%%%%%%%%%%%%% Macros %%%%%%%%%%%%%%%%%%%
\documentstyle[12pt]{article}

\def\doit#1#2{\ifcase#1\or#2\fi}

\doit0{
% Horizontal alignment of accents: bars, hats, tildes, etc.
\skewchar\fivmi='177 \skewchar\sixmi='177 \skewchar\sevmi='177
\skewchar\egtmi='177 \skewchar\ninmi='177 \skewchar\tenmi='177
\skewchar\elvmi='177 \skewchar\twlmi='177 \skewchar\frtnmi='177
\skewchar\svtnmi='177 \skewchar\twtymi='177
\def\@magscale#1{ scaled \magstep #1}
}

% For framing we need to put 1 in \doit just below as well as in \framing{1}.
\doit{0}{
% Framing 
\def\framingfonts#1{
\doit{#1}{\font\twfvmi  = ammi10   \@magscale5 % math italic
\skewchar\twfvmi='177 \skewchar\fivsy='60 \skewchar\sixsy='60
\skewchar\sevsy='60 \skewchar\egtsy='60 \skewchar\ninsy='60
\skewchar\tensy='60 \skewchar\elvsy='60 \skewchar\twlsy='60
\skewchar\frtnsy='60 \skewchar\svtnsy='60 \skewchar\twtysy='60
\font\twfvsy  = amsy10   \@magscale5 % math symbols
\skewchar\twfvsy='60
% Fonts
\font\go=font018			% Gothic
\font\sc=font005			% script
\def\Go#1{{\hbox{\go #1}}}	% Gothic for single characters in equations
\def\Sc#1{{\hbox{\sc #1}}}	% script for single characters in equations
\def\Sf#1{{\hbox{\sf #1}}}	% sans serif for single characters in equations
\font\oo=circlew10	      % thick circles (hollow: ` and a-n , solid:  p-z)
\font\ooo=circle10			% thin circles   
\font\ro=manfnt				% font with rope
\def\kcl{{\hbox{\ro 6}}}		% left-handed rope
\def\kcr{{\hbox{\ro 7}}}		% right-handed rope
\def\ktl{{\hbox{\ro \char'134}}}	% top end for left-handed rope
\def\ktr{{\hbox{\ro \char'135}}}	% " right
\def\kbl{{\hbox{\ro \char'136}}}	% " bottom left
\def\kbr{{\hbox{\ro \char'137}}}	% " right
}}  
}

% Underline for text or math
\catcode`@=11
%\def\un#1{\relax\ifmmode\@@underline#1\else
%	$\@@underline{\hbox{#1}}$\relax\fi}
\catcode`@=12

% Accents and foreign (in text):
\let\under=\b			% bar-under (but see \un above)
\let\ced=\c			% cedilla
\let\du=\d			% dot-under
\let\um=\H			% Hungarian umlaut
\let\sll=\l			% slashed (suppressed) l (Polish)
\let\Sll=\L			% " L
\let\slo=\o			% slashed o (Scandinavian)
\let\Slo=\O			% " O
\let\tie=\t			% tie-after (semicircle connecting two letters)
\let\br=\u			% breve

% Abbreviations for Greek letters
\def\a{\alpha} \def\b{\beta} \def\c{\chi} \def\d{\delta}
\def\e{\epsilon} \def\f{\phi} \def\g{\gamma}
\def\h{\eta} \def\i{\iota} \def\j{\psi} \def\k{\kappa}
\def\l{\lambda} \def\m{\mu} \def\n{\nu} \def\o{\omega}
\def\p{\pi} \def\q{\theta} \def\r{\rho} \def\s{\sigma}
\def\t{\tau} \def\u{\upsilon} \def\x{\xi} \def\z{\zeta}
\def\D{\Delta} \def\F{\Phi} \def\G{\Gamma} \def\J{\Psi}
\def\L{\Lambda} \def\O{\Omega} \def\P{\Pi} \def\Q{\Theta}
\def\S{\Sigma} \def\U{\Upsilon} \def\X{\Xi}

% Math symbols
\def\dvp{\raisebox{-.45ex}{\rlap{$=$}} \raisebox{-.45ex}{$\hskip .48ex { 
|}$}}	
\def\dvm{\raisebox{-.45ex}{\rlap{$=$}} }
\def\uvp{\raisebox{.45ex}{\rlap{$=$}} \raisebox{.45ex}{$\hskip .48ex { 
|}$}}	
\def\uvm{\raisebox{.45ex}{\rlap{$=$}} }
\def\DP{{\scriptsize{\dvp}}~~}\def\DM{{\scriptsize{\dvm}}~~}
\def\UP{{\scriptsize{\uvp}}~~}
\def\UM{{\scriptsize{\uvm}}~~}                          % 2-d vector indices
%\def\bo{{\raise.15ex\hbox{\large$\Box$}}}		% D'Alembertian
\def\bo{{\raise-.46ex\hbox{\large$\Box$}}}		% D'Alembertian
\def\cbo{\Sc [}						% curly "
\def\pa{\partial}					% curly d
\def\dell{\bigtriangledown}				% hi ho the dairy-o
\def\pr{\prod}						% product
\def\iff{\leftrightarrow}				% <-->
\def\conj{{\hbox{\large *}}}				% complex conjugate
\def\ltap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}}	  % < or ~
\def\gtap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}}	  % > or ~
\def\TH{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-4.7mu \llap H \;}}
\def\face{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-2.2mu \llap {$\ddot
	\smile$}}}					% happy face
\def\dg{\sp\dagger}					% hermitian conjugate
\def\ddg{\sp\ddagger}					% double dagger
\font\tenex=cmex10 scaled 1200

% Math stuff with one argument
\def\sp#1{{}^{#1}}				% superscript (unaligned)
\def\sb#1{{}_{#1}}				% sub"
\def\oldsl#1{\rlap/#1}				% poor slash
\def\sl#1{\rlap{\hbox{$\mskip 1 mu /$}}#1}	% good slash for lower case
\def\Sl#1{\rlap{\hbox{$\mskip 3 mu /$}}#1}	% " upper
\def\SL#1{\rlap{\hbox{$\mskip 4.5 mu /$}}#1}	% " fat stuff (e.g., M)
\def\PMMM#1{\rlap{\hbox{$\mskip 2 mu | $}}#1}	% 
\def\PMM#1{\rlap{\hbox{$\mskip 4 mu ~ \mid $}}#1}	% 

%\def\Tilde#1{{\widetilde{#1}}\hskip 0.03in}			
\def\Tilde#1{{\widetilde{#1}}\hskip 0.015in}	 % big tilde
\def\Hat#1{\widehat{#1}}			% big hat
\def\Bar#1{\overline{#1}}			% big bar
\def\bra#1{\left\langle #1\right|}		% < |
\def\ket#1{\left| #1\right\rangle}		% | >
\def\VEV#1{\left\langle #1\right\rangle}	% < >
\def\abs#1{\left| #1\right|}			% | |
\def\leftrightarrowfill{$\mathsurround=0pt \mathord\leftarrow \mkern-6mu
	\cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill
	\mkern-6mu \mathord\rightarrow$}
\def\dvec#1{\vbox{\ialign{##\crcr
	\leftrightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}
	$\hfil\displaystyle{#1}\hfil$\crcr}}}		% <--> accent
\def\dt#1{{\buildrel {\hbox{\LARGE .}} \over {#1}}}	% dot-over for sp/sb
\def\dtt#1{{\buildrel \bullet \over {#1}}}		% alternate "
\def\der#1{{\pa \over \pa {#1}}}		% partial derivative
\def\fder#1{{\d \over \d {#1}}}			% functional derivative

