\l@tter \catcode`\4\l@tter \catcode`\5\l@tter
  \catcode`\6\l@tter \catcode`\7\l@tter \catcode`\8\l@tter
  \catcode`\9\l@tter \catcode`\'\l@tter}

\def\quote@all#1{\leavevmode\hbox{\mathcode`\-\dq 707B$#1$}}
\def\use{\num@lett\@use}
\def\@use{\setbox\z@\hbox}

%***  equations  ***

\newtoks\e@names   \e@names={}
\newcount\eqn@m   \eqn@m=0

\outer\def\equall{\glet\eq@acs A\glet\eq@pref\empty
  \glet\def@eq\@eq \glet\eq@reset\relax \glet\eq@@reset\relax}
\outer\def\equchap{\glet\eq@acs C\gdef\eq@pref{\dot@pref}%
  \gdef\def@eq{\chap@@eq}\glet\eq@reset\eqz@ \glet\eq@@reset\relax}
\outer\def\equsect{\glet\eq@acs S\gdef\eq@pref{\sect@pref}%
  \gdef\def@eq{\sect@@eq}\glet\eq@reset\eqz@ \glet\eq@@reset\eqz@}
\def\eqz@{\gz@\eqn@m}

\outer\def\equfull{\glet\eq@fs F\gdef\eq@@fs{\let\test@eq\full@eq}}
\outer\def\equshort{\glet\eq@fs S\glet\eq@@fs\relax}

\def\@eq(#1){#1}
\def\@chap@eq{\noexpand\chap@eq\@eq}
\def\@sect@eq{\noexpand\sect@eq\@eq}
\def\@chap@sect@eq{\noexpand\chap@sect@eq\@eq}

\def\chap@eq{\test@eq\empty\dot@pref}
\def\sect@eq{\test@eq\empty\sect@dot@pref}
\def\chap@sect@eq{\test@eq\sect@eq\dot@pref}
\def\test@eq#1#2#3.{\def\n@xt{#3.}\ifx#2\n@xt \let\n@xt#1\fi \n@xt}
\def\full@eq#1#2{}
\def\short@eq#1#2#3.{#1}

\outer\def\equleft{\glet\eq@lrn L\glet\eqtag\leqno
  \glet\eq@tag\leq@no}
\outer\def\equright{\glet\eq@lrn R\glet\eqtag\eqno
  \glet\eq@tag\eq@no}
\outer\def\equnone{\glet\eq@lrn N\glet\eqtag\n@eqno
  \glet\eq@tag\neq@no}

\begingroup
  \catcode`\$=\active \catcode`\*=3 \lccode`\*=`\$
  \lowercase{\gdef\n@eqno{\catcode`\$\active
                \def$${\egroup **}\setbox\z@\hbox\bgroup}}
\endgroup
\def\eq@no{\llap{$\@lign##$}\tabskip\z@skip}
\def\leq@no{\kern-\displaywidth \rlap{$\@lign##$}\tabskip\displaywidth}
\def\neq@no{\@use{$\@lign##$}\tabskip\z@skip}

\def\displaylines{\afterassignment\display@lines \@eat}
\def\display@lines{\displ@y
  \halign\n@xt\hbox to\displaywidth{$\@lign\hfil\displaystyle##\hfil$}%
    &\span\eq@tag\crcr}

\def\eqalignno{\let\eq@@tag\eq@no \eqalign@tag}
\def\leqalignno{\let\eq@@tag\leq@no \eqalign@tag}
\def\eqaligntag{\let\eq@@tag\eq@tag \eqalign@tag}
\def\eqalign@tag{\afterassignment\eqalign@@tag \@eat}
\def\eqalign@@tag{\displ@y
  \tabskip\centering \halign to\displaywidth\n@xt
    \hfil$\@lign\displaystyle{##}$\tabskip\z@skip
    &$\@lign\displaystyle{{}##}$\hfil\tabskip\centering
    &\span\eq@@tag\crcr}

\def\fulltag#1{{\let\test@eq\full@eq#1}}
\def\shorttag#1{{\let\test@eq\short@eq#1}}

\def\eq{\g@ne\eqn@m \make@eq\empty}
\def\make@eq#1{(\eq@pref\the\eqn@m #1)}
\def\EQ{\eq \num@lett\eq@save}
\def\eq@save#1{\def@name\e@names#1{\expandafter\def@eq\make@eq\empty}}

\def\eqn{\eqtag\eq}
\def\EQN{\eqtag\EQ}

\def\eqadv{\g@ne\eqn@m}
\def\EQADV{\eqadv \num@lett\eq@save}

\newcount\seqn@m   \seqn@m=96

\def\subeqbegin{\global\seqn@m96 \subeq}
\def\SUBEQBEGIN{\global\seqn@m96 \SUBEQ}
\def\subeq{\g@ne\seqn@m \make@eq{\char\seqn@m}}
\def\SUBEQ{\num@lett\@SUBEQ}
\def\@SUBEQ#1{\subeq \def@name\e@names#1{\char\the\seqn@m}}

\def\subeqnbegin{\eqtag\subeqbegin}
\def\SUBEQNBEGIN{\eqtag\SUBEQBEGIN}
\def\subeqn{\eqtag\subeq}
\def\SUBEQN{\eqtag\SUBEQ}

\def\eqapp{\num@lett\@eqapp}
\def\@eqapp#1#2{(\fulltag#1#2)}
\def\eqnapp{\eqtag\eqapp}

\def\queq{\num@lett\@queq}
\def\@queq#1{\quote@all{\eq@@fs(#1)}}
\def\qeq{\case@abbr\eqabbr\queq}
\def\qeqs{\case@abbr\eqsabbr\queq}

\outer\def\eqrestore{\all@restore\e@names}
%***  end of file PHYSEQU  ***
%\input physstor
%***  this is file PHYSSTOR as of 03.12.87  ***
%***  storage management for figures, tables, references,  ***
%***  table of contents and (possibly) footnotes  ***

% Note: The text for figure captions, table captions, references,
%       table of contents and (if applicable) footnotes must be read
%       from the input file, i.e. tokenized, via \read@store.
%       These texts must be written to external files via \write@store
%       in order to avoid problems with very long strings.

\newtoks\toks@store
\newtoks\file@list \file@list={1234567}
\def\@sysut{\jobname.sysut}
\let\ext@ft=F

\begingroup
  \let\storebox=\relax \let\refnam=\relax  % allow input to be
  \let\RFfile=\relax \let\RFext=\relax     %   read twice
  \newhelp\opt@help{The options \string\refnam, \string\RFfile\space
    and \string\RFext\space are incompatible with \string\storebox.
    Your request will be ignored.}
  \global\opt@help=\opt@help % make this definition global
  \gdef\opt@err{{\errhelp\opt@help \errmessage{Incompatible options}}}
\endgroup

% Freeze option \storebox, \storelist or \storefile and eliminate
%   code not needed for that option
\begingroup
  \let\storebox=\relax \let\storelist=\relax  % allow input to be
  \let\storefile=\relax \let\RFfile=\relax    %   read twice
  \gdef\case@store{%
    \glet\storebox\undefined
    \if B\store@blf \glet\storebox\empty \glet\case@store\case@box
      \else \glet\box@store\undefined
      \glet\box@out\undefined \glet\box@print\undefined
      \glet\box@save\undefined \glet\box@kill\undefined \fi
    \glet\storelist\undefined
    \if L\store@blf \glet\storelist\empty \glet\case@store\case@list
      \else \glet\list@store\undefined
      \glet\list@out\undefined \glet\list@print\undefined
      \glet\list@save\undefined \glet\list@kill\undefined \fi
    \glet\storefile\undefined
    \if F\store@blf \glet\storefile\empty \glet\case@store\case@file
      \else \store@setup
      \glet\file@out\undefined \glet\file@print\undefined
      \glet\filef@rm@t\undefined \glet\fil@f@rm@t\undefined
      \glet\file@save\undefined \glet\file@kill\undefined \fi
    \glet\case@box\undefined \glet\case@list\undefined
    \glet\case@file\undefined \glet\store@setup\undefined
    \case@store}
  \gdef\store@setup{\ifx \RFfile\undefined \glet\file@store\undefined
    \glet\file@open\undefined \glet\file@close\undefined
    \glet\file@wlog\undefined \glet\file@free\undefined
    \glet\file@copy\undefined \glet\file@read\undefined \fi}
\endgroup

\outer\def\storebox{\if T\ext@ft \opt@err \else
    \if L\RF@lfe \if N\ref@sbn \opt@err
    \else \glet\store@blf B\fi \else \opt@err \fi \fi}
\outer\def\storelist{\glet\store@blf L}
\outer\def\storefile{\glet\store@blf F}

\def\read@store{\bgroup \@read@store}
\def\read@@store{\bgroup \catcode`\@\l@tter \@read@store}
\def\@read@store#1{\def\after@read{\egroup \toks@store\toks@i
    #1\after@read \ignorespaces}%
  \catcode`\^^M\active \afterassignment\after@read \global\toks@i}
\def\afterread#1{\bgroup \def\after@read{\egroup #1\after@read}}
\let\after@read=\relax
\begingroup
  \catcode`\:=\active
  \gdef\write@save#1{\write@store{:restore\@type{#1}}}
\endgroup
\def\write@store#1{\s@ve{#1{\the\toks@store}}}

\def\f@rm@t#1#2{\bgroup \ignorefoot
  \leftskip\z@skip \rightskip\z@skip \f@rmat
  \ifx @#1@\everypar{\b@format}\else
    \varitem\@indent{#1}\b@format \fi #2\e@format}
\let\f@rmat=\nointerlineskip
\def\form@t{\unskip \strut \par \@break \eform@t}
\def\b@format{\glet\e@format\form@t \strut}
\def\eform@t{\egroup \glet\e@format\eform@t}
\let\e@format=\eform@t

\def\@store{\case@store\box@store\list@store\file@store}
\def\@sstore#1#2#3{\par \noindent \bgroup \captionstyle
  \case@abbr#2#3:\enskip \the\toks@store \egroup \par \@store#1{#3.}}
\def\@add#1{\read@store{\@store#1{}}}
\def\@out#1#2#3#4#5{\case@store\box@out\list@out\file@out#1\begingroup
  \if T#2\let\chap@break\superendpage \fi \chap@break
  \chap@init{\case@language#3}%
  \if C\page@ac \skippagenum T\fi
  \page@reset6#1%
  \@style \he@d{\strut\case@language#4}\@break \@print#1%
  \ifx\chap@break\superendpage \superendpage \fi
  \def\\##1{\glet##1\undefined}\the#5\global#5\emptyt@ks
  \endgroup \fi}
\def\@print{\case@store\box@print\list@print\file@print}
\def\@save{\case@store\box@save\list@save\file@save}
\def\@kill{\case@store\box@kill\list@kill\file@kill}
\def\@ext#1#2#3 {\if B\store@blf \opt@err \else
  \glet\ext@ft T\@add#1{#2#3 }\bgroup
   \def\@store##1##2{}\input#3 \egroup \fi}
\def\@@ext#1#2{\let#1#2\everypar\emptyt@ks
  \def\read@store##1{\relax##1}\def\@store##1{\f@rm@t}\input}

\def\case@box#1#2#3{\case@@store#1%
  \fig@box\tab@box\ref@box\toc@box\foot@box}
\def\case@list#1#2#3{\case@@store#2%
  \fig@list\tab@list\ref@list\toc@list\foot@list}
\def\case@file#1#2#3{\case@@store{\expandafter#3}%
  \fig@file\tab@file\ref@file\toc@file\foot@file}

\def\case@@store#1#2#3#4#5#6#7{\ifcase#7%
  \toks@{\fig@type#1#2}\or
  \toks@{\tab@type#1#3}\or
  \toks@{\ref@type#1#4}\or
  \toks@{\toc@type#1#5}\or
  \toks@{\foot@type#1#6}\fi
  \expandafter\let\expandafter\@type\the\toks@}
\def\@style{\csname\@type style\endcsname}
\def\@indent{\csname\@type indent\endcsname}
\def\@break{\csname\@type break\endcsname}

\def\box@store#1#2{\global\setbox#1\vbox
  {\ifvbox#1\unvbox#1\fi \@style \f@rm@t{#2}{\the\toks@store}}}
\def\box@out{\ifvbox}
\def\box@print{\vskip\baselineskip \unvbox}
\begingroup
  \catcode`\:=\active \catcode`\;=\active
  \gdef\box@save#1{\wlog{; Unable to save text for
    \@type's with option :storebox}}
\endgroup
\def\box@kill#1{{\setbox\z@\box#1}}