% Math stuff with more than one argument
\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise.20ex
	\hbox{$\scriptstyle{#2}$}}}}}			% fraction
\def\sfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{\small$#1$}}\over
	\vphantom1\smash{\raise.4ex\hbox{\small$#2$}}}}	% alternate fraction
\def\bfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over
	\vphantom1\smash{\raise.3ex\hbox{$#2$}}}}	% "
\def\afrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over#2}}    % "
\def\partder#1#2{{\partial #1\over\partial #2}}	  % partial derivative of
\def\parvar#1#2{{\d #1\over \d #2}}	          % variation of
\def\secder#1#2#3{{\partial^2 #1\over\partial #2 \partial #3}}	% second "
\def\on#1#2{\mathop{\null#2}\limits^{#1}}		% arbitrary accent
\def\bvec#1{\on\leftarrow{#1}}			% backward vector accent
\def\oover#1{\on\circ{#1}}				% circle accent

% Aligned equations
\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\caja{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup2\jot \caja
	\ialign{\strut \hfil$\displaystyle{##}$&$
	\displaystyle{{}##}$\hfil\crcr#1\crcr}}\,}
\newif\ifdtup
\def\panorama{\global\dtuptrue \openup2\jot \caja
	\everycr{\noalign{\ifdtup \global\dtupfalse
	\vskip-\lineskiplimit \vskip\normallineskiplimit
	\else \penalty\interdisplaylinepenalty \fi}}}
\def\li#1{\panorama \tabskip=\humongous				% eqalignno
	\halign to\displaywidth{\hfil$\displaystyle{##}$
	\tabskip=0pt&$\displaystyle{{}##}$\hfil
	\tabskip=\humongous&\llap{$##$}\tabskip=0pt
	\crcr#1\crcr}}
\def\eqalignnotwo#1{\panorama \tabskip=\humongous
	\halign to\displaywidth{\hfil$\displaystyle{##}$
	\tabskip=0pt&$\displaystyle{{}##}$
	\tabskip=0pt&$\displaystyle{{}##}$\hfil
	\tabskip=\humongous&\llap{$##$}\tabskip=0pt
	\crcr#1\crcr}}

% The following is to be skipped when using harvmac.tex
\doit0{
\def\ref#1{$\sp{#1)}$}
}

% Text style parameters
\topmargin=0in				% top margin (less 1") (LaTeX)
\headheight=0in				% height of heading (LaTeX)
\headsep=0in			% separation of heading from body (LaTeX)
\textheight=9in				% height of body (LaTeX)
\footheight=3ex				% height of foot (LaTeX)
\footskip=4ex		% distance between bottoms of body & foot (LaTeX)
\textwidth=6in				% width of body (LaTeX)
\hsize=6in				% " (TeX)
\parskip=\medskipamount			% space between paragraphs (LaTeX)
\lineskip=0pt				% minimum box separation (TeX)
\abovedisplayskip=1em plus.3em minus.5em	% space above equation (either)
\belowdisplayskip=1em plus.3em minus.5em	% " below
\abovedisplayshortskip=.5em plus.2em minus.4em	% " above when no overlap
\belowdisplayshortskip=.5em plus.2em minus.4em	% " below
\def\baselinestretch{1.2}	% magnification for line spacing (LaTeX)
\thicklines			    % thick straight lines for pictures (LaTeX)

% Title page
\thispagestyle{empty}		    % no heading or foot on title page (LaTeX)
\def\oldheadpic{				% old UM heading
	\setlength{\unitlength}{.4mm}
	\thinlines
	\par
	\begin{picture}(349,16)
	\put(325,16){\line(1,0){4}}\put(330,16){\line(1,0){4}}
 \put(340,16){\line(1,0){4}}\put(335,0){\line(1,0){4}}
	\put(340,0){\line(1,0){4}}\put(345,0){\line(1,0){4}}
	\put(329,0){\line(0,1){16}}\put(330,0){\line(0,1){16}}
	\put(339,0){\line(0,1){16}}\put(340,0){\line(0,1){16}}
	\put(344,0){\line(0,1){16}}\put(345,0){\line(0,1){16}}
	\put(329,16){\oval(8,32)[bl]}\put(330,16){\oval(8,32)[br]}
	\put(339,0){\oval(8,32)[tl]}\put(345,0){\oval(8,32)[tr]}
	\end{picture}
	\par
	\thicklines
	\vskip.2in}
\def\border{						% border
	\setlength{\unitlength}{1mm}
	\newcount\xco
	\newcount\yco
	\xco=-24
	\yco=12
	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcl$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbl$}}
	\xco=158
	\yco=12
	\put(\xco,\yco){$\ktr$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcr$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbr$}}
        \put(-20,11){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics University of
Maryland Elementary Particle Physics}
	\put(-20,-241.5){\tiny University of Maryland Elementary
Particle Physics University of Maryland Elementary Particle Physics
University of Maryland Elementary Particle Physics}
	\end{picture}
	\par\vskip-8mm}
\def\bordero{						% alternate border
	\setlength{\unitlength}{1mm}
	\newcount\xco
	\newcount\yco
	\xco=-24
	\yco=12
	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcl$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbl$}}
	\xco=158
	\yco=12
	\put(\xco,\yco){$\ktr$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcr$}
	\advance\yco by-2
	\ifnum\yco>-240
	\repeat
	\put(\xco,\yco){$\kbr$}}
	\put(-20,12){\ooo bacdefghidfghghdhededbihdgdfdfhhdheidhdhebaaahjhhdahbahgdedgehgfdiehhgdigicba}
	\put(-20,-241.5){\ooo ababaighefdbfghgeahgdfgafagihdidihiidhiagfedhadbfdecdcdfagdcbhaddhbgfchbgfdacfediacbabab}
	\end{picture}
	\par\vskip-8mm}
\def\headpic{						% UM heading
	\indent
	\setlength{\unitlength}{.4mm}
	\thinlines
	\par
	\begin{picture}(29,16)
	\put(165,16){\line(1,0){4}}
	\put(170,16){\line(1,0){4}}
	\put(180,16){\line(1,0){4}}
	\put(175,0){\line(1,0){4}}
	\put(180,0){\line(1,0){4}}
	\put(185,0){\line(1,0){4}}
	\put(169,0){\line(0,1){16}}
	\put(170,0){\line(0,1){16}}
	\put(179,0){\line(0,1){16}}
	\put(180,0){\line(0,1){16}}
	\put(184,0){\line(0,1){16}}
	\put(185,0){\line(0,1){16}}
	\put(169,16){\oval(8,32)[bl]}
	\put(170,16){\oval(8,32)[br]}
	\put(179,0){\oval(8,32)[tl]}
	\put(185,0){\oval(8,32)[tr]}
	\end{picture}
	\par\vskip-6.5mm
	\thicklines}