\def\list@store#1#2{\toks@\expandafter{#1\\}%
  \xdef#1{\the\toks@ {#2}{\the\toks@store}}}
\def\list@out#1{\ifx #1\empty \else}
\def\list@print#1{\let\\\f@rm@t #1\glet#1\empty}
\def\list@save{\def\\##1##2{\toks@store{##2}\write@save{##1}}%
  \newlinechar`\^^M}
\def\list@kill#1{\glet#1\empty}

\begingroup
  \catcode`\:=\active
  \gdef\file@store#1#2#3#4{%
    \if0#3\expandafter\file@open\the\file@list\@@#1#2\fi
    {\newlinechar`\^^M\let\save@write#2\write@store{::{#4}}}}
\endgroup
\def\file@open#1#2\@@#3#4{\immediate\openout#4\@sysut#1
  \gdef#3{#3#4#1}\global\file@list{#2}\file@wlog{open}#1}
\def\file@wlog#1#2{\wlog{#1 \@sysut#2 for \@type's}}
\def\file@out#1#2#3{\if0#3\else}
\def\file@print#1#2#3{\file@close#2#3\let\\\filef@rm@t
  \file@copy#1#2#3}
\def\filef@rm@t#1{\bgroup \catcode`\@\l@tter \fil@f@rm@t{#1}}
\def\fil@f@rm@t#1#2{\egroup \f@rm@t{#1}{#2}}
\def\file@save#1#2#3{\if0#3\else \file@close#2#3%
  \def\\##1{\read@@store{\expandafter\file@store#1{##1}%
    \write@save{##1}}}%
  \newlinechar`\^^M\file@copy#1#2#3\fi}
\def\file@kill#1#2#3{\if0#3\else \file@close#2#3\file@free#1#2#3\fi}
\def\file@close#1#2{\immediate\closeout#1\file@wlog{close}#2}
\def\file@copy#1#2#3{\file@free#1#2#3\file@read#3}
\def\file@read#1{\input\@sysut#1 }
\def\file@free#1#2#3{\gdef#1{#1#20}%
  \global\file@list\expandafter{\the\file@list#3}}
%***  end of file PHYSSTOR  ***
%\input physfig
%***  this is file PHYSFIG as of 07.03.87  ***
%***  figures and tables  ***

\def\if@t#1#2#3#4#5#6{\glet#1#6\gdef#2{\dot@pref}\glet#3#5\glet#4\relax
  \if#6A\glet#2\empty \glet#3\relax \fi
  \if#6S\gdef#2{\sect@pref}\glet#4#5\fi}
\def\bf@t#1{\let\f@t#1\num@l@tt}
\def\ef@t#1#2#3#4#{\ifx @#4@\g@ne#1\xdef\thef@tn@m{#2\the#1}\else
  \gdef\thef@tn@m{#4}\fi #3\read@store\f@t}
\def\@extf@t#1#2#{\gdef\thef@tn@m{#1}\f@t}

%***  figures  ***

\newtoks\f@names   \f@names={}
\newcount\fign@m   \fign@m=0
\def\fig@type{fig}
\newbox\fig@box
\let\fig@list=\empty
\newwrite\fig@write  \def\fig@file{\fig@file\fig@write0}

\outer\def\figall{\fig@init A}
\outer\def\figchap{\fig@init C}
\outer\def\figsect{\fig@init S}
\def\fig@init{\if@t\fig@acs\fig@pref\fig@reset\fig@@reset\figz@}
\def\figz@{\gz@\fign@m}

\outer\def\figpage{\glet\fig@page T}
\outer\def\nofigpage{\glet\fig@page F}

\def\fig{\bf@t\fig@\@fig}
\def\FIG{\bf@t\fig@\@FIG}
\def\ffig{\bf@t\ffig@\@fig}
\def\FFIG{\bf@t\ffig@\@FIG}

\def\@fig{\ef@t\fign@m\fig@pref\relax}
\def\@FIG#1{\ef@t\fign@m\fig@pref{\def@name\f@names#1{\thef@tn@m}}}

\def\fig@{\@store0{\thef@tn@m .}}
\def\ffig@{\@sstore0\figpref{\thef@tn@m}}
\def\figadd{\@add0}

\def\qufig{\case@abbr\figabbr\num@lett\quote@all}

\outer\def\figout{\@out0\fig@page\figpref\fighead\emptyt@ks}
\outer\def\figkill{\@kill0}
\outer\def\restorefig#1{\read@store{\@store0{#1}}}
\outer\def\figrestore{\all@restore\f@names}

\outer\def\FIGext{\@ext0\FIG@ext}
\def\FIG@ext{\@@ext\@FIG\@extf@t}
%***  end of file PHYSFIG  ***
%\input physpict
%***  this is file PHYSPICT as of 28.04.87  ***
%***  picture macros  ***

\newdimen\spictskip  \spictskip=2.5pt

\def\pict{\bf@t\pict@\@fig}
\def\PICT{\bf@t\pict@\@FIG}

\def\pict@{\vskip\the\toks@store \bpargroup
  \raggedright \captionstyle \varitem{\qufig{\thef@tn@m\,}:}%
  \let\@spict\spict@ \spacefactor998\ignorespaces}

\def\spict#1{\ifnum\spacefactor=998\else \parvskip\spictskip \fi
  \@spict{#1\enskip}\ignorespaces}
\def\spict@#1{\setbox\z@\hbox{#1}\advance\hangindent\wd\z@
  \box\z@ \let\@spict\llap}

%***  graphic macros  ***

\outer\def\graphics{\glet\graphic\gr@phic}
\outer\def\nographics{\glet\graphic\nogr@phic}

\def\gr@phic#1{\vbox\bgroup \def\gr@@@ph{#1}\bfr@me\gr@ph}
\def\nogr@phic#1{\vbox\bgroup \write\m@ne{Insert plot #1}\bfr@me\fr@me}
\def\frame{\vbox\bgroup \bfr@me\fr@me}
\def\bfr@me#1#2#3#4{\tfr@me\z@\dimen@iv#2\relax % set and test \wd
  \dimen@#3\relax \tfr@me\dimen@\dimen@ii#4\relax % set and test \dp
  \setbox\z@\hbox to\dimen@iv{#1}\ht\z@\dimen@ \dp\z@\dimen@ii
  \box\z@ \egroup}
\def\tfr@me#1#2#3\relax{#2#3\relax \ifdim#2<-#1\errhelp\fr@mehelp
  \errmessage{Invalid box size}#2-#1\fi}
\newhelp\fr@mehelp{The \string\wd\space and \string\ht+\string\dp\space
  of a \string\frame\space or \string\graphic\space \string\box\space
  must not be negative and will be changed to 0pt.}

\def\gr@ph{\lower\dimen@ii\gr@@ph b\hfil
  \raise\dimen@\gr@@ph e}
\def\gr@@ph#1{\hbox{\special{^X\gr@@@ph^A}}}         % temporary version
\def\gr@@ph#1{\hbox{\special{#1plot GKSM \gr@@@ph}}} % standard interface
\def\gr@@ph#1{\hbox{\special{^X#1plot GKSM \gr@@@ph^A}}} % temporary fix
\def\fr@me{\vrule\fr@@@me
  \bgroup \dimen@ii-\dimen@ \fr@@me\dimen@ii \hfilneg
  \bgroup \dimen@-\dimen@ii \fr@@me\dimen@ \vrule\fr@@@me}
\def\fr@@me#1{\advance#1.4\p@ \leaders \hrule\fr@@@me \hss \egroup}
\def\fr@@@me{\@height\dimen@ \@depth\dimen@ii}
%***  end of file PHYSPICT  ***
%\input phystab
%***  this is file PHYSTAB as of 07.03.87  ***
%***  tables  ***

\newtoks\t@names   \t@names={}
\newcount\tabn@m   \tabn@m=0
\def\tab@type{tab}
\newbox\tab@box
\let\tab@list=\empty
\newwrite\tab@write  \def\tab@file{\tab@file\tab@write0}

\outer\def\taball{\tab@init A}
\outer\def\tabchap{\tab@init C}
\outer\def\tabsect{\tab@init S}
\def\tab@init{\if@t\tab@acs\tab@pref\tab@reset\tab@@reset\tabz@}
\def\tabz@{\gz@\tabn@m}

\outer\def\tabpage{\glet\tab@page T}
\outer\def\notabpage{\glet\tab@page F}

\def\tab{\bf@t\tab@\@tab}
\def\TAB{\bf@t\tab@\@TAB}
\def\ttab{\bf@t\ttab@\@tab}
\def\TTAB{\bf@t\ttab@\@TAB}

\def\@tab{\ef@t\tabn@m\tab@pref\relax}
\def\@TAB#1{\ef@t\tabn@m\tab@pref{\def@name\t@names#1{\thef@tn@m}}}

\def\tab@{\@store1{\thef@tn@m .}}
\def\ttab@{\@sstore1\tabpref{\thef@tn@m}}
\def\tabadd{\@add1}

\def\qutab{\case@abbr\tababbr\num@lett\quote@all}

\outer\def\tabout{\@out1\tab@page\tabpref\tabhead\emptyt@ks}
\outer\def\tabkill{\@kill1}
\outer\def\restoretab#1{\read@store{\@store1{#1}}}
\outer\def\tabrestore{\all@restore\t@names}

\outer\def\TABext{\@ext1\TAB@ext}
\def\TAB@ext{\@@ext\@TAB\@extf@t}
%***  end of file PHYSTAB  ***
%\input phystabl
%***  this is file PHYSTABL as of 02.05.88  ***
%***  table macro  ***

\newskip\htabskip   \htabskip=1em plus 2em minus .5em
\newdimen\vtabskip  \vtabskip=2.5pt
\newbox\tab@top   \newbox\tab@bot

\let\@hrule=\hrule
\let\@halign=\halign
\let\@valign=\valign
\let\@span=\span
\let\@omit=\omit

\def\@@span{\@span\@omit\@span}
\def\@@@span{\@span\@omit\@@span}

\def\sp@n{\span\@omit\advance\mscount\m@ne} % for multispan

\def\table#1#{\vbox\bgroup\offinterlineskip
  \toks@ii{#1\bgroup \unhcopy\tab@top \unhcopy\tab@bot
    ##}\afterassignment\tab@preamble \@eat}

%***  scan preamble of table

\def\tab@preamble#1\cr{\let\tab@@vrule\tab@repeat
  \let\tab@amp@\empty \let\tab@amp\empty \let\span@\@@span
  \toks@{\tab@space#1&\cr}\the\toks@}
\def\tab@space{\tab@test{ }{}\tab@vrule}                 % blank
\def\tab@vrule{\tab@test\vrule{\tab@add\vrule}%          % \vrule
  \tab@@vrule}
\def\tab@repeat{\tab@test&{\tab@add&%                    % & (repeat)
    \let\tab@@vrule\tab@template \let\tab@amp@\tab@@amp
    \let\tab@amp&\let\span@\@@@span}\tab@template}
\def\tab@template#1&{\tab@add{\@span\tab@amp@
    \tabskip\htabskip&\tab@setup#1&\tabskip\z@skip##}%   % <template>
  \tab@test\cr{\let\tab@@vrule\tab@exec}\tab@space}      % \cr
\def\tab@@amp{&##}

\def\tab@test#1#2#3{\let\tab@comp= #1\toks@{#2}\let\tab@go#3%
  \futurelet\n@xt \tab@@test}
\def\tab@@test{\ifx \tab@comp\n@xt \the\toks@
    \afterassignment\tab@go \expandafter\@eat \else
  \expandafter\tab@go \fi}

\def\tab@add#1{\toks@ii\expandafter{\the\toks@ii#1}}

%*** execute preamble of table

\def\tab@exec{\tab@r@set\everycr{\tab@body}%
  \def\halign{\tab@r@set \halign}\def\valign{\tab@r@set \valign}%
  \def\omit{\@omit \tab@setup}%
  \def\n@xt##1{\hbox{\dimen@\ht\strutbox\dimen@ii\dp\strutbox
    \advance##1\vtabskip \vrule \@height\dimen@ \@depth\dimen@ii
    \@width\z@}}%
  \setbox\tab@top\n@xt\dimen@ \setbox\tab@bot\n@xt\dimen@ii
  \def\ml##1{\relax                                      % multi
    \ifmmode \let\@ml\empty \else \let\@ml$\fi           %   line
    \@ml\vcenter{\hbox\bgroup\unhcopy\tab@top            %   mode
    ##1\unhcopy\tab@bot\egroup}\@ml}%
  \def\nl{\egroup\hbox\bgroup\strut}%                    % new line
  \tabskip\z@skip\@halign\the\toks@ii\cr}

\def\tab@r@set{\let\cr\endline \everycr\emptyt@ks
  \let\halign\@halign \let\valign\@valign
  \let\span\@span \let\omit\@omit}
\def\tab@setup{\relax \iffalse {\fi \let\span\span@ \iffalse }\fi}

%***  scan and execute body of table

\def\tab@body{\noalign\bgroup \tab@@body}           % scan inside
\def\tab@@body{\futurelet\n@xt \tab@end}            %   noalign
\def\tab@end{\ifcat\egroup\noexpand\n@xt
    \expandafter\egroup \expandafter\egroup \else        % }
  \expandafter\tab@blank \fi}
\def\tab@blank{\ifcat\space\noexpand\n@xt
    \afterassignment\tab@@body \expandafter\@eat \else   % blank
  \expandafter\tab@hrule \fi}
\def\tab@hrule{\ifx\hrule\n@xt
    \def\hrule{\@hrule\egroup \tab@body}\else            % \hrule
  \expandafter\tab@noalign \fi}
\def\tab@noalign{\ifx\noalign\n@xt
    \aftergroup\tab@body \expandafter\@eat@ \else        % \noalign
  \expandafter\tab@row \fi}
\def\tab@row#1\cr{\toks@ii{\toks@ii{\egroup}%            % one row
    \tab@item#1&\cr}\the\toks@ii}
\def\tab@item#1&{\tab@add{\tab@amp&#1&}%                 % one item
  \futurelet\n@xt \tab@cr}
\def\tab@cr{\ifx\cr\n@xt
    \expandafter\the\expandafter\toks@ii \else           % \cr
  \expandafter\tab@item \fi}
%***  end of file PHYSTABL  ***
%\input physref
%***  this is file PHYSREF as of 13.10.87  ***
%***   references  ***

\newtoks\r@names   \r@names={}
\newcount\refn@m   \refn@m=0
\newcount\ref@temp
\def\ref@type{ref}
\newbox\ref@box
\let\ref@list=\empty
\newwrite\ref@write  \def\ref@file{\ref@file\ref@write0}

\begingroup \let\refnam=\relax  % allow input to be read twice
  \newhelp\ref@help{The option \string\refnam\space allows predefined
    references only and is incompatible with \string\qref(s).
    Your request will be ignored.}
  \global\ref@help=\ref@help % make this definition global
  \gdef\ref@err{{\errhelp\ref@help \errmessage{Invalid request}}}
\endgroup

% Freeze option \refsup, \refsqb or \refnam and eliminate
%   code not needed for that option
\begingroup
  \let\refsup=\relax \let\refsqb=\relax  % allow input to be
  \let\refnam=\relax                     %   read twice
  \gdef\ref@setup{%
    \glet\refsup\undefined
    \if S\ref@sbn \glet\refsup\empty \glet\therefn@m\suprefn@m \fi
    \glet\refsqb\undefined
    \if B\ref@sbn \glet\refsqb\empty \glet\therefn@m\sqbrefn@m \fi
    \glet\refnam\undefined
    \if N\ref@sbn \glet\refnam\empty \glet\the@quref\nam@quref
      \glet\@@ref\ref@err \gdef\qref{\ref@err \quref}\glet\qrefs\qref
      \glet\RF@def@\RF@def@nam
      \glet\RF@find\undefined \glet\RF@search\undefined
      \glet\RF@locate\undefined \glet\@RFread\undefined
      \glet\qurefsup\ref@err \glet\sup@quref\undefined
      \glet\qurefsqb\ref@err \glet\sqb@quref\undefined
      \glet\qurefnum\ref@err \glet\num@quref\undefined
      \glet\ref@restore\ref@err
      \else \gdef\@@ref{\@store2\therefn@m}%
      \gdef\qref{\case@abbr\refabbr\num@lett\quote@all}%
      \gdef\qrefs{\case@abbr\refsabbr\num@lett\quote@all}%
      \glet\RF@def@\RF@def@num \glet\RF@print\undefined
      \gdef\ref@restore{\all@restore\r@names}\fi
    \glet\nam@quref\undefined
    \gdef\quref{\ref@unskip \num@lett\the@quref}%
    \glet\RF@def@num\undefined \glet\RF@def@nam\undefined
    \gdef\RF@restore{\all@restore\R@names}%
    \glet\ref@setup\undefined}
\endgroup

\outer\def\refsup{\glet\ref@sbn S\global\qurefsup}
\outer\def\refsqb{\glet\ref@sbn B\global\qurefsqb}
\outer\def\refnam{\if B\store@blf \opt@err \else \glet\ref@sbn N\fi}

\def\qurefsup{\let\the@quref\sup@quref}
\def\qurefsqb{\let\the@quref\sqb@quref}
\def\qurefnum{\let\the@quref\num@quref}

\outer\def\refpage{\glet\ref@page T}
\outer\def\norefpage{\glet\ref@page F}

\outer\def\refkeep{\glet\ref@kc K}
\outer\def\refclear{\glet\ref@kc C}

\def\ref{\ref@advance \refend \@ref}
\def\REF{\num@lett\@REF}
\def\@REF#1{\ref@name#1\@ref}
\def\refend{\quref{\the\refn@m}}

\def\refs{\ref@advance \ref@temp\refn@m \@ref}
\def\REFS{\num@lett\@REFS}
\def\@REFS#1{\ref@name#1\ref@temp\refn@m \@ref}
\def\refscon{\ref@advance \@ref}
\let\REFSCON=\REF
\def\refsend{\quref{\the\ref@temp -\the\refn@m}}

\def\ref@advance{\ref@unskip \g@ne\refn@m}
\def\ref@name{\ref@@name\r@names}
\def\ref@@name#1#2{\ref@advance \def@name#1#2{\the\refn@m}}
\def\ref@unskip{\ifhmode \unskip \fi}

\def\suprefn@m{\the\refn@m .}
\def\sqbrefn@m{$\lbrack \the\refn@m \rbrack$}
\def\@ref{\read@store\@@ref}
\def\@@ref{\ref@setup \@@ref}
\def\refadd{\@add2}


\def\sup@quref#1{\leavevmode \nobreak \quote@all{^{#1}}}
\def\sqb@quref#1{\ \quote@all{\lbrack #1\rbrack}}
\def\num@quref{\ \quote@all}
\def\nam@quref{\@use}
\def\quref{\ref@setup \quref}
\def\qref{\ref@setup \qref}
\def\qrefs{\ref@setup \qrefs}

\outer\def\refout{{\if K\ref@kc \@out2\ref@page\refpref\refhead\emptyt@ks
  \else \@out2\ref@page\refpref\refhead\r@names
  \let\\\@RFdef \the\R@names \global\R@names\emptyt@ks
  \if L\RF@lfe \else \glet\RF@list\empty \gz@\RF@high \fi
  \gz@\refn@m \fi}}
\outer\def\refkill{\@kill2}
\outer\def\restoreref#1{\read@@store{\@store2{#1}}}
\outer\def\refrestore{\ref@restore}
\outer\def\RFrestore{\RF@restore}
\def\ref@restore{\ref@setup \ref@restore}
\def\RF@restore{\ref@setup \RF@restore}

\outer\def\REFext{\@ext2\REF@ext}
\def\REF@ext{\@@ext\ref@name\refn@m}

%***  predefined references  ***

\newtoks\R@names   \R@names={}
\newcount\RFn@m   \newcount\RF@high
\newcount\RFmax   \RFmax=50  % maximum number of RF's held in storage
\def\RF@type{RF}
\let\RF@list=\empty
\newwrite\RF@write  \def\RF@file{\RF@file\RF@write0}
\let\RF@noc=N

% Freeze option \RFlist, \RFfile or \RFext and eliminate
%   code not needed for that option
\begingroup \let\storefile=\relax        % allow input
  \let\RFlist=\relax \let\RFfile=\relax  %   to be read
  \let\RFext=\relax \let\RF=\relax       %   twice
  \gdef\@RF{%
    \glet\RFlist\undefined
    \if L\RF@lfe \glet\RFlist\empty \glet\@RF\@RFlist
      \gdef\RF@input{\RF@list}%
      \else \glet\@RFlist\undefined \fi
    \glet\RFfile\undefined
    \if F\RF@lfe \glet\RFfile\empty \glet\@RF\@RFfile
      \else \RF@setup
      \glet\@RFfile\undefined \glet\@RFcopy\undefined
      \glet\RF@store\undefined \glet\RF@copy\undefined
      \glet\RF@@input\undefined \fi
    \glet\RFext\undefined
    \if E\RF@lfe \gdef\RFext##1 {}\glet\@RF\@RFext
      \else \glet\@RFext\undefined \fi
    \glet\RF@setup\undefined \@RF}
  \gdef\RF@setup{\ifx \storefile\undefined \glet\file@store\undefined
    \glet\file@open\undefined \glet\file@close\undefined
    \glet\file@wlog\undefined \glet\file@free\undefined
    \glet\file@copy\undefined \glet\file@read\undefined \fi}
\endgroup

\outer\def\RFlist{\glet\RF@lfe L}
\outer\def\RFfile{\if B\store@blf \opt@err \else \glet\RF@lfe F\fi}
\outer\def\RFext#1 {\if B\store@blf \opt@err \else
  \glet\RF@lfe E\gdef\RF@input{\input#1 }\RF@input \fi}

\def\@RFdef#1{\gdef#1{\RF@def#1}}
\def\@RF@list#1{\toks@\expandafter{\RF@list\RF@#1}%
  \xdef\RF@list{\the\toks@ {\the\toks@store}}}
\def\@RFcopy#1{\RF@store{\noexpand#1}}
\def\RF@store{\let\@type\RF@type \if C\RF@noc \glet\RF@noc N%
  \RF@copy \fi \glet\RF@noc O\expandafter\file@store\RF@file}
\def\RF@input{\expandafter\RF@@input\RF@file}
\begingroup \let\RF=\relax  % allow input to be read twice
  \gdef\RF@copy{{\let\\\RF \let\@RF\@RFcopy
    \expandafter\file@copy\RF@file}}
  \gdef\RF@@input#1#2#3{\if O\RF@noc \let\@type\RF@type
    \file@close#2#3\glet\RF@noc C\fi \let\\\RF \file@read#3}
\endgroup

\def\RF@def#1{\ref@@name\R@names#1\RF@def@ #1}
\def\RF@def@{\ref@setup \RF@def@}
\def\RF@def@num{\toks@store\expandafter{\expandafter\RF@find
  \expandafter{\the\refn@m}}\@@ref}
\def\RF@def@nam{\ifnum\refn@m=\@ne \refadd{\RF@print}\fi}
\def\RF@test#1{\z@}

\def\RF@print{\let\@RF\@RF@print \let\RF@def\RF@test
  \let\RF@first T\RF@input}
\def\@RF@print#1{\ifnum#1>\z@
  \if\RF@first T\let\RF@first F\setbox\z@\lastbox
  \else \form@t\f@rmat \noindent \strut \fi
  \hangindent\namrefindent \the\toks@store \fi}

\def\RF@find#1{\RFn@m#1\bgroup \let\RF@def\RF@test
  \if L\RF@lfe \else \ifnum\RFn@m<\RF@high \else \RF@search \fi \fi
  \let\RF@\RF@locate \RF@list \egroup}

\def\RF@search{\global\RF@high\RFn@m \global\advance\RF@high\RFmax
  \glet\RF@list\empty \let\@RF\@RFread \let\par\relax \RF@input}
\def\@RFread#1{\ifnum#1<\RFn@m \else \ifnum#1<\RF@high
  \@RF@list#1\fi \fi}
\def\RF@locate#1#2{\ifnum#1=\RFn@m #2\fi}

\def\@RFlist#1{\@RFdef#1\@RF@list#1}
\def\@RFfile#1{\@RFdef#1\@RFcopy#1}
\let\@RFext=\@RFdef

\outer\def\RF{\num@lett\RF@}
\def\RF@#1{\ref@unskip \read@store{\@RF#1}}
%***  end of file PHYSREF  ***
%\input physjrnl
%***  this is file PHYSJRNL as of 11.02.86  ***
%***  physics journals  ***

\outer\def\yearpage{\glet\yearpage@yp Y}
\outer\def\pageyear{\glet\yearpage@yp P}