\def\endtitle{\end{quotation}\newpage}			% end title page

% Section heading and reference stuff
\def\sect#1{\bigskip\medskip \goodbreak \noindent{\bf {#1}} \nobreak \medskip}
\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
\def\Item{\par\hang\textindent}
\def\Itemitem{\par\indent \hangindent2\parindent \textindent}
\def\makelabel#1{\hfil #1}
\def\topic{\par\noindent \hangafter1 \hangindent20pt}
\def\Topic{\par\noindent \hangafter1 \hangindent60pt}
\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil}
\def\]{\rfloor{\hskip 0.35pt}\!\!\!\rceil}
%\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil\,}
%\def\]{\,\rfloor{\hskip 0.35pt}\!\!\!\rceil}
\def\delsl{{{\partial\!\!\! /}}}
\def\caldsl{{\calD\!\!\! /}}    
\def\calO{{\cal O}}
\def\asym{({\scriptstyle 1\leftrightarrow \scriptstyle 2})}
\def\Lag{{\cal L}}
\def\du#1#2{_{#1}{}^{#2}}
\def\ud#1#2{^{#1}{}_{#2}}
\def\dud#1#2#3{_{#1}{}^{#2}{}_{#3}}
\def\udu#1#2#3{^{#1}{}_{#2}{}^{#3}}
\def\calA{{\cal A}}\def\calB{{\cal B}}\def\calC{{\cal C}}\def\calD{{\cal D}}
\def\calE{{\cal E}}\def\calF{{\cal F}}\def\calG{{\cal G}}\def\calH{{\cal H}}
\def\calI{{\cal I}}\def\calJ{{\cal J}}\def\calK{{\cal K}}\def\calL{{\cal L}} 
\def\calM{{\cal M}}\def\calN{{\cal N}}\def\calO{{\cal O}}\def\calP{{\cal P}}
\def\calQ{{\cal Q}}\def\calR{{\cal R}}\def\calS{{\cal S}}\def\calT{{\cal T}}
\def\calU{{\cal U}}\def\calV{{\cal V}}\def\calW{{\cal W}}\def\calX{{\cal X}}
\def\calY{{\cal Y}}\def\calZ{{\cal Z}}
\def\rma{{\rm a}} \def\rmb{{\rm b}} \def\rmc{{\rm c}} \def\rmd{{\rm d}} 
\def\rme{{\rm e}} \def\rmf{{\rm f}} \def\rmg{{\rm g}} 
\def\tildef{{\tilde f}}
\def\calDsl{{\calD\!\!\!\! /}}
\def\plpl{{{\hskip0.03em}{}{+{\hskip -0.563em}{\raise -0.03em\hbox{$_+$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}} 
\def\mimi{{{\hskip0.03em}{}{-{\hskip -0.563em}{\raise -0.05em\hbox{$_-$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}}   
\def\E{{\cal E}}
\def\Re{{\cal R}e\,} 
\def\Im{{\cal I}m\,}
\def\order#1#2{{\cal O}({#1}^{#2})}
\def\alp{\alpha{\hskip 0.007in}'}
\def\oalp#1{\alp^{\hskip 0.007in {#1}}}
\def\naive{{{na${\scriptstyle 1}\!{\dot{}}\!{\dot{}}\,\,$ve}}}
\def\pl#1#2#3{Phys.~Lett.~{\bf {#1}B} (19{#2}) #3}
\def\np#1#2#3{Nucl.~Phys.~{\bf B{#1}} (19{#2}) #3}
\def\prl#1#2#3{Phys.~Rev.~Lett.~{\bf #1} (19{#2}) #3}
\def\pr#1#2#3{Phys.~Rev.~{\bf D{#1}} (19{#2}) #3}
\def\cqg#1#2#3{Class.~and Quant.~Gr.~{\bf {#1}} (19{#2}) #3} 
\def\cmp#1#2#3{Comm.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\jmp#1#2#3{Jour.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\ap#1#2#3{Ann.~of Phys.~{\bf {#1}} (19{#2}) #3} 
\def\prep#1#2#3{Phys.~Rep.~{\bf {#1}C} (19{#2}) #3}
\def\ptp#1#2#3{Prog.~Theor.~Phys.~{\bf {#1}} (19{#2}) #3}
\def\ijmp#1#2#3{Int.~Jour.~Mod.~Phys.~{\bf A{#1}} (19{#2}) #3}
\def\nc#1#2#3{Nuovo Cim.~{\bf {#1}} (19{#2}) #3}
\def\ibid#1#2#3{{\it ibid.}~{\bf {#1}} (19{#2}) #3}
\def\grg#1#2#3{Gen.~Rel.~Grav.~{\bf{#1}} (19{#2}) {#3} }
\def\pla#1#2#3{Phys.~Lett.~{\bf A{#1}} (19{#2}) {#3}}
\def\mpl#1#2#3{Mod.~Phys.~Lett.~{\bf A{#1}} (19{#2}) #3} 
\def\zp#1#2#3{Zeit.~f\"ur Phys.~{\bf{#1}C} (19{#2}) {#3}} 
\def\jgtp#1#2#3{Jour.~of Group Theory for Physicists, {\bf{#1}} (19{#2}) {#3}}
\def\rmp#1#2#3{Rev.~Mod.~Phys.~{\bf {#1}} (19{#2}) {#3}}
\def\hepth#1{hep-th/{#1}} \def\heplat#1{hep-lat/{#1}} 
\def\hepph#1{hep-ph/{#1}} 
\def\szet{{${\scriptstyle \b}$}}
\def\ula{{\underline a}} \def\ulb{{\underline b}} 
\def\ulc{{\underline c}} \def\uld{{\underline d}} 
\def\ule{{\underline e}} \def\ulf{{\underline f}} 
\def\ulg{{\underline g}} \def\ulm{{\underline m}} 
\def\ulp{{\underline p}} \def\uln{{\underline n}}
\def\ulq{{\underline q}} \def\ulr{{\underline r}}
\def\ul{\underline} 
\def\un{\underline} 
\def\ulA{{\un A}} \def\ulM{{\underline M}} 
\def\Check#1{{\raise0.6pt\hbox{\Large\v{}}{\hskip -10pt}{#1}}}
\def\cdm{{\Sc D}_{--}} \def\cdp{{\Sc D}_{++}}
\def\Pisl{{\Pi\!\!\!\! /}}
\def\eqques{{~\,={\hskip -11.5pt}\raise -1.8pt\hbox{\large ?}
{\hskip 4.5pt}\,}}
%\def\fracmm#1#2{{{#1}\over{#2}}}
%\def\fracm#1#2{\hbox{\large{${\frac{{#1}}{{#2}}}$}}}
\def\fracm#1#2{\,\hbox{\large{${\frac{{#1}}{{#2}}}$}}\,}
\def\fracmm#1#2{\,{{#1}\over{#2}}\,}
\def\half{{\fracm12}}
\def\ha{\half}
\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise -.20ex
	\hbox{$\scriptstyle{#2}$}}}}}			% fraction
\def\dt#1{\Dot{#1}}
\def\sqrttwo{{\sqrt2}}
\def\scst{\scriptstyle}
\def\itrema{$\ddot{\scriptstyle 1}$}
\def\Bo{\bo{\hskip 0.03in}}
\def\lrad#1{ \left( A {\buildrel\leftrightarrow\over D}_{#1} B\right) } 
\def\derx{\partial_x} \def\dery{\partial_y} \def\dert{\partial_t} 
\def\Vec#1{{\overrightarrow{#1}}}
\def\.{.$\,$}
\def\-{{\hskip 1.5pt}\hbox{-}}
\def\kd#1#2{\d\du{#1}{#2}}
% There are more than one line for the definition of \footnotew below:
\def\footnotew#1{\footnote{{\hsize=7.0in {\lineskip=-5pt #1}}}
\baselineskip 16pt\oddsidemargin=0.03in 
\evensidemargin=0.01in\hsize=6.5in\textwidth=6.5in} 
\def\low#1{\hskip0.01in{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}}
\def\ip{{=\!\!\! \mid}}
%\def\Dot#1{\buildrel{_{_{\hskip 0.01in}\bullet}}\over{#1}}
%\def\Dot#1{\buildrel{_{_{\hskip 0.01in}{_\bullet}}}\over{#1}}
\def\Dot#1{\buildrel{\hskip2.5pt_{\scriptscriptstyle\bullet}}\over{#1}}
\def\dt#1{\Dot{#1}}
\def\gg{{\hbox{\sc g}}}
\def\tr{\,\,{\rm tr}\,} \def\Tr{\,\,{\rm Tr}\,}
\def\det{{\rm det}\,} 
\def\Norm{|\hskip -2.0pt|}
\def\Sgn{\hbox{\rm Sgn}\,}
\def\hati{{\hat i}} \def\hatj{{\hat j}} \def\hatk{{\hat k}} 
\def\hatl{{\hat l}} \def\hatm{\hat m} \def\hatn{\hat n} \def\hatr{\hat r}
\def\hats{\hat s} \def\hatt{\hat t}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%% Additional Fonts %%%%%%%%%%%%%%%%%%%%%%%%%%%
\font\tenmib=cmmib10
\font\sevenmib=cmmib10 at 7pt % =cmmib7 % if you have it
\font\fivemib=cmmib10 at 5pt  % =cmmib5 % if you have it
\font\tenbsy=cmbsy10
\font\sevenbsy=cmbsy10 at 7pt % =cmbsy7 % if you have it
\font\fivebsy=cmbsy10 at 5pt  % =cmbsy5 % if you have it
 %
\def\BMfont{\textfont0\tenbf \scriptfont0\sevenbf
                              \scriptscriptfont0\fivebf
            \textfont1\tenmib \scriptfont1\sevenmib
                               \scriptscriptfont1\fivemib
            \textfont2\tenbsy \scriptfont2\sevenbsy
                               \scriptscriptfont2\fivebsy}
 %
\def\rlx{\relax\leavevmode}                  
 % Guess what this is for...
\def\BM#1{\rlx\ifmmode\mathchoice
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\scriptstyle\BMfont#1$}}
                      {\hbox{$\scriptscriptstyle\BMfont#1$}}
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{\bf Preliminary Version (FOR YOUR EYES ONLY!) \hfill \today} \vskip -0.04in  
}
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{\hbox to\hsize{
%\today
% April 1998
\hfill UMDEPP 98--099}}
{\hbox to\hsize{
\doit1{\hfill{hep-th/9804162}} 
}
\vskip 0.05in
{\hbox to\hsize{
\hfill(Revised Version)
}} 
\vskip -10pt