\def\journal#1{{\journalstyle{#1}}\j@urnal{}}
\def\journalp#1{{\journalstyle{#1}}\j@urnal}
\def\journalf#1#2#3({{\journalstyle{#1}}\j@urnal{#3}#2(}

\def\j@urnal#1#2(#3)#4*{\unskip
  \ {\volumestyle{#1\ifx @#1@\else\ifx @#2@\else
  \kern.2em\fi \fi#2}}\unskip
  \ifx @#3@\else\ifx @#4@ (#3)\else\if Y\yearpage@yp\ (#3) #4\else
  , #4 (#3)\fi \fi \fi}

\def\Lett{Lett.\ }
\def\Math{Math.\ }
\def\Nucl{Nucl.\ }
\def\Phys{Phys.\ }
\def\Rev{Rev.\ }

\def\NPA{\journalp{\Nucl\Phys}A}
\def\NPB{\journalp{\Nucl\Phys}B}
\def\PLA{\journalf{\Phys\Lett}A}
\def\PLB{\journalf{\Phys\Lett}B}
\def\PR{\journal{\Phys\Rev}}
\def\PRA{\journalp{\Phys\Rev}A}
\def\PRB{\journalp{\Phys\Rev}B}
\def\PRC{\journalp{\Phys\Rev}C}
\def\PRD{\journalp{\Phys\Rev}D}
\def\PRL{\journal{\Phys\Rev\Lett}}
%***  end of file PHYSJRNL  ***
%\input phystoc
%***  this is file PHYSTOC as of 02.02.87  ***
%***  table of contents  ***

\newcount\tocn@m   \tocn@m=0
\newcount\auto@toc   \auto@toc=-1
\let\toc@saved\empty

\def\toc@type{toc}
\newbox\toc@box
\let\toc@list=\empty
\newwrite\toc@write  \def\toc@file{\toc@file\toc@write0}

\outer\def\tocpage{\glet\toc@page T}
\outer\def\notocpage{\glet\toc@page F}

\outer\def\tocnone{\gm@ne\auto@toc}
\outer\def\tocchap{\gz@\auto@toc}
\outer\def\tocsect{\global\auto@toc\@ne}

\def\toc#1{\read@store{\@toc{#1}}}
\def\tocadd{\@add3}
\def\@toc{\g@ne\tocn@m
  \expandafter\@@toc\csname toc@\romannumeral\tocn@m\endcsname}
\def\@@toc#1{\pagelabel#1%
  \toks@store\expandafter{\the\toks@store\toc@fill#1}\@store3}
\def\toc@fill{\rightskip4em\@plus1em\@minus1em\parfillskip-\rightskip
  \unskip\vadjust{}\leaders\hbox to1em{\hss.\hss}\hfil}

\outer\def\tocout{\@out3\toc@page\tocpref\tochead\emptyt@ks}
\outer\def\tockill{\@kill3}
\outer\def\restoretoc#1{\read@@store{\@store3{#1}}}
%***  end of file PHYSTOC  ***
%\input physfoot
%***  this is file PHYSFOOT as of 31.12.86  ***
%***  footnotes  ***

\newcount\footn@m   \footn@m=0
\def\foot@type{foot}
\newbox\foot@box
\let\foot@list=\empty
\newwrite\foot@write  \def\foot@file{\foot@file\foot@write0}

\outer\def\footsqb{\glet\foot@bp B\glet\thefootn@m\sqbfootn@m}
\outer\def\footpar{\glet\foot@bp P\glet\thefootn@m\parfootn@m}

\outer\def\footbot{\glet\foot@be B\glet\vfootnote\vfootn@te}
\outer\def\footend{\glet\foot@be E\glet\vfootnote\foot@store}

\outer\def\footpage{\glet\foot@page T}
\outer\def\nofootpage{\glet\foot@page F}

\def\sqbfootn@m{\lbrack \the\footn@m \rbrack}
\def\parfootn@m{\the\footn@m )}

\def\foot{\hfoot \vfootnote\footid}
\def\hfoot{\g@ne\footn@m \edef\n@xt{{$^{\thefootn@m}$}}%
  \expandafter\hfootnote\n@xt}
\def\footnote#1{\hfootnote{#1}\vfootnote\footid}
\def\hfootnote#1{\let\@sf\empty
  \ifhmode\unskip\edef\@sf{\spacefactor\the\spacefactor}\/\fi
  #1\@sf \gdef\footid{#1}}
\def\vfootn@te#1{\insert\footins\bgroup \foot@style
  \llap{#1}\after@arg\@foot}
\let\fo@t=\undefined
\let\f@@t=\undefined
\let\f@t=\undefined

\def\foot@style{\footstyle  % initializiation for footnote insertions
  \interlinepenalty\interfootnotelinepenalty
  \baselineskip\footnotebaselineskip
  \splittopskip\interfootnoteskip % top baseline for broken footnotes
  \splitmaxdepth\dp\strutbox \floatingpenalty\@MM
  \leftskip.05\hsize \rightskip\z@skip
  \spaceskip\z@skip \xspaceskip\z@skip \noindent \footstrut}

\def\foot@store#1{\read@store{\@store4{#1}}}
\def\footadd{\@add4}

\outer\def\footout{\@out4\foot@page\footpref\foothead\emptyt@ks}
\outer\def\footkill{\@kill4}
\outer\def\restorefoot#1{\read@store{\@store4{#1}}}

\def\ignorefoot{\let\foot\eat \let\hfoot\eat  % ignore footnotes
  \def\footnote{\expandafter\eat\eat}\let\hfootnote\footnote}
%***  end of file PHYSFOOT  ***
%\input physitem
%***  this is file PHYSITEM as of 07.02.87  ***
%***  item  ***

\let\plainitem=\item     % rename \item from PLAIN

\def\varitem{\afterassignment\v@ritem \setbox\z@\hbox}
\def\v@ritem{\hss \bgroup \aftergroup\v@@ritem}
\def\v@@ritem{\enskip \egroup \endgraf \noindent
  \hangindent\wd\z@ \box\z@ \ignorespaces}

\def\hvskip{\afterassignment\h@vskip \skip@}
\def\h@vskip{\unskip\nobreak \vadjust{\vskip\skip@}\lb \ignorespaces}

\def\parvskip{\bgroup \afterassignment\par@vskip \parskip}
\def\par@vskip{\parindent\hangindent \endgraf \indent \egroup
  \ignorespaces}

\def\item{\varitem to2.5em}
\def\sitem{\varitem to4.5em}
\def\ssitem{\varitem to6.5em}

\def\itemcon{\item{}}
\def\sitemcon{\sitem{}}
\def\ssitemcon{\ssitem{}}

%***  point  ***

\newcount\pointn@m   \pointn@m=0
\def\pointbegin{\gz@\pointn@m \point}
\def\point{\g@ne\pointn@m
  \xdef\the@label{\the\pointn@m}\item{\the@label.}}

\newcount\spointn@m   \spointn@m=96
\def\spointbegin{\global\spointn@m96 \spoint}
\def\spoint{\g@ne\spointn@m
  \xdef\the@label{\char\the\spointn@m}\sitem{(\the@label)}}

\newcount\sspointn@m   \sspointn@m=0
\def\sspointbegin{\gz@\sspointn@m \sspoint}
\def\sspoint{\g@ne\sspointn@m
  \xdef\the@label{\romannumeral\sspointn@m}\ssitem{\the@label)}}

\let\pointcon=\itemcon
\let\spointcon=\sitemcon
\let\sspointcon=\ssitemcon

%***  bull-, star- and dashitem  ***

\def\bullitem{\item{$\bullet $}}
\def\staritem{\item{$\ast $}}
\def\dashitem{\item{---}}
\def\sbullitem{\sitem{$\bullet $}}
\def\sstaritem{\sitem{$\ast $ }}
\def\sdashitem{\sitem{---}}
\def\ssbullitem{\ssitem{$\bullet $}}
\def\ssstaritem{\ssitem{$\ast $ }}
\def\ssdashitem{\ssitem{---}}
%***  end of file PHYSITEM  ***
%\input physmath
%***  this is file PHYSMATH as of 14.04.88  ***
%***  matrices  ***

\def\matc{\let\mat@lfil\hfil \let\mat@rfil\hfil}
\def\matl{\let\mat@lfil\relax \let\mat@rfil\hfil}
\def\matr{\let\mat@lfil\hfil \let\mat@rfil\relax}

\def\matrix#1{\null\,\vcenter{\normalbaselines\m@th
    \ialign{$\mat@lfil##\mat@rfil$&&\quad$\mat@lfil##\mat@rfil$\crcr
      \mathstrut\crcr\noalign{\kern-\baselineskip}%
      #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,}

\def\bordermatrix#1{\begingroup \m@th
  \setbox\z@\vbox{%
    \def\cr{\crcr\noalign{\kern2\p@\glet\cr\endline}}%
    \ialign{$##\hfil$\kern2\p@\kern\p@renwd&\thinspace$\mat@lfil##%
      \mat@rfil$&&\quad$\mat@lfil##\mat@rfil$\crcr
      \omit\strut\hfil\crcr\noalign{\kern-\baselineskip}%
      #1\crcr\omit\strut\cr}}%
  \setbox\tw@\vbox{\unvcopy\z@\global\setbox\@ne\lastbox}%
  \setbox\tw@\hbox{\unhbox\@ne\unskip\global\setbox\@ne\lastbox}%
  \setbox\tw@\hbox{$\kern\wd\@ne\kern-\p@renwd\left(\kern-\wd\@ne
    \global\setbox\@ne\vbox{\box\@ne\kern2\p@}%
    \vcenter{\kern-\ht\@ne\unvbox\z@\kern-\baselineskip}\,\right)$}%
  \null\;\vbox{\kern\ht\@ne\box\tw@}\endgroup}

\let\mat=\pmatrix

%***  additional math symbols  ***

\mathchardef\smallsum=\dq1006
\mathchardef\smallprod=\dq1005

\def\b@mmode{\relax\ifmmode \expandafter\c@mmode \else $\fi}
\def\c@mmode#1\e@mmode{#1}
\def\e@mmode{$}
\def\defmmode#1#2{\def#1{\b@mmode#2\e@mmode}}

\defmmode\{{\lbrace}
\defmmode\}{\rbrace}

\defmmode\,{\mskip\thinmuskip}
\defmmode\>{\mskip\medmuskip}
\defmmode\;{\mskip\thickmuskip}

\defmmode{\Mit#1}{\mit#1}
\defmmode{\Cal#1}{\cal\uppercase\expandafter{#1}}

\def\dotii#1{{\mathop{#1}\limits^{\vbox to -1.4\p@{\kern-2\p@
   \hbox{\tenrm..}\vss}}}}
\def\dotiii#1{{\mathop{#1}\limits^{\vbox to -1.4\p@{\kern-2\p@
   \hbox{\tenrm...}\vss}}}}
\def\dotiv#1{{\mathop{#1}\limits^{\vbox to -1.4\p@{\kern-2\p@
   \hbox{\tenrm....}\vss}}}}
\def\zero#1{{\mathop{#1}\limits^\circ}}

\let\vecsymbol=\rightarrow
\let\barsymbol -
\mathchardef\tildesymbol=\dq0218
\def\hatsymbol{{\mathchoice{\null}{\null}{\,\,\hbox{\lower 10\p@\hbox
    {$\widehat{\null}$}}}{\,\hbox{\lower 20\p@\hbox
       {$\hat{\null}$}}}}}
\def\dotiisymbol{{\nonscript\,\hbox{\tenrm..}}}
\def\dotiiisymbol{{\nonscript\,\hbox{\tenrm...}}}
\def\dotivsymbol{{\nonscript\,\hbox{\tenrm....}}}
\def\dotsymbol{{\nonscript\,\hbox{\tenrm.}}}
\let\zerosymbol=\circ

%***  mathematical statements  ***

\def\begin@stmt{\par\noindent\bpargroup\stmttitlestyle}
\def\adv@stmt#1#2#3#4{\begin@stmt
  \count@\ifx#2#3#1 \else\z@ \glet#2#3\fi \advance\count@\@ne
  \xdef#1{\the\count@}\edef\the@label{#4#1}}

\def\make@stmt{\ \the@label \make@@stmt}
{\catcode`\:\active
  \gdef\make@@stmt{\ \catcode`\:\active \let:\end@stmt}
}
\def\end@stmt{\catcode`\:\@ther \unskip :\stmtstyle
  \enskip \ignorespaces}

\def\defstmt#1#2#3{\expandafter\def@stmt \csname#1\endcsname
  {#2}{#1@stmt@}#3@}
\def\def@stmt#1#2#3#4#5@{\bgroup
  \toks@{\begin@stmt #2\make@@stmt}\toks@ii{#2\make@stmt}%
  \if#4n\xdef#1{\the\toks@}\else
    \xdef#1{\csname#3adv\endcsname \the\toks@ii}%
    \if#4=\edef\n@xt{\expandafter\noexpand
      \csname#5@stmt@adv\endcsname}\else
      \toks@{\empty}\toks@ii\toks@ \if#4c\toks@{\dot@pref}\fi
      \if#4s\toks@{\sect@pref}\if#5c\toks@ii{\sect@dot@pref}\fi \fi
      \if#5a\toks@ii\toks@ \fi
      \edef\n@xt{\noexpand\adv@stmt \csname#3num\endcsname
        \csname#3save\endcsname \the\toks@ \the\toks@ii}\fi
    \expandafter\glet\csname#3adv\endcsname\n@xt \fi
  \egroup}

\def\Prf{\par\noindent\bpargroup\prftitlestyle \case@language\prfhead
  \let\stmtstyle\prfstyle \make@@stmt}
%***  end of file PHYSMATH  ***
%\input physsave
%***  this is file PHYSSAVE as of 13.10.87  ***
%***  save and restore  ***

\def\save@type{.texsave }  % file type for save and restore
\newread\test@read
\newwrite\save@write

\def\s@ve{\immediate\write\save@write}
\begingroup \catcode`\:=\active \catcode`\;=\active
  \outer\gdef\save#1 {{\let\,\space
    \immediate\openout\save@write#1\save@type
    \s@ve{;* definitions for :restore #1 \date\space- \thetime\space*}%
    \s@ve{:comment}%
    \s@ve{:\case@language{german\else english}}\save@page \save@chap
    \save@equ \save@fig \save@tab \save@ref \save@toc \save@foot
    \bgroup \def\n@xt##1.##2{\advance##2\@ne \s@ve{:start##1\the##2}}%
      \n@xt chap.\chapn@m \n@xt sect.\sectn@m
      \n@xt appendix.\appn@m \n@xt equ.\eqn@m
      \n@xt fig.\fign@m \n@xt tab.\tabn@m
      \n@xt ref.\refn@m \n@xt toc.\tocn@m
      \n@xt foot.\footn@m \egroup
    \s@ve{:\ifx\chap@@eq\sect@@eq app\else
      \ifnum\chapn@m=\z@ sect\else chap\fi \fi init}%
    \@save0\@save1\@save2\@save3\@save4%
    \if K\ref@kc \s@ve{:endcomment}\fi
    \save@restore ref \r@names  \save@restore RF \R@names
    \if C\ref@kc \s@ve{:endcomment}\fi
    \save@restore lbl \l@names  \save@restore eq \e@names
    \save@restore fig \f@names  \save@restore tab \t@names
    \s@ve{;*  end of definitions  *}\immediate\closeout\save@write
    }\wlog{* file #1\save@type saved *}}
  \gdef\save@restore#1 {\def\\{\s@ve{:#1restore}%
      \let\\\save@@restore \\}\the}
  \gdef\save@@restore#1{\toks@\expandafter{#1}%
    \s@ve{:dorestore\string#1{\the\toks@}}}
  \gdef\save@page{\s@ve{:\@opt\page@tbn Ttop Bbot Nno *pagenum%
      :\@opt\head@lrac Llef Rrigh Aal Ccen *thead%
      :\@opt\foot@lrac Llef Rrigh Aal Ccen *tfoot%
      :page\@opt\page@ac Aall Cchap *%
      :\@opt\ori@pl Pportrait Llandscape *}%
    \s@ve{:startpage\@opt\page@ac C\the\pageno@pref. *\the\pageno}}
  \gdef\save@chap{\s@ve{:\@opt\chap@page Fno *chappage%
      :\@opt\chap@yn Nno *chapters%
      :\@opt\chap@ar Aarabic Rroman *chapnum}}
  \gdef\save@equ{\s@ve{:equ\@opt\eq@acs Aall Cchap Ssect *%
      :equ\@opt\eq@lrn Lleft Rright Nnone *%
      :equ\@opt\eq@fs Ffull Sshort *}}
  \gdef\save@fig{\s@ve{:\@opt\fig@page Fno *figpage%
      :fig\@opt\fig@acs Aall Cchap Ssect *}}
  \gdef\save@tab{\s@ve{:\@opt\tab@page Fno *tabpage%
      :tab\@opt\tab@acs Aall Cchap Ssect *}}
  \gdef\save@ref{\s@ve{:\@opt\ref@page Fno *refpage%
      :ref\@opt\ref@kc Kkeep Cclear *%
      :ref\@opt\ref@sbn Ssup Bsqb Nnam *%
      :\@opt\yearpage@yp Yyearpage Ppageyear *}}
  \gdef\save@toc{\s@ve{:\@opt\toc@page Fno *tocpage%
      :toc\ifcase\save@@toc \else none\fi}}
  \gdef\save@foot{\s@ve{:\@opt\foot@page Fno *footpage%
      :foot\@opt\foot@be Bbot Eend *%
      :foot\@opt\foot@bp Bsqb Ppar *}}
\endgroup
\def\@opt#1#2#3 #4{\if #1#2#3\fi \if #4*\else
  \expandafter\@opt\expandafter#1\expandafter#4\fi}
\def\save@@toc{\auto@toc chap\or sect}

\newif\ifcr@ss
\begingroup \let\comment=\relax
  \outer\gdef\restore{\@kill0\@kill1\@kill2\@kill3\@kill4%
    {\def\\##1{\glet##1\undefined}%
      \def\n@xt##1{\the##1\global##1\emptyt@ks}%
      \n@xt\l@names \n@xt\e@names \n@xt\f@names \n@xt\t@names
      \n@xt\r@names \let\\\@RFdef \n@xt\R@names}
    \bgroup \let\comment\relax \let\d@rest@re\dorest@re \@restore}
\endgroup
\def\crossrestore#1 {\bgroup \openin\test@read#1\save@type
  \ifeof\test@read \let\n@xt\egroup
    \message{* file #1\save@type missing *}%
  \else \closein\test@read
    \def\n@xt{\@restore#1 }\let\d@rest@re\docr@ss \fi \n@xt}
\def\@restore#1 {\input #1\save@type \egroup}
\def\all@restore{\glet\name@list}
\outer\def\dorestore{\bgroup \catcode`\@\l@tter \num@lett\d@restore}
\def\d@restore#1#2{\egroup \toks@{#2}\d@rest@re#1}
\def\dorest@re#1{\def@name\name@list#1{\the\toks@}}
\def\docr@ss#1{\ifx#1\undefined \cr@sstrue
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\def\testcr@ss#1\cr@ss#2\@@{\ifx @#2@\toks@{\cr@ss#1}\cr@ssfalse
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\let\cr@ss=\empty

\def\def@name#1#2{\let\name@list#1\add@name#2\xdef#2}
\def\del@name#1{\bgroup
  \def\n@xt##1\\#1##2\\#1##3\@@##4{\global##4{##1##2}}%
  \def\del@@name##1{\expandafter\n@xt\the##1\\#1\\#1\@@##1}%
  \del@@name\l@names \del@@name\e@names \del@@name\f@names
  \del@@name\t@names \del@@name\r@names \del@@name\R@names \egroup}
\def\add@name#1{\del@name#1%
  {\global\name@list\expandafter{\the\name@list\\#1}}}

\def\kill{\num@lett\@k@ll}
\def\@k@ll#1{\def\k@ll##1{\ifx##1\k@ll \let\k@ll\relax \else
    \del@name##1\glet##1\undefined \fi \k@ll}\k@ll#1\k@ll}

%***  start macros  ***

\outer\def\startchap{\st@rt\chapn@m}
\outer\def\startsect{\st@rt\sectn@m}
\outer\def\startappendix{\st@rt\appn@m}
\outer\def\startequ{\st@rt\eqn@m}
\outer\def\startfig{\st@rt\fign@m}
\outer\def\starttab{\st@rt\tabn@m}
\outer\def\startref{\st@rt\refn@m}
\outer\def\starttoc{\st@rt\tocn@m}
\outer\def\startfoot{\st@rt\footn@m}

\def\st@rt#1{\gm@ne#1 \global\advance#1}
%***  end of file PHYSSAVE  ***
%\input physinst
%***  this is file PHYSINST as of 13.03.86  ***
%***  installation dependent parameters  ***

%***  \hoffset@corr, \hoffset@corrm, \voffset@corr and \voffset@corrm
%     are corrections to \hoffset  and  \voffset  chosen such that
%     the output pages are centered correctly on the paper.

%***  the values used are:
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%       \hoffset@corr@l etc. for the landscape format are

\hoffset@corr@p=86mm   \voffset@corr@p=98mm      % absolute
\hoffset@corrm@p=-5mm   \voffset@corrm@p=4mm      % magnified
\hoffset@corr@l=134mm   \voffset@corr@l=70mm     % absolute
\hoffset@corrm@l=-5mm   \voffset@corrm@l=-4mm     % magnified
%***  end of file PHYSINST  ***

%\input physmpp   % get MPP macros
%***  this is file PHYSMPP as of 03.10.88  ***

\def\wlog#1{} % suppress allocation messages
\catcode`\@=11

%***  additional macros for MPP  ***
%***  (Max-Planck-Institut fuer Physik und Astrophysik)  ***
%***  (Werner-Heisenberg-Institut fuer Physik)  ***

\def\({\relax\ifmmode[\else$[$\nobreak\hskip.3em\fi}
\def\){\relax\ifmmode]\else\nobreak\hskip.2em$]$\fi}

\def\gappr{\mathpalette\under@rel{>\approx}}
\def\lappr{\mathpalette\under@rel{<\approx}}
\def\gsim{\mathpalette\under@rel{>\sim}}
\def\lsim{\mathpalette\under@rel{<\sim}}
\def\under@rel#1#2{\under@@rel#1#2}

% version 1 (PEB)
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% version 2 (UFK)
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\def\mppae@text{{Max-Planck-Institut f\"ur Physik und Astrophysik}}
\def\mppwh@text{{Werner-Heisenberg-Institut f\"ur Physik}}

\def\mppaddresstext{Postfach 40 12 12, D-8000 M\"unchen 40\else
  P.O.Box 40 12 12, Munich (Fed.^^>Rep.^^>Germany)}

\def\mppaddress{\address{\mppae@text \nl -- \mppwh@text\space --\nl
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\def\mppnum#1{\topright{MPI--PAE/#1}}

%***  mpp letterhead  ***

\font\fourteenssb=cmssdc10 scaled \magstep2 % for letterhead
\font\seventeenssb=cmssdc10 scaled \magstep3 % for letterhead

\def\letter#1#2{\b@lett@r{26}%
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    \lftline{\case@language{TELEFON\else PHONE}: (089) 32 308
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    \addressspacing \dimen@\baselineskip \dimen@ii-2.79ex%
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  \vskip1cm\rtline{\thedate}\vskip1cm\@plus1cm\@minus.5cm\endgroup}

\let\addressspacing=\empty

%***  private letterhead  ***

\def\myname{Dr.\ Xxxx Xxxxxxxxxx\nl Physiker}
\def\myaddress{Xxxxxxx Stra\ss e  ??\nl
    \llap{D--8000\quad}M\"unchen ??\nl
    Tel:\ (089) \vtop{\hbox{?? ?? ?? (privat)}%
                      \hbox{3 18 93-??? (B\"uro)}}}
\def\myletter{\b@lett@r{26}\line{\let\nl\cr \use@nl \caps
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  \vtop to25mm{\tenpoint
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%***  firstpageoutput to print first page only ***

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%***  macros to leave room for equations

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\def\veqn{\afterassignment\v@eqn \dimen@}
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\def\heq{\afterassignment\h@eq \dimen@}
\def\h@eq{$\hbox to\dimen@{}$ }

\def\wlog{\immediate\write\m@ne} % restore PLAIN's definition
\catcode`\@=12 % @ signs are no longer letters
%***  end of file PHYSMPP  ***
%\input physthe   % get THE macros
%***  this is file PHYSTHE as of 27.07.87  ***
\def\wlog#1{} % suppress allocation messages
\catcode`\@=11

\outer\def\pthnum#1{\mppnum{PTh #1}}
\let\pthdate=\pubdate

\def\wlog{\immediate\write\m@ne} % restore PLAIN's definition
\catcode`\@=12 % @ signs are no longer letters