\par 

\hsize=6.5in
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\begin{center}
\vglue 0.15in

\baselineskip 18pt 
 
{\large\bf Parity Conservation in} \\  
{\large\bf Supersymmetric Vector-Like Theories}$\,$\footnote{This
work is supported in part by NSF grant \# PHY-93-41926.} \\  

\baselineskip 10pt 

\vskip 0.5in

Hitoshi~ N{\small ISHINO}

\vskip 0.08in

{\it Department of Physics} \\[.015in]
{\it University of Maryland} \\[.015in]
{\it College Park, MD 20742-4111, USA} \\[.020in]   
{E-Mail: nishino@nscpmail.physics.umd.edu}

\vskip 2.7in

{\bf Abstract} \\[0.1in]  
\end{center}

\baselineskip 16pt 

\begin{quotation}

~~~We show that parity is conserved in vector-like
supersymmetric theories, such as supersymmetric QCD with massive quarks
with no cubic couplings among chiral multiplets, based on fermionic
path-integrals, originally developed by Vafa and Witten.  
We also look into the effect of supersymmetric breaking 
through gluino masses, and see that the 
parity-conservation is intact also in this case.   
Our conclusion is valid, when only bosonic parity-breaking
observable terms are considered in path-integrals like the original Vafa-Witten
formulation.  

\endtitle

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\vskip 0.1in                                                       
\centerline{\bf 1.~~Introduction}          
         
Non-perturbative chiral symmetry breaking \ref\thooft{G.~'t Hooft, 
in {\it `Recent Developments in Gauge Theories'}, G.~'t Hooft {\it et al.}~eds. 
(Plenum Press, NY 1980); S.~Coleman and E.~Witten, \prl{45}{80}{100}.} is an
important aspect for phenomenological model building based on vector-like
theories, such as in composite particle models \ref\nussinov{S.~Nussinov,
\prl{51}{83}{2081}; \ibid{52}{84}{966}; D.~Weingarten, \prl{51}{83}{1830};
E.~Witten, \prl{51}{83}{2351}.}\ref\bps{J.C.~Pati, \pl{228}{89}{228}; 
K.S.~Babu, J.C.~Pati and H.~Stremnitzer,
\pl{256}{91}{206}; \prl{67}{91}{1688}.}.  
It has been proven that parity symmetry 
is conserved in (non-supersymmetric) vector-like QCD theories, and
is not broken spontaneously even non-perturbatively \ref\vw{C.~Vafa and
E.~Witten, \prl{53}{84}{535}.}\ref\vwnp{C.~Vafa and E.~Witten, 
\np{234}{84}{173}.}.  This proof \vw\ is based on the 
evaluation of fermionic path-integral, which gives always non-negative  
vacuum energy after adding parity-breaking terms.  However, 
the question has risen whether the Vafa-Witten constraint \vw\vwnp\ for the
non-supersymmetric case can be avoided in supersymmetric
vector-like theories because of the new interactions among
gluino-quark-squarks \ref\cvetic{M.~Cveti\v c, Maryland
preprint, PP \#85-23 (Aug.~1984).}\ref\pati{J.C.~Pati, {\it private
communications}.}, and whether parity is broken like other 
global symmetries \ref\kotcheff{A.C.W.~Kotcheff and
G.M.~Shore, \np{301}{88}{267}; G.M.~Shore, Nucl.~Phys.\newline    {\bf B271}
(1986) 589.}.  These particular interaction terms with scalar-dependence 
seem to be the main obstruction for the proof for the 
positive definiteness of the determinant in the fermionic path-integral
\vwnp\cvetic\pati.  Moreover, the results about gauge symmetry breaking for
massless supersymmetric QCD, when the number of flavor $~N_f$~ is smaller 
than the number of colors: $~N_f < N$~ \ref\ads{I.~Affleck, M.~Dine 
and N.~Seiberg, \np{241}{84}{493}.} also seem to suggest the 
parity-breaking in supersymmetric vector-like theories.  Although these
observations seem reasonable, it also seems to contradict with the other
universal wisdom about supersymmetry that supersymmetric vacuum is stable when
the Witten index $~\Tr (-1)^F$~ \ref\wittenindex{E.~Witten, \np{202}{82}{253}.}
is non-zero, {\it e.g.,} $~\Tr (-1)^F =N$~ for the gauge group $~SU(N)$, and
therefore the vacuum energy most probably stays zero with no parity breaking. 
It has been also recently point out \ref\ks{A.~Kovner and M.~Shifman, 
\pr{56}{97}{2346}.} that supersymmetric QCD has condenstate-free phase with no
gluino condenstate.  In our present paper we reconsider this subtle problem of
parity breaking, and give a proof for parity  conservation in supersymmetric
vector-like theories with massive quarks.  

The model we deal with in this paper is a globally supersymmetric
vector-like theory with massive chiral multiplets coupled to a non-Abelian vector
multiplet with no cubic coupling among chiral multiplets.  
Our proof is based on three major 
assumptions:  First one for the massiveness of all the quark chiral
multiplets, the second one about the absence of the Yukawa-couplings among 
chiral multiplets, and the third one that we rely on the method in \vw\ for
purely bosonic parity-breaking observables.  Therefore our method does not 
cover the fermionic parity-breaking observables like Wilson fermions
\ref\ag{S.~Aoki and A.~Gocksch,
\pl{231}{89}{449}; \ibid{243B}{90}{409}; \pr{45}{92}{3845}; 
S.~Sharpe and R.~Singleton, \heplat{9804208}.} treted 
in lattice QCD.\footnotew{For reviews for supersymmetric QCD
on lattice, see, e.g., \ref\montvay{I.~Montvay, \heplat{9709080}, 
{\it and references therein}.}}  The