%***  additional physics journals  ***

\def\AP{\journal{Ann.\ \Phys}}
\def\CMP{\journal{Commun.\ \Math\Phys}}
\def\JMP{\journal{J.\ \Math\Phys}}
\def\NC{\journal{Nuov.\ Cim.\ }}
\def\NCA{\journalp{Nuov.\ Cim.\ }A}
\def\NCB{\journalp{Nuov.\ Cim.\ }B}
\def\PRep{\journal{\Phys Rep.\ }}
\def\PRepC{\journalf{\Phys Rep.\ }C}
\def\RMP{\journal{\Rev Mod.\ \Phys}}
\def\TMF{\journal{Teor.\ Mat.\ Fiz.\ }}

%***  end of file PHYSTHE  ***
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%          Modifications by Robin Stuart, January, 1990
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\wlog#1{} % suppress allocation messages
\catcode`\@=11

%  Define the fonts that will be needed

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\font\headlinefont=cmssdc10 scaled \magstep1
\font\headeraddressfont=cmss10

\def\CERNlogo{\hbox{\CERNfont C}}            % logo 21mm in size
\def\smallCERNlogo{\hbox{\CERNfont c}}       % logo 13mm in size

%  Define the headers

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                           \headlinefont
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\def\letterhead{\setbox0=\vtop{\logo}

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               ORGANISATION EUROP\'EENNE POUR LA RECHERCHE NUCL\'EAIRE\cr
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                               \headeraddressfont
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\def\CERNheader{\let\logo=\CERNlogo\header}
\def\CERNletterhead{\let\logo=\CERNlogo\letterhead}

\def\smallCERNheader{\let\logo=\smallCERNlogo\header}
\def\smallCERNletterhead{\let\logo=\smallCERNlogo\letterhead}

%  Define macros for letters

\def\letter#1#2{\b@lett@r{23}%
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    \ifx\EARN\undefined\else\vskip5\p@\lftline{EARN/BITNET: \EARN
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\def\e@lett@r#1#2{\skippagenum T\skipheadline T\skipfootline T%
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 \def\sincerely#1{\nobreak\par\nobreak\vskip 7mm%
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%  Define some things for the preprint title page

\def\preprintheader{\vglue -23mm\line{\hglue -5mm\CERNheader\hss}%
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\def\pubnum#1{\topright{CERN-TH.#1}\def\thepubnum{#1}}
\def\pubdate#1{\def\thepubdate{#1}}
\def\CERNaddress{\address{Theory Division, CERN\nl
                          CH-1211 Geneva 23, Switzerland}}
\def\endtitlepage{\footline={\vtop{\null\vfil
                     \hbox{\ifx\thepubnum\undefined
                           \else CERN-TH.\thepubnum\fi}
                     \hbox{\ifx\thepubdate\undefined
                             \ifcase\month\or
                                 January\or February\or March\or
                                 April\or May\or June\or
                                 July\or August\or September\or
                                 October\or November\or December\fi,
                           \space\number\year
                           \else\thepubdate\fi}\vss}
                           \hfil}\endpage
\footline={\footlinestyle \head@foot\skip@foot\foot@lrac
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% Takes account that the format for quoting PLB changes after vol 170

\def\PLB#1(#2)#3*{\ifnum #1<171
                      \journalf{\Phys\Lett}B#1(#2)#3*%
                  \else
                      \journalp{\Phys\Lett}B#1(#2)#3*%
                  \fi}
\def\ZPC{\journalp{Z.\ \Phys}C}               % Zeitschrift fur Physik
\def\CPC{\journal{Comput.\ Phys.\ Commun.\ }} % Computer Physics Commun.

% Some useful mathematical symbols
%
\def\ltapprox{\underrel\textstyle\sim\under<}
\def\gtapprox{\underrel\textstyle\sim\under>}

% This is the current date in European format (day first)
%
\def\thedate{\number\day\case@language{.\ \else\ }%
             \themonth\case@language{\else,} \number\year}

% The current English date

\def\today{\number\day\space\ifcase\month\or
 January\or February\or March\or April\or May\or June\or
 July\or August\or September\or October\or November\or December\fi,
 \number\year}

% Insert blank page
\long\def\blankpage{
     \skippagenum=T   \skipheadline=T   \skipfootline=T
     \null\vfill\eject
     \advance\pageno -1
     \skippagenum=F   \skipheadline=F   \skipfootline=F}

\def\wlog{\immediate\write\m@ne} % restore PLAIN's definition
\catcode`\@=12 % @ signs are no longer letters
%***  end of file THEMAC  ***
\catcode`\@=12 % @ signs are no longer letters

\message{and default options.}

%***  default options  ***

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\newsect        % define \secs

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\botpagenum     % page numbers at bottom of page
\centhead       % headlines not alternating,
                % page numbers (if any) centered
\centfoot       % footlines not alternating,
                % page numbers (if any) centered
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\portrait       % portrait orientation ( vsize > hsize )
\titlepage      % title etc. on a separate page (e.g. preprint)
\nochappage     % chapters and appendices not on separate pages
\arabicchapnum  % arabic chapter numbers
\chapters       % paper has chapters
\equchap        % equations numbered per chapter and appendix
\equshort       % equation numbers quoted in abbreviated form
\equright       % equation numbers on right side
\storelist      % figure captions etc. stored in lists (macros)
\figall         % figures numbered throughout the paper
\figpage        % figure captions on a separate page
\nographics     % graphic yields an empty frame
\taball         % tables numbered throughout the paper
\tabpage        % table captions on a separate page
\refsup         % quotations of references as superscripts
\refpage        % references on a separate page
\refkeep        % references are kept after \refout
\RFlist         % predefined ref's stored in list (macros)
\yearpage       % journals quoted in the form `vol (year) page'
\tocpage        % table of contents on a separate page
\tocnone        % no automatic entries in toc
\footsqb        % footnote numbers in square brackets
\footbot        % footnotes at bottom of page (as footnotes)
\footpage       % footnotes on a separate page (for \footend only)
\matc           % matrix elements are centered

\wlog{summary of allocations:}
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\def\Fmtname{physe}   \def\Fmtversion{2.0}
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%\everyjob{%\input phystime %\input physupdt}
%***  end of file PHYS  ***
%
%
%    version of the paper as of April 3rd 17H15.
%
%    this is the version we sent to PLB and give to Vascotto
%
%
%
%
%\batchmode
%\input themac
\equfull
\refsqb
\overfullrule 0pt
\preprintheader
\pubnum{6449/92}
\pubdate{April, 1992}
\title{Two--Body Decays of $B_s$ Mesons }
\author{J. Bijnens}
\autcon{F. Hoogeveen}
\CERNaddress
\abstract{
We have calculated the decay rates of the $B_s$ meson in a number of
exclusive two--body decay channels using
 the Bauer--Stech--Wirbel model for
current matrix elements.
The influence of the free parameters of the model on the predictions is
studied.
The total branching ratio of the $B_s$ into
 final states which only contain
stable charged particles is found to be about $10^{-3}$. }
\endtitlepage
%===================REFERENCES=========================================
\RF\BSW{M.Bauer, B.Stech, M.Wirbel, \ZPC34(1987)103*}
\RF\WSB{M.Wirbel, B.Stech, M.Bauer, \ZPC29(1985)637*}
\RF\ARGUS{H.Albrecht et al. [ARGUS collab.], \ZPC 48(1990)543*}
\RF\CLEO{ D.Borteletto et al. [CLEO collab.],
 Inclusive and exclusive decays of $B$--mesons to final states
 including charm and charmonium mesons, CLNS-91-1102}
\RF\CPVIOL{CP violation, ed. C.Jarlskog, World Scientific,
           Singapore, 1989}
\RF\PDG{Part. Data Group, \PLB 239(1990)1*}
\RF\FIT{M.Neubert et al., Exclusive Weak decays of B--Mesons,
        HD-THEP-91-28}
%======================================================================
\def\Dsxp{$D_s^{*+}$}
\def\Etap{$\eta^\prime$}
\def\Kxp{$K^{*+}$}
\def\Kxm{$K^{*-}$}
\def\Dsp{$D_s^+$}
\def\Etaa{$\eta$}
\def\Dsxm{$D_s^{*-}$}
\def\Phii{$\phi$}
\def\Kz{$K^0$}
\def\Aim{$a_1^-$}
\def\Kxz{$K^{*0}$}
\def\Kp{$K^+$}
\def\Rhom{$\rho^-$}
\def\Dsm{$D_s^-$}
\def\Pim{$\pi^-$}
\def\Pip{$\pi^+$}
\def\Jpsi{$J/\psi$}
\def\Dmx{$D^{*-}$}
\def\Kmx{$K^{*-}$}
\def\Dm{$D^-$}
\def\Dp{$D^+$}
\def\Km{$K^-$}
\def\Dzx{$D^{*0}$}
\def\Dz{$D^0$}
\def\Etac{$\eta_c$}
\def\Aiz{$a_1^0$}
\def\Rhoz{$\rho^0$}
\def\Dzbar{$\overline{D^0}$}
\def\Omeg{$\omega$}
\def\Dzxbar{$\overline{D^{*0}}$}
\def\Dm{$D^-$}
\def\Piz{$\pi^0$}
\def\Rhoz{$\rho^0$}
\def\Kzbar{$\overline{K^0}$}
%======================================================================
\chap{ Introduction.}
It does not have to be emphasized that the study of $B$--mesons is
worthwhile.
The observation of mixing and the prospect of observing  $CP$--violation
in the \hbox to 1 cm{}  \hbox{$B$--system}
 are exciting and may give deep insight into the inner
mechanism of the standard model.
Of the pseudoscalar $B$--mesons only the $B^\pm$ and the $B^0$ have been
seen in fully reconstructed events \quref{\ARGUS,\CLEO}.
So far the $B_s$--meson has remained elusive.
Although it would be highly surprising if this state didn't exist, it is
very important to actually find it.
The $B_s$ meson is expected to mix strongly with its anti particle and
many proposed schemes\quref\CPVIOL\ %
for detecting $CP$--violation in the $B$--system use this particle as
initial state.

In this paper we calculate a
number of branching ratios of the $B_s$--meson
into two--body channels.
To do so we employ the model of
Bauer, Stech and Wirbel\quref{\BSW,\WSB} which these authors
have
used to calculate the two body
decays of $D^\pm,D^0,B^\pm$ and $B^0$ mesons.

The organization of this paper is as follows.
In section 2 we briefly review the BSW model and discuss the
input parameters of the model.
In section 3 we give the results of our calculation together with a
discussion of the sensitivity of our results with respect to variations
in the input parameters.
Finally in section 4 we discuss goldplated decay chains where the
decay products at the end of the decay chain
 are all charged pions, kaons,
muons or electrons.
These can be used for the direct detection of the $B_s$.

\chap{ The Bauer--Stech--Wirbel model.}
With the current state of the art it is impossible to calculate matrix
element for processes involving strongly interacting particles from
first principles using the standard model.
Therefore one has to resort to models.
The model used here is a model by Bauer, Stech and Wirbel
\quref{\BSW,\WSB}.
This model can be used to calculate the decay of a spin $0$ meson in
two mesons of spin $0$ or $1$, as well as its semileptonic decay.
The two main ingredients are factorization and an {\it Ansatz\/} for the
current matrix elements.