massiveness of quarks are also important for non-perturbative conservation of
supersymmetry,  due to well-defined non-zero Witten index $~\Tr(-1)^F$~ in such
cases \wittenindex.  This is because 
supersymmetry is conserved, only if the vacuum energy is zero.  Therefore the  
non-perturbative breaking of supersymmetry would cause the shift of
vacuum energy, causing the breaking of parity
symmetry \vw\vwnp.  Interestingly, we will find that parity is conserved also for a
vector-like supersymmetric theory, like non-supersymmetric vector-like theory. 
We also look into the effect of gluino masses, which will not disturb the main
body of the proof for supersymmetric case, and therefore parity is also
conserved in broken supersymmetric vector-like theories.

   

\bigskip\bigskip\bigskip


\centerline{\bf 2.~Review for Non-Supersymmetric Vector-Like Theory} 

We start with reviewing the parity conservation in non-supersymmetric case
\vw\vwnp\ first, in order also to elucidate our notation.  Suppose the total
lagrangian $~\Lag(\l)\equiv \Lag - \l X$~  with a parameter $~\l$~ is a
generalization of the parity-conserving lagrangian $~\Lag$, 
such as that of QCD,
with a parity-non-conserving observable $~X$, 
such as the $~F\Tilde F\-$term, with a real constant $~\l$.  
If parity is broken in the vacuum and $~\langle X\rangle\neq0$, 
then the theory can choose a vacuum state in which $~\l\langle X \rangle <0$ 
due to the signature
ambiguity of $~\langle X \rangle$.  Hence the vacuum  energy $~E(\l)$~ can be
lower than $~E(0)$~ of the parity-conserving vacuum energy: $~E(\l\neq0)
< E(0)$.  

However, an explicit evaluation of path-integral reveals that this would not
happen, {\it i.e.,} there is no such vacuum whose energy is lower than that 
of the parity-conserving one \vw.  Consider the path-integral in Euclidian
space for the vacuum energy $~E(\l)$:
$$ e^{- VE(\l)} = \int \[ d A\du a I\,\] \[d\chi\] \[d \Bar\chi\,\] 
\[d\xi\] \[d \Bar\xi\,\] \exp
\left[  - \int d^4 x \left( \Lag + i \l X \right)  \right] ~~, 
\eqno(2.1) $$ 
where $~V$~ is the Euclidian volume, $~A\du a I$~ is the gluon field.  
The indices $~{\scst I,~J,~\cdots}$~ are for adjoint 
representations of the gauge group $~G$.  For example for
$~G=SU(N)$, we have $~{\scst I,~J,~\cdots~=~1, ~2,~\cdots,~N^2-1}$. 
In order to clarify basic constituents of our system, we use 2-component spinors
in this paper.  Since we are dealing with a vector-like theory, our two-component
Weyl spinors  $~\chi\low i$~ and $~\xi^i$~ with the flavor indices $~{\scst
i,~j,~\cdots~=~1,~2,~\cdots,~N}$~ for quarks are in the conjugate
representations to each other.   The factor of $~i$~ in the
$~\l X\-$term is due to the usual Wick rotation.  We specify the lagrangian as 
$$\li{ & \Lag = - \frac14 \big(F_{a b}{}^I \big)^2 + \Lag_{\rm F}~~, \cr  
&\Lag_{\rm F} = +i \big( \Bar\chi{\,}^{\Dot\a\, i} \Dsl\ud\b{\Dot\a}
     \chi\low{\b i} \big) + i \big( \xi{\,}^{\a\, i} \Dsl\du\a{\Dot\b}
     \Bar\xi_{\Dot\b i} \big)  
     + m\du i j \big(\Bar\chi{\,}^{\Dot\a i} \Bar\xi_{\Dot\a j} \big)
     + m\du j i \big(\xi^{\a j} \chi\low{\a i} \big)  ~~,  
&(2.2) \cr } $$ 
where $~m\equiv\big(m\du i j\big)$~ is an $~N\times N$~ 
hermitian mass matrix: $~\big(m\du i j\big)^* 
= m\du j i$, which can be arranged to have only positive eigenvalues.  
Since we are dealing in this paper only with a vector-like theory, 
the quark fermions $~\chi\low i$~
and $~\xi^i$~ are in the representations conjugate to each other, {\it e.g.,}
$~{\bf N}$~ and $~{\bf N}^*\-$representations of $~SU(N)$, respectively.  
Accordingly, our covariant derivative $~D_a$~ contains the minimal coupling 
of the gauge field to these fermions.  We
are using the notations similar to that in \ref\ggrs{S.J.~Gates Jr., 
M.T.~Grisaru, M.~Ro\v cek and W.~Siegel, {\it `Superspace'}, Benjamin/Cummings,
Reading, MA, 1983.}, {\it e.g.,} we use the Minkowskian four-dimensional (4D)
vector indices $~{\scst a,~b,~\cdots ~=~ 0,~1,~2,~3}$, with the signature
$~\big(\eta_{a b} \big)  = \hbox{diag.}~(+,-,-,-)$, while
$~{\scst \a,~\b,~\cdots~=~1,~2}$~ and $~{\scst \Dot\a,~\Dot\b,~\cdots~=~\Dot
1,~\Dot 2}$~ for the 2-component spinors.  Other relevant
relations are such as  
$$\li{ & \Dsl_{\a\Dot\b} \equiv \big(\s^c \big)_{\a\Dot\b} D_c~~, ~~~~
      \Big[\big(\s^c)_{\a\Dot\b} \Big]^* = (\s^c)_{\b\Dot\a} ~~, ~~~~
      \psi^\a = C^{\a\b} \psi_\b ~~, ~~~~ 
      \Bar\psi_{\Dot\a} = \Bar\psi{}^{\Dot\b} 
     C_{\Dot\b\Dot\a} ~~, \cr 
& \big(\psi^\a\big)^\dagger = + \Bar\psi{}^{\Dot\a}~~, ~~~~
      \big(\psi_\a\big)^\dagger = -  \Bar\psi{}_{\Dot\a}~~, ~~~~
    \big(\psi_1^{\a_1} \cdots \psi_j^{\a_j} 
      \Bar\chi{}_1^{\Dot\b_1} \cdots \Bar\chi{}_k^{\Dot\b_k} \big)^\dagger 
   = \chi_k^{\b_k} \cdots  \chi_1^{\b_1} \Bar\psi{}_j^{\Dot\a_j} \cdots
   \Bar\psi{}_1^{\Dot\a_1}~~, \cr 
& \partial_a^\dagger = - \partial_a~~, ~~~~ 
      \big( C_{\a\b} \big) = \big( C_{\Dot\a\Dot\b} \big) 
      = \pmatrix {{\scst 0} & {\scst - i} \cr {\scst + i} & {\scst 0} \cr} ~~. 
&(2.3) \cr } $$
Based on these, it is easy to confirm the reality of each term in our
lagrangian.  

As was shown by Vafa-Witten in a vector-like theory \vwnp, the fermionic
space for the path-integral (2.1) can be a direct sum of the positive and
negative eigenstates of the Dirac operator in a finite volume $~V$.  
Let $~\chi\low{\a i\m}$~ and 
$~\Bar\xi_{\Dot\a i \m}$~ correspond to such eigenstates \ref\amati{{\it For 
using two-component spinors for eigenstates of Dirac operator, see,
e.g,} D.~Amati, K.~Konishi, Y.~Meurice, G.C.~Rossi and G.~Veneziano,
\prep{162}{88}{169}.}:   
$$\Dsl^{\b\Dot\a} \chi\low{\b i \m} 
     = +\m \,\Bar\xi{}\ud{\Dot\a}{i\m}~~,~~~~
\Dsl^{\a\Dot\b} \Bar\xi_{\Dot\b i
\m} = +\m \,\chi\ud\a{i\m}~~.  
\eqno(2.4) $$
We then
easily see that the set of $~\chi\low{\a(-\m)} \equiv\chi\low{\a(+\m)},~
\Bar\xi_{\Dot\a i(-\m)} \equiv - \Bar\xi_{\Dot\a i(+\m)}$~ corresponds to the
eigenvalue $~-\m$:  
$$\Dsl^{\b\Dot\a} \chi\low{\b i (-\m)} = -\m \,
              \Bar\xi\ud{\Dot\a}{i(-\m)}~~,~~~~
\Dsl^{\a\Dot\b} \Bar\xi_{\Dot\b i
(-\m)} = -\m \,\chi\ud\a{i(-\m)}~~.   
\eqno(2.5) $$
Therefore the whole fermionic space is not only a direct sum of positive and
negative eigenstates, but also they are always paired up between $~+\m>0$~ and 
$~-\m<0$.  

Or equivalently, in terms of a four-component Dirac spinor
$$ \li{&\psi\equiv\big( \psi_{\un\a i}\big) 
     \equiv \pmatrix{{\scst\chi_{\a i}} \cr
     {\scst\Bar\xi_{\Dot\a i}} \cr}~~, ~~~~
      \Bar\psi\equiv\big(\Bar\psi{}^{\un\a i} \big)  
     \equiv \pmatrix{{\scst\Bar\chi{}^{\Dot\a i}} \cr {\scst\xi^{\a i}}
     \cr}~~,
&(2.6) \cr } $$ 
with the four-component spinorial indices $~{\scst
\a~\equiv~(\a,\Dot\a),~\b~\equiv~(\b,\Dot\b),~\cdots}$, we have  
$$ \li{& \Tilde
\Dsl \psi = \pmatrix{ O & \Dsl\du\a{\Dot\b} \cr
                                 \Dsl\ud\b{\Dot\a} & O \cr }       
     \pmatrix{ \chi_{\b i} \cr \Bar\xi_{\Dot\b i} \cr }  ~~, ~~~~ 
     \Tilde \Dsl \equiv \pmatrix{ 0 & \Dsl\du\a{\Dot\b} \cr
                                 \Dsl\ud\b{\Dot\a} & 0 \cr} {~~.~~~~~ ~~~~~}
&(2.7) \cr } $$
We can also introduce the $~2N\times 2N$~ mass matrix $~\Tilde m$~ 
and the usual $~4\times 4$~ $~\g_5\-$matrix for the four-component notation by 
$$ \li{&\Tilde m \equiv \pmatrix{ m\du i j & 0 \cr 0 & m\du i j \cr } ~~, ~~~~
       \g_5 \equiv \pmatrix { \d\du\a\b & 0 \cr 
        0 & - \d\du{\Dot\a}{\Dot\b} \cr } ~~. 
&(2.8) \cr } $$ 
Note also that $~\Tilde m$~ is hermitian: $~\Tilde m^\dagger = \Tilde m$. 
The eigenstates in (2.6) are much transparent now for $~\Tilde\Dsl$~ as 
$$ \li{&\Tilde\Dsl\psi_\m = + \m \psi_\m ~~, ~~~~
     \Tilde\Dsl \psi_{(-\m)} = -\m \psi_{(-\m)} ~~,  
&(2.9) \cr } $$
The eigenstates $~\psi_{(-\m)}$~ correspond to 
$~\psi_{(-\m)} \equiv \g_5 \psi_\m$, because $~\g_5$~ satisfies
$~\{ \g_5,\Tilde \Dsl\}= 0 $, and therefore 
$$ \li{&\Tilde\Dsl\psi_{(-\m)}  
        = \Tilde\Dsl\big(\g_5 \psi_\m\big) = - \g_5 \Tilde \Dsl\psi_\m 
       = - \m \g_5 \psi_\m = (- \m) \psi_{(-\m)} ~~.  
&(2.10) \cr }$$ 
Therefore the pairing between $~\m$~ and $~-\m$~ eigenstates in (2.6) 
is clear.  Accordingly, the lagrangian $~\Lag_{\rm F}$~ is simply
$$ \li{& \Lag_{\rm F} = \Bar\psi \big( i\Tilde\Dsl + \Tilde m \big) \psi ~~, 
&(2.11) \cr} $$ 

Our path-integral (2.1) is now
$$ \li{ e^{- V E(\l) } & = \int \[ d A\du a I \] 
      \int \[ d\psi \] \[ d\Bar\psi \] 
       \, e^{-\int d^4 x \big( \Lag_{\rm F} + \Lag_{\rm B} + i \l X \big)} \cr 
& = \int \[ d A \du a I \] \int \[ d\psi \] \[ d\Bar\psi \] 
     \, e^{- \int d^4 x \big(\Lag_{\rm B} + i \l X  \big)} \,  
      \exp \left[ -\int d^4 x 
     \Bar\psi \big( i\Tilde\Dsl + \Tilde m \big) \psi \, \right] \cr 
& = \int \[ d A \du a I \] \, I_{\rm F} \,  
    \exp \left[ \, - \int d^4 x \big( \Lag_{\rm B} + i \l X \big) \,
\right] ~~. 
&(2.11) \cr } $$ 
Here $~I_{\rm F}$~ is the 
fermionic determinant from the fermionic path-integral
$$\li{I_{\rm F} & \equiv 
    \int \[ d\psi \] \[ d\Bar\psi \] \exp\left[ \, - \int d^4 x \, 
    \Bar\psi \, \big( i\Tilde\Dsl + \Tilde m \big) \, \psi \, \right] \cr 
& = \Det \big( i\Tilde\Dsl + \Tilde m \big) \cr 
& = \prod_\m \det\big( i\m I_{2N} + \Tilde m \big) \cr 
& = \big(\det\Tilde m \big)^{n_{\rm L 0} + n_{\rm R0}} 
     \left[\, \prod_{\m>0} \det\big(i\m I_{2N} + \Tilde m\big) \, \right]
      \left[\, \prod_{\m<0} \det\big(i\m I_{2N}  + \Tilde m\big) \, \right]\cr 
& = \big(\det\Tilde m  \big)^{2 n_0} 
      \left[\, \prod_{\m>0} \det\big(i\m I_{2N}  + \Tilde m\big) \, \right]
      \left[\, \prod_{\m>0} \det\big(\! - i\m I_{2N}  + \Tilde m\big) \, \right] 
&(2.12\rma) \cr  
& = \big( \det\Tilde m \big)^{2 n_0} 
      \prod_{\m>0} \det\big(i\m I_{2N} + \Tilde m\big) \, 
      \det\big(\! - i\m I_{2N} + \Tilde m\big)  \cr 
& = \big(\det\Tilde m \big)^{2 n_0} 
      \prod_{\m>0}\, \det\big(i\m I_{2N} + \Tilde m\big)  \, 
      \det\big(\! + i\m I_{2N} + \Tilde m\big)^\dagger 
&(2.12\rmb) \cr 
& = \big(\det\Tilde m \big)^{2 n_0}  
      \prod_{\m>0} \left[ \, \det\big(i\m I_{2N} +  \Tilde m\big) \,\right]\,  
      \left[ \,\det\big(\! + i\m I_{2N} + \Tilde m\big)\, \right]^* \cr 
& = \big(\det\Tilde m \big)^{2 n_0} \,   
      \bigg| \,\prod_{\m>0} \det\big(i\m I_{2N} + \Tilde m\big) \,\bigg|^2  > 0 