The short distance effective action is given by
$$\eqalign{
 {\cal L}_{eff}=&
 {{G_F}\over \sqrt{2}}
   \big\{ c_1(\mu)({\bar u}d^\prime)({\bar s}^\prime c)
        + c_2(\mu)({\bar s}^\prime d^\prime)({\bar u} c)
         \big\} \cr
& + ((\bar{u}d^\prime)\to(\bar{c}s^\prime)) + \cdots  \cr}
                                               \EQN\ShortDistance  $$
where the primes on the quark fields denote the interaction eigenstates.
The factorization assumption amounts to replacing the short distance
effective action
by the following product of meson currents
$$ {\cal L}_{eff}= {{G_F}\over \sqrt{2}}
  : \big\{ a_1({\bar u}d^\prime)_H({\bar s}^\prime c)_H
        + a_2({\bar s}^\prime d^\prime)_H({\bar u} c)_H
         +\cdots
        \big\}:
                                              \EQN\LongDistance $$
where the index $H$ means that the quarks denoted inside the bracket are
to be contracted with one and the same meson.
In this way the problem of calculating decay matrix elements reduces to
the calculation of current matrix elements between a one meson state
and the vacuum or between two one meson states.
The former matrix elements are simply given by the corresponding
decay constants:
$$\langle 0 \vert J_\mu \vert A\,0^\pm \rangle = i f_A p_\mu \qquad
  \langle 0 \vert J_\mu \vert A\,1^\pm \rangle = \epsilon_\mu m_A F_A
                                                       \EQN\DecayConst $$
for a spin zero and a spin one particle respectively.
The latter kind of matrix element can be expressed in terms of
Lorentz--scalar formfactors:
$$\eqalign{
  \big\langle B \,0^\pm \vert J_\mu \vert A \,0^\pm \big\rangle=&
  \left(p_A+p_B-{{m_A^2-m_B^2}\over{q^2}}q\right)_\mu\, F_1(q^2) \cr
&+{{m_A^2-m_B^2}\over {q^2}}\,q_\mu \,F_0(q^2) \cr
  \big\langle B\, 1^\pm \vert J_\mu \vert A \,0^\pm \big\rangle &=
{2\over{m_A+m_B}}\epsilon_{\mu\nu\rho\sigma}\,\epsilon^{*\nu}
        p_A^\rho p_B^\sigma\, V(q^2) \cr
 &+i(m_A+m_B) \left(\epsilon^*_\mu-{{\epsilon^*\cdot q}\over{q^2}}
            q_\mu\right)\, A_1(q^2)\cr
  &-i\epsilon^*\cdot q
  \left({{p_{A\mu}-p_{B\mu}}\over{m_A+m_B}}-{{m_A-m_B}\over{q^2}}
  q_\mu\right)\,
      A_2(q^2) \cr
  &+i{{\epsilon^*\cdot q}\over{q^2}}2m_B\,q_\mu\, A_0(q^2)   }
                                                  \EQN\FormFactorDef $$
where $q=p_A-p_B$, and $\epsilon^\nu$ is the polarization vector of
the spin one meson.
The $q^2$ behaviour is assumed to be dominated
by the nearest pole which has
the correct quantum numbers.
At this point the third  ingredient
of the BSW model makes its appearance.
These authors assume that using the wavefunctions
of a relativistic harmonic
oscillator model provides a reasonable approximation for the wavefunction
of a physical meson.
This amounts to taking the following expressions
for the formfactors at zero momentum transfer :
$$\eqalign{
F_0(0)&=A_0(0)={f({{a_A+a_B}\over 2},{{b_A+b_B}\over 2} )\over
          \sqrt{f(a_A,b_A)f(a_B,b_B)}} \cr
A_1(0)&={{m_{A1}+m_{B1}}\over{m_A+m_B}}\cdot
        {{2\omega_A\omega_B}\over{\omega_A^2+\omega_B^2}}\cdot
               {g({{a_A+a_B}\over 2},{{b_A+b_B}\over 2} )\over
          \sqrt{f(a_A,b_A)f(a_B,b_B)}} \cr
V(0)&={{m_{A1}-m_{B1}}\over{m_A-m_B}}\cdot
        {{2\omega_A\omega_B}\over{\omega_A^2+\omega_B^2}}\cdot
               {g({{a_A+a_B}\over 2},{{b_A+b_B}\over 2} )\over
          \sqrt{f(a_A,b_A)f(a_B,b_B)}} \cr
A_2(0)&={{m_A+m_B}\over{m_A-m_B}}A_1(0)-{{2m_B}\over{m_A-m_B}}A_0(0).}
      \EQN\hi  $$
In this expression $m_{A,B}$ are the masses of the mesons
in the matrix element,
$m_{A1,B1}$ are  the masses of the nonspectator quarks, and $m_2$ is the
spectator quark mass.
The numbers $a_A$,$b_A$ are given by
$$ a_A={{m_A^2}\over{\omega_A^2}} \quad\quad
    b_A=-\,{{m_A^2+m_{A1}^2-m_2^2}\over {\omega_A^2}} \EQN\ab   $$
with similar expressions for $a_B$,$b_B$.
The functions $f$ and $g$, given by
$$\eqalign{
f(a,b)&=\int^1_0\!dx\,x(1-x)e^{-ax^2-bx} \cr
g(a,b)&=\int^1_0\!dx\,(1-x)e^{-ax^2-bx}} \EQN\fg $$
can be readily expressed in terms of the error function.
The parameters $\omega_A$ and $\omega_B$
define the spatial extensions
of the two mesons sandwiching the currents. (Small $\omega$ means a large
meson and vice versa).

Final state interactions play an important r\^ole in the analogous decays
of the $D$--meson, but for the $B_s$ meson they are expected to be small
due to the large mass of the $b$--quark.
In the following we have neglected all final state interactions.
Also neglected are the effects of weak annihilation contributions,
 penguin  diagrams and loop contributions.

Short distance QCD radiative corrections are in principle incorporated
in the coefficients $a_1,a_2$, but rather than calculating these numbers
from
first principles we are using the values which have been obtained
from a comprehensive fit to nonstrange $B$--meson decays\quref\FIT
$$  a_1 = 1.11   \qquad a_2 = 0.21      .\EQN\aNumbers $$

The values of the decay constants used can be found in table 2.1, whereas
the polemasses are listed in table 2.2 .
Many of these pole masses correspond to unobserved states.
In principle the
same values are being used as those of BSW,
unless more precise information
has become available in the mean time.

As constituent quark masses we assume 350 MeV for the up and down
quarks, 550 MeV for the
strange quarks and 1.7 GeV for the charm quark.
The b--quark mass was taken to be 4.9 GeV.

The values of the Kobayashi--Maskawa matrix elements were taken from
reference \quref\PDG, for $V_{ud},V_{us},V_{cd}$ and $V_{cs}$, whereas
$V_{cb}$ and $V_{ub}$ were taken to be $0.05$ and $0.005$ respectively.


Finally for the $B_s$ mass itself we adopted 5.4 GeV as a standard
value.

\chap{Results}
Going through the formalism
 described above and assuming that the lifetime
of the $B_s$--meson is the same as the average lifetime of the nonstrange
$B$--mesons $11.8\ 10^{-13}\rm \ sec$
we have calculated the branching ratios
for the $B_s$ into a number of two meson decay channels.
The results of our calculation can be found in table 3.1
There we list the branching ratio for
the decay, the KM matrix elements which
appear in the amplitude.
All two--body decay channels in the list
add up to a branching ratio of 11.4\%.
The branching ratio simply scales quadratically
with the KM matrix elements.
The decays into a pair of charged particles scale as $a_1^2$ and the
decays into a pair of neutral particles scale like $a_2^2$.
In this calculation all effects of mixing and/or CP violation are ignored.
If the $B_s$--meson is mixing strongly, as it is expected to do, then the
branching ratios for a mass eigenstate to
decay in a particular channel is just
the average of the branching ratios of
this channel and its charge conjugate.

%Discussion of the behaviour as m $B_s$ changes.
The mass of the $B_s$--meson is presently not known.
For the numerical estimates in table 1 a
value of $5.4\rm\ GeV$ has been used.
Increasing this value to $5.5\rm\ GeV$ leads to an increase of all
branching  ratios considered.
The largest increase is by 29\% and occurs for the channel \Etap\Jpsi ,
whereas the smallest increase is 17\% which occurs in the channel
 \Etaa\Piz .
The changes in the decay channels with branching ratios over one promille
are more or less uniformly distributed between +27\% and +18\%.


%Discuss changing the omegas
For every particle the parameter $\omega$ is a free parameter
 defining the
spatial extension of the meson.
A value of $400\rm\ MeV$ gives a reasonable
description of $D$ and non--strange
$B$--meson decays, and has been adopted here for all particles.
Increasing the $\omega$ of the $B_s$--meson to $500\rm\ MeV$
 while keeping
the rest fixed has drastic consequences for some decay channels.
The branching ratio of the \Phii\Jpsi\ channel increases by 39\%.
The other important decay channels increase by between 20\% and 30\%.

%Discussion of the behaviour if the pole masses change
The $q^2$ dependence of the formfactors is modeled using the assumption
that the nearest resonance with the correct quantum numbers dominates.
Many of the resonances which are needed are themselves not yet detected
however and consequently their mass is unknown.
The values from table 2.2 are used to produce the numbers in table 3.1.
To see the effect of this uncertainty,
we increased all pole masses by 10\%.
The resulting change in the branching
ratios was very small, most important
channels decreased by a few percent.
The \Phii\Jpsi\ channel was among a few
exceptions and decreased by 13\%.

The overall error of any calculation is very hard to estimate.
{}From the uncertainties in the input
parameters discussed above we know that
the inaccuracy can be at least as large as a factor of two for the decay
$B_s \to\phi J/\psi$ and as large as
 $1.5$ for the important decay channels
involving $D$--mesons.
The relative importance of the different
decay channels is however much less
influenced by the uncertainty in the input parameters.
The order of importance of the largest channels is not changed, even by
rather drastic changes in input parameters.

\chap{Goldplated decay chains.}
Decay chains in which the stable particles at the end of the chain are all
charged are especially suited for reconstructions of the $B_s$ meson.
So far not a single fully reconstructed $B_s$ has been seen.
The numerical values below have been taken from \quref\PDG.

First consider the decays of the $D_s^\pm$--meson.
In $2.7 \%$ of the time it decays in
$\phi\pi^\pm$ and $1.3\%$ of its decays
produce $\phi\pi^\pm\pi^\pm\pi^\mp$.
Of these roughly half are followed by the decay $\phi\to K^+K^-$.
The decay channel $K_SK^+$ followed by the decay of the $K_S$ into two
charged pions has a branching ratio of about 0.9\%.
Furthermore the decay channels $\overline{K^{*0}}K^+$
and $K^{*+}K_S$ have a probability of
$2.6\%$ and $1.6\%$ respectively.
The charged $K^*$ decays into a charged
pion and $K_S$ one third of
the time,
whereas the $K^{*0}$ decays into $K^+\pi^-$
 two thirds of the time.
Finally the direct decay $\pi^\pm\pi^\pm\pi^\mp$ has a branching ratio
of $1.2$\%.
The decay mode $K^+K^-\pi^+$ adds another 0.67 \%.
Taking everything together we notice that
the $D_s^\pm$ decays $6.7 \%$ of the
time into a final state in which all the particles are charged.

The $a_1^\pm$ meson decays with a branching ratio of fifty percent to
$\pi^\pm\pi^\pm\pi^\mp$  through  $\rho^0\pi^\pm$.

The \Jpsi\ decays into $e^+e^-$ or $\mu^+\mu^-$ with a branching ratio of
13.8\%.

Another interesting possibility is the $\psi^\prime$ which decays either
in charged leptons or into \Jpsi$\,\pi^+\pi^-$
followed by \Jpsi\ decaying into $e^+e^-$ or $\mu^+\mu^-$.
These channels contribute 6.1 \% to the all charged branching ratio
of the $\psi^\prime$.
The branching ratio $B_s \to \phi\psi^\prime$ has been calculated
using the same methods as in section 3.

All together we find
the branching ratios for goldplated decays
as they are displayed in table 4.1 .

Adding everything together we note that roughly one promille of the $B_s$
decays goes into a goldplated decay chain.
Furthermore to all these decay channels one may add $\pi^+\pi^-$ and or
$K^+K^-$ pairs ad libitum (or at least upto the kinematical limit).
This may bring a substantial improvement as a comparison with the
$D$--system shows.
$$
{{\Gamma(D^+\to\overline{K^0}\pi^+\pi^+\pi^-)
}\over{\Gamma(D^+\to\overline{K^0}\pi^+)}}=2.5
\quad\quad
{{\Gamma(D^0\to K^-\pi^+\pi^+\pi^-)}
\over{\Gamma(D^0\to K^-\pi^+)}}=2.1\ . \EQN\dratio
   $$.
\ack
The authors like to thank A.Pich and P.Kluit for a number of pleasant
discussions.
\refout
\break\vfil
\line{{\bf Table 2.1 :} The decay constants in MeV. \hfil }
\vskip 3 mm
\vbox{\offinterlineskip\hrule
\halign{
  \vrule\strut\quad # \hfil&\quad # \hfil\vrule&\quad # \hfil &\quad
      # \hfil\vrule \cr
  \noalign{\hrule}
  $\pi^\pm$           & 131.7 & $\rho,a_1,K^*,D^*,D_s^*$ & 221 \cr
  $K,D,D_s,\eta_c$    & 160.6 & $\omega$                 & 156 \cr
  $\eta\ (uu) $       &  94   & $\phi$                   & 233 \cr
  $\eta^\prime\ (uu)$ &  65   & \Jpsi                    & 382 \cr
  \noalign{\hrule} }}
\vskip 1 cm
\line{{\bf Table 2.2 :} The pole masses in MeV. \hfil}
\vskip 3 mm
\vbox{\offinterlineskip\hrule
 \halign{\vrule\strut\quad # \quad\vrule&
         \quad\hfil # \hfil\quad &
         \quad\hfil # \hfil\quad &
         \quad\hfil # \hfil\quad &
         \quad\hfil # \hfil\quad\vrule\cr
 & $0^+$ & $0^-$ & $1^+$ & $1^-$ \cr
 \noalign{\hrule}
 cb & 6800 & 6300   & 6730 & 6340   \cr
 ub & 5780 & 5277.6 & 5710 & 5331.3 \cr
 bs & 5890 & 5400   & 5820 & 5430\cr
 \noalign{\hrule} }}
\vfil
\break
\def\bsline#1#2#3#4#5#6{#1#2 & $#3\cdot 10^{#4}$ &
                        $\vert  V_{#5}V_{#6} \vert^2$ \cr}
\vbox{\halign{ # & # \cr
\bf Table 3.1 :& Branching ratios for two--body $\overline{B_s}$ decays
      \hfil \cr  \noalign{\vskip 1 mm}
     & assuming $a_1=1.11$, $a_2=0.21$,
      $\tau_{B_s}= 11.8\cdot 10^{-13}{\rm sec}$ , \cr
     & $V_{cb}=0.05$ and $V_{ub}=0.005$.\hfill \cr}}
\vskip 3 mm
\vbox{\offinterlineskip
\hrule
\halign{
\vrule\strut\quad # \quad\hfil\vrule &
            \quad # \quad\hfil\vrule &
            \quad # \quad\hfil\vrule\cr
\noalign{\hrule}
 \bsline{ \Dsxp}{ \Dsxm}{0.22}{-1}{cb}{cs}
 \bsline{ \Dsxp}{ \Aim} {0.16}{-1}{cb}{ud}
 \bsline{ \Dsp }{ \Rhom}{0.14}{-1}{cb}{ud}
 \bsline{ \Dsxp}{ \Rhom}{0.13}{-1}{cb}{ud}
 \bsline{ \Dsp }{ \Aim} {0.13}{-1}{cb}{ud}
 \bsline{ \Dsp }{ \Dsxm}{0.84}{-2}{cb}{cs}
 \bsline{ \Dsp }{ \Dsm} {0.76}{-2}{cb}{cs}
 \bsline{ \Dsp }{ \Pim} {0.55}{-2}{cb}{ud}
 \bsline{ \Dsxp}{ \Pim} {0.41}{-2}{cb}{ud}
 \bsline{ \Dsxp}{ \Dsm} {0.34}{-2}{cb}{cs}
 \bsline{ \Phii}{ \Jpsi}{0.13}{-2}{cb}{cs}
 \bsline{ \Dsxp}{ \Dmx} {0.11}{-2}{cb}{cd}
 \bsline{ \Dsp }{ \Kmx} {0.72}{-3}{cb}{us}
 \bsline{ \Dsxp}{ \Kmx} {0.71}{-3}{cb}{us}
 \bsline{ \Dsp }{ \Dmx} {0.46}{-3}{cb}{cd}
 \bsline{ \Dsp }{ \Km}  {0.42}{-3}{cb}{us}
 \bsline{ \Dsp }{ \Dm}  {0.40}{-3}{cb}{cd}
 \bsline{ \Dsxp}{ \Km}  {0.30}{-3}{cb}{us}
 \bsline{ \Kxz }{ \Dzx} {0.23}{-3}{cb}{ud}
 \bsline{ \Dsxp}{ \Dm}  {0.19}{-3}{cb}{cd}
 \bsline{ \Etap}{ \Jpsi}{0.15}{-3}{cb}{cs}
 \bsline{ \Etaa}{ \Jpsi}{0.13}{-3}{cb}{cs}
 \bsline{ \Kz  }{ \Dzx} {0.12}{-3}{cb}{ud}
 \bsline{ \Kz  }{ \Dz}  {0.83}{-4}{cb}{ud}
 \bsline{ \Kxp }{ \Dsxm}{0.67}{-4}{ub}{cs}
 \bsline{ \Etap}{ \Etac}{0.59}{-4}{cb}{cs}
 \bsline{ \Kxz }{ \Jpsi}{0.57}{-4}{cb}{cd}
 \bsline{ \Etaa}{ \Etac}{0.46}{-4}{cb}{cs}
 \bsline{ \Kxz }{ \Dz}  {0.45}{-4}{cb}{ud}
}%end of halign
\hrule
}%end of vbox
\vfill\eject
\vbox{\offinterlineskip
\hrule
\halign{
\vrule\strut\quad # \quad\hfil\vrule &
            \quad # \quad\hfil\vrule &
            \quad # \quad\hfil\vrule\cr
\noalign{\hrule}
 \bsline{ \Kxp }{ \Aim} {0.43}{-4}{ub}{ud}
 \bsline{ \Phii}{ \Etac}{0.40}{-4}{cb}{cs}
 \bsline{ \Kp  }{ \Rhom}{0.39}{-4}{ub}{ud}
 \bsline{ \Kp  }{ \Aim} {0.37}{-4}{ub}{ud}
 \bsline{ \Kp  }{ \Dsxm}{0.34}{-4}{ub}{cs}
 \bsline{ \Kxp }{ \Rhom}{0.33}{-4}{ub}{ud}
 \bsline{ \Kp  }{ \Dsm} {0.23}{-4}{ub}{cs}
 \bsline{ \Phii}{ \Dzx} {0.14}{-4}{cb}{us}
 \bsline{ \Kz  }{ \Jpsi}{0.14}{-4}{cb}{cd}
 \bsline{ \Kp  }{ \Pim} {0.14}{-4}{ub}{ud}
 \bsline{ \Kxp }{ \Dsm} {0.12}{-4}{ub}{cs}
 \bsline{ \Kxp }{ \Pim} {0.10}{-4}{ub}{ud}
 \bsline{ \Kz  }{ \Etac}{0.49}{-5}{cb}{cd}
 \bsline{ \Etap}{ \Dzx} {0.39}{-5}{cb}{us}
 \bsline{ \Kxp }{ \Dmx} {0.33}{-5}{ub}{cd}
 \bsline{ \Etaa}{ \Dzx} {0.31}{-5}{cb}{us}
 \bsline{ \Phii}{ \Dz}  {0.30}{-5}{cb}{us}
 \bsline{ \Phii}{ \Dzxbar}{0.28}{-5}{ub}{cs}
 \bsline{ \Etap}{ \Dz}  {0.27}{-5}{cb}{us}
 \bsline{ \Etaa}{ \Dz}  {0.21}{-5}{cb}{us}
 \bsline{ \Kp } {\Kmx}  {0.20}{-5}{ub}{us}
 \bsline{ \Kxp }{ \Kmx} {0.18}{-5}{ub}{us}
 \bsline{ \Kp }{ \Dmx} {0.18}{-5}{ub}{cd}
 \bsline{ \Kxz }{ \Etac}{0.17}{-5}{cb}{cd}
 \bsline{ \Kp }{ \Dm}  {0.12}{-5}{ub}{cd}
 \bsline{ \Kp }{ \Km}  {0.11}{-5}{ub}{us}
 \bsline{ \Etap}{ \Dzxbar}{0.76}{-6}{ub}{cs}
 \bsline{ \Kxp }{ \Km}  {0.76}{-6}{ub}{us}
 \bsline{ \Kxz }{ \Aiz} {0.75}{-6}{ub}{ud}
 \bsline{ \Kz  }{ \Rhoz}{0.70}{-6}{ub}{ud}
 \bsline{ \Kz  }{ \Omeg}{0.70}{-6} {ub}{ud}
 \bsline{ \Kz  }{ \Aiz} {0.67}{-6}{ub}{ud}
 \bsline{ \Kxp }{ \Dm}  {0.65}{-6}{ub}{cd}
}%end of halign
\hrule
}%end of vbox
\vfill\eject
\vbox{\offinterlineskip
\hrule
\halign{
\vrule\strut\quad # \quad\hfil\vrule &
            \quad # \quad\hfil\vrule &
            \quad # \quad\hfil\vrule\cr
\noalign{\hrule}
 \bsline{ \Etaa}{ \Dzxbar}{0.60}{-6}{ub}{cs}
 \bsline{ \Kxz }{ \Omeg}{0.60}{-6} {ub}{ud}
 \bsline{ \Kxz }{ \Rhoz}{0.59}{-6}{ub}{ud}
 \bsline{ \Phii}{ \Dzbar}{0.58}{-6}{ub}{cs}
 \bsline{ \Etap}{ \Dzbar}{0.53}{-6}{ub}{cs}
 \bsline{ \Etaa}{ \Dzbar}{0.40}{-6}{ub}{cs}
 \bsline{ \Kz  }{ \Etaa}{0.26}{-6} {ub}{ud}
 \bsline{ \Kz  }{ \Piz} {0.25}{-6} {ub}{ud}
 \bsline{ \Kxz }{ \Etaa}{0.18}{-6} {ub}{ud}
 \bsline{ \Kxz }{ \Piz} {0.18}{-6} {ub}{ud}
 \bsline{ \Kxz }{ \Dzxbar}{0.12}{-6} {ub}{cd}
 \bsline{ \Kz  }{ \Etap}{0.94}{-7} {ub}{ud}
 \bsline{ \Kz  }{ \Dzxbar}{0.63}{-7}{ub}{cd}
 \bsline{ \Kxz }{ \Etap}{0.62}{-7} {ub}{ud}
 \bsline{ \Phii}{ \Aiz} {0.49}{-7}{ub}{us}
 \bsline{ \Kz  }{ \Dzbar}{0.42}{-7}{ub}{cd}
 \bsline{ \Phii}{ \Omeg}{0.39}{-7}{ub}{us}
 \bsline{ \Phii}{ \Rhoz}{0.39}{-7}{ub}{us}
 \bsline{ \Etap}{ \Rhoz}{0.24}{-7}{ub}{us}
 \bsline{ \Etap}{ \Omeg}{0.24}{-7}{ub}{us}
 \bsline{ \Kxz }{ \Dzbar}{0.23}{-7}{ub}{cd}
 \bsline{ \Etap}{ \Aiz} {0.22}{-7}{ub}{us}
 \bsline{ \Etaa}{ \Rhoz}{0.18}{-7}{ub}{us}
 \bsline{ \Etaa}{ \Omeg}{0.18}{-7}{ub}{us}
 \bsline{ \Etaa}{ \Aiz} {0.17}{-7}{ub}{us}
 \bsline{ \Etaa}{ \Etaa}{0.13}{-7}{ub}{us}
 \bsline{ \Phii}{ \Etaa}{0.12}{-7}{ub}{us}
 \bsline{ \Phii}{ \Piz} {0.12}{-7}{ub}{us}
 \bsline{ \Etap}{ \Piz} {0.86}{-8}{ub}{us}
 \bsline{ \Etaa}{ \Piz} {0.65}{-8}{ub}{us}
 \bsline{ \Etap}{ \Etap}{0.63}{-8}{ub}{us}
 \bsline{ \Phii}{ \Etap}{0.41}{-8}{ub}{us}
 \bsline{ \Etaa}{ \Etap}{0.20}{-8}{ub}{us}
\noalign{\hrule}
} % end of halign
} % end of vbox
\vfil\break
\vfil
\vbox{\halign{ # & # \cr
\bf Table 4.1 :&The contribution of several two--body channels to \cr
&the branching ratio $B_s\to$ stable charged particles only. \cr }}
\vskip 3 mm
\vbox{\offinterlineskip\hrule\halign{
  \vrule\strut\quad # \quad\hfil&\vrule \quad $ #\cdot 10^{-4}$
   \quad \vrule\cr
  \Dsp\Aim      & 4.37 \cr
  \Dsp\Pim      & 3.70 \cr
  \Phii\Jpsi    & 0.88 \cr
  \Dsp\Dsm      & 0.34\cr
  \Dsp\Km       & 0.28 \cr
  \Phii$\psi^\prime$&  0.24   \cr
  \Kp\Aim       & 0.19 \cr
  \Kp\Pim       & 0.14 \cr
  \Dsp\Kxm      & 0.11 \cr
\noalign{\hrule}
  Sum           &  10.3\cr
\noalign{\hrule}}}
\vfil\eject\end