&(2.12\rmc) \cr } $$ 
Here $~I_{2N}$~ is an $~2N \times 2N$~ unit matrix, and 
in (2.12a), $~n_0$~ is the number of $~\m=0$~ modes, 
satisfying $~n\low{\rm L 0} 
= n\low{\rm R 0} \equiv n_0$.  
This is because we have to consider only the instanton
number zero background $~n\low{\rm L 0} - n\low{\rm R 0} = 0$~ that is 
connected with the original vacuum with energy $~E(\l=0)$~ 
\ref\wittenprivate{E.~Witten, {\it private  communication.}}.  The
previously-mentioned parings $~\m\leftrightarrow -\m$~ are used also in
(2.12a).  The determinants 
in (2.12a) are taken for the $~2N\times 2N$~ matrix for flavour indices,
distinguished from the symbol ~`$\Det$'~ for the fermionic path-integral.   In
(2.12b)  we have also used the hermiticity of $~\Tilde m$.  Now the 
positive definiteness of $~I_{\rm F}$~ is clear from (2.12).  For the case of 
$~N=1$, eq.~(2.12c) is in agreement with \vw.   

Once the fermionic determinant (2.12) is positive, we see that 
the path-integral (2.1) is
positive, except for the phase factor $~\exp\,(i\l \int d^4 x X)$, which does
not lower the ground state energy.  This is why $~E(\l)$~ must have a minimum
only at $~\l=0$~ \vw.  

Before ending this section, we give the following lemma which will be of 
importance in the next section.  Note that the lagraigian $~\Lag_{\rm F}$~ 
is rewritten as
$$ \Lag_{\rm F} = \sum_\m \Lag_\m  
\equiv \sum_\m \Big[ \, 
     + i\m\big(\Bar\chi{}\ud{\Dot\a i}\m\Bar\xi_{\Dot\a i\m} \big)  
     + i \m \big(\xi\ud{\a i}\m \chi\low{\a i \m}\big) 
     + m\du i j \big( \Bar\chi{}\ud{\Dot\a i}\m
       \Bar\xi_{\Dot\a j\m} \big) 
     + m\du i j \big(\xi{}\ud{\a i}\m\chi\low{\a j\m} \big) \, \Big]  ~~.  
\eqno(2.13) $$
Accordingly, the path-integral (2.12) is also equivlent to  
$$ \li{ I_{\rm F} & = \left(\prod_\m \int \[ d \chi_\m \] \[ d \Bar\chi_\m \] 
     \[ d \xi_\m \] \[ d \Bar\xi_\m \] \right) \cr  
& ~~~~~ ~~~~~ ~~~~~  \times\exp\bigg[ \,  \int d^4 x\, \sum_\m \bigg\{ \xi_{\a
      \m} C^{\a\b} \big( m  + i \m I_N \big) \chi\low{\b\m} 
     + \Bar\chi_{\Dot\a\m} C^{\Dot\a\Dot\b} 
      \big( m + i \m I_N \big) \Bar\xi_{\Dot\b \m} \bigg\} \, \bigg]  \cr 
& = \big(\det\Tilde m \big)^{2 n_0} \,   
      \bigg| \,\prod_{\m>0} \det\big(i\m I_{2N} + \Tilde m\big) \,\bigg|^2  > 0 
     ~~. 
&(2.14) \cr } $$ 
In other words, $~I_{\rm F}$~ in (2.12) can be computed in terms of 
2-component spinors in (2.6).  This relatinship will be helpful when 
we consider complicated mixed lagrangians between the gaugini and the
quark/lepton fields in the next section.     
 



\bigskip\bigskip\bigskip

\centerline{\bf 3.~~Vector-Like Theory with Supersymmetry} 

We now generalize the above method to supersymmetric theories with 
no Yukawa couplings among quarks.  
Suppose we have the massive quark fermions
$~\chi\low i,~\Bar\chi{\,}^i,~{\xi}^i,~\Bar\xi_i~$ together with the massless
gluino Majorana fields $~\l\du\a I,~\Bar\l\du{\Dot\a}I $.  As before, 
the indices  
$~{\scst I,~J,~\cdots~=~1,~2,~\cdots, ~g~=~ \dim G}$~ are for the adjoint
representations of the gauge group $~G$.  All the
fermion-dependent terms in our lagrangian are       
$$\li{ \Lag_{\rm F} = & +i \big( \Bar\chi{}^{\Dot\a\, i} \Dsl\ud\b{\Dot\a}
     \chi\low{\b i} \big)  + i \big( \xi{}^{\a\, i} \Dsl\du\a{\Dot\b}
     \Bar\xi_{\Dot\b i} \big)
      + i \big( \Bar\l{}^{\Dot\a I} \Dsl\ud\b{\Dot\a} \l\du\b I \big)  
     + m\du i j \big(\Bar\chi^{\Dot\a i} \Bar\xi_{\Dot\a j} \big)
     + m\du j i \big(\xi^{\a j} \chi\low{\a i} \big)  \cr   
& + i\big(T^I\big)\du i j  \left[ \, z^ {*i} \big( \l^{\a I} \chi\low{\a j}\big)
     - z_j \big(\Bar\l{}^{\Dot\a I} \Bar\chi\du{\Dot\a}j \big) \, \right] 
    - i \big(T^I\big)\du i j  
      \left[ \, u^i \big(\Bar\l{}^{\Dot\a I} \Bar\xi_{\Dot\a j} \big)
     - u^*_j \big(\l^{\a I} \xi\du{\a}i \big) \, \right] {~~.~~~~~ ~~~~~}
&(3.1)  \cr} $$ 
The $~z_i$~ and $~u^i$~ are the
spin $~0$~ fields (squarks) in the chiral multiplets $~(z\low i, \chi\low i)$~
and $~(u^i, \xi^i)$~ with $~{\scst i,~j,~\cdots~=~1,~2,~\cdots,~N}$, 
in the representations $~{\bf N}$~ and $~{\bf N^*}$. 
The $~\big(T^I\big)\du i j$~ are 
hermitian generators of the gauge group.  As in (2.2), we can assume that
$~m\du i j$~ is hermitian only with positive eigenvalues.  
The presence of these mixing
terms with (pseudo)scalar-dependence have been considered 
to be the main obstruction for the 
parity-conservation in supersymmetric theories in the past \vwnp\cvetic\pati, 
because they seem to prevent us from proving the positive definiteness 
of the fermionic determinant.  
However, we will see that this is not the obstruction.  After  
the above prescription, there is no fermion-dependent term in the
supersymmetric lagrangian $~\Lag$~ left over
other than $~\Lag_{\rm F}$: $~\Lag = \Lag_{\rm F} + \Lag_{\rm B}$~ with a purely
bosonic lagrangian $~\Lag_{\rm B}$.  

We now consider the eigenstates for $~\chi$~ and $~\xi$~ as in (2.4), and
rewrite all the $~\chi$~ and $~\xi\-$depenent terms in $~\Lag_{\rm F}$, as
$$ \li{\Lag_{\chi,\xi} = \sum_\m \Lag_{\chi,\xi,\m} 
\equiv \sum_\m \Big[ & \! -\xi\dud\a i\m C^{\a\b} \big( m\du i j +
     i\m\d\du i j \big) \chi\low{\b j \m} 
     - \Bar\chi\dud{\Dot\a} i\m C^{\Dot\a\Dot\b} \big( m\du i j +
     i\m\d\du i j \big) \Bar\xi_{\Dot\b j \m} \cr 
& \!- \xi\dud\a i\m C^{\a\b} \r\low{\b i} 
        - \Bar\chi\dud{\Dot\a} i\m C^{\Dot\a\Dot\b}\Bar\omega_{\Dot\b i} 
       - \Bar\xi_{\Dot\a i\m} C^{\Dot\a\Dot\b} \Bar\r\du{\Dot\b} i 
          - \chi\low{\a i\m} C^{\a\b} \omega\du\b i \, \Big] {~,~~~~~ ~~~~~}
&(3.2) \cr } $$
where 
$$ \li{& \r\low{\a i} \equiv +i \big( T^I u^*\big)_i \l\du\a I ~~, ~~~~
       \o\du\a i \equiv +i \big( z^* T^I \big)_i \l\du\a I ~~, \cr 
& \Bar\r\du{\Dot\a}i \equiv -i \big(u T^I \big)^i \Bar\l\du{\Dot\a} I ~~, ~~~~
\Bar\o_{\Dot\a i} \equiv - i \big(T^I z \big)_i \Bar\l\du{\Dot\a} I ~~. 
&(3.3) \cr } $$

As usual in path-integral, we can redefine the fields in such a way that the
linear terms in $~\chi$~ or $~\xi$~ disappear.  In our case, this can be done
by the field redefinitions\footnotew{Note that (3.4b) is not necessarily the 
hermitian conjugate of (3.4a).  This is related to the hermiticity only 
by the combination of $~+\m$~ and $~-\m$.}  
$$ \li{&\Tilde\xi\dud\a i\m \equiv \xi\dud\a i \m + \omega\du\a j \big( m+
     i\m I_N \big)^{-1}{}\du j i ~~, ~~~~
\Tilde\chi\low{\a i\m} \equiv \chi\low{\a i \m} + \big( m+
     i\m I_N \big)^{-1}{}\du i j \r_{\a j} ~~, 
&(3.4{\rm a})  \cr 
&\Tilde{\Bar\xi}_{\Dot\a i\m} \equiv \Bar\xi_{\Dot\a i\m} 
      + \big( m+ i\m I_N \big)^{-1}{}\du i j \Bar\omega_{\Dot\a j} ~~,
~~~~ \Tilde{\Bar\chi}\dud{\Dot\a} i\m \equiv \Bar\chi\dud{\Dot\a} i \m 
     + \Bar\r\du{\Dot\a} j \big( m+
     i\m I_N \big)^{-1}{}\du j i ~~, 
&(3.4{\rm b}) \cr } $$
to have 
$$\li{ \Lag_{\xi,\chi,\m} = \, & -\Tilde\xi\dud\a i\m C^{\a\b} 
      \big(m+i\m I_N \big)\du i j\Tilde\chi\low{\b j\m} 
      - \Tilde{\Bar\chi}\dud{\Dot\a}i\m
     C^{\Dot\a\Dot\b} \big(m+i\m I_N \big)\du i j 
     \Tilde{\Bar\xi}_{\Dot\b j\m} \cr 
& + \omega\du\a i C^{\a\b} \big(m+i\m I_N \big)^{-1}{}\du i j \r\low{\b j} 
     + \Bar\r\du{\Dot\a} i C^{\Dot\a\Dot\b} \big(m+i\m I_N \big)^{-1}{}\du i j 
     \Bar\omega_{\Dot\b j} ~~.   
&(3.5) \cr } $$
Since $~m$~ is hermitian only with positive eigenvalues, $~m + i\m I_N$~ is 
also diagonalizable only with non-zero eigenvalues, and there is no problem for
defining the inverse $~\big(m + i\m I_N\big)^{-1}$.  
After this, $~\Lag_{\rm F}$~ is now 
$$\Lag_{\rm F} = \sum_\m \Lag\dashs_{\!\chi,\xi,\m} + \sum_\m \Lag_{\l^2,\m} 
     +\Lag_{\l \Dsl\l} ~~, 
\eqno(3.6) $$
where $~\Lag\dashs_{\!\chi,\xi,\m}$~ is the first line of (3.5) which coincides
with the non-supersymmetric case (2.7), 
$~\Lag_{\l \Dsl\l}$~ is the gluino kinetic term, while $~\Lag_{\l^2,\m}$~
is the $~\l^2$~ and $~\Bar\l{}^2\-$terms after the field redefinition (3.4).  
Let us collect all of these $~\l\-$dependent terms into $~\Lag_\l$:  
$$\li{ \Lag_\l & \equiv -i \Bar\l\du{\Dot\a} I \Dsl^{\b\Dot\a} \l\du\b I 
     - \sum_\m \l\du\a I C^{\a\b} M\du\m{I J} \l\du\b J 
     - \sum_\m \Bar\l\du{\Dot\a} I C^{\Dot\a\Dot\b} \Bar M\du\m{I J} 
      \Bar\l\du{\Dot\b} J \cr 
&\equiv -i \Bar\l\du{\Dot\a} I \Dsl^{\b\Dot\a} \l\du\b I 
     - \l\du\a I C^{\a\b} M^{I J} \l\du\b J 
     - \Bar\l\du{\Dot\a} I C^{\Dot\a\Dot\b} \Bar M{}^{I J} 
      \Bar\l\du{\Dot\b} J ~~, 
&(3.7) \cr  } $$
where the matrices $~M_\m\equiv \big( M\du\m{I J}\big),~
\Bar M_\m\equiv \big( M\du\m{I J}\big),~ 
M\equiv \big( M^{I J}\big)$~ and $~\Bar M\equiv \big(\Bar M{}^{I J}\big)$~ 
are defined
by $$ \li{& M\du\m{I J} \equiv 
     -\big( z^* \, T^{(I|} \,\Tilde m_\m^{-1} T^{|J)} \, u^* \big)~~,~~~~
      \Bar M\du\m{I J} \equiv -\big( u\, T^{(I|} \,  
     \Tilde m_\m^{-1} T^{|J)} \, z\big)~~,\cr    
&M^{I J} \equiv \sum_\m M\du\m{I J} ~~, ~~~~
    \Bar M{}^{I J} \equiv \sum_\m \Bar M\du\m{I J} ~~, ~~~~
    \Tilde m_\m \equiv m + i\m I_N ~~.  
&(3.8) \cr }$$ 
Due to the antisymmetry of $~C^{\a\b}$~ and $~~C^{\Dot\a\Dot\b}$, 
the matrices $~M_\m,~\Bar M_\m,~M$~ and $~\Bar M$~ are all symmetric in $~{\scst
I \leftrightarrow J}$.  Note also that   
$$ M\du\m\dagger = M\du\m * = \Bar M_{-\m}~~, ~~~~ M^\dagger = M^* = \Bar M~~,  
\eqno(3.9) $$ 
the latter of which is confirmed by the former under $~\sum_\m$, which is
symmetric between $~+\m\leftrightarrow -\m$.  
The $~M$~ is not necessarily 
hermitian, and it has both real and imaginary part.  
Now $~\Lag_\l$~ is rewritten as 
$$ \li{ & \Lag_\l = - \big( \l^{\a \,I} , \Bar\l{}^{\Dot\a\,I} \big) 
     \pmatrix{\d\du\a\b M^{I J} &~& i \d^{I J} \Dsl\du\a{\Dot\b} \cr 
                i \d^{I J} \Dsl\ud\b{\Dot\a} &~& \d\du{\Dot\a}{\Dot\b}
              \Bar M^{I J} \cr }
     \pmatrix{\l\du\b J \cr \Bar\l\du{\Dot\b}J \cr } 
     = \Bar\L \, \big( i \calDsl + \calM \big) \L ~~,      
&(3.10{\rm a})  \cr
& \calDsl \equiv \pmatrix {O & \Dsl\du\a{\Dot\b} \cr  
                           \Dsl\ud\b{\Dot\a} & O \cr }\otimes I_g ~~, ~~~~
           \calM \equiv I_2\otimes\pmatrix {M & O \cr O & \Bar M \cr }~~,   
&(3.10{\rm b}) \cr 
& \Bar\L \equiv -\big( \l^\a, \Bar\l{}^{\Dot\a} \big) 
     = \big(\l_\b, \Bar\l_{\Dot\b}  \big) 
     \pmatrix{C^{\b\a} & O \cr O & C^{\Dot\b\Dot\a} \cr } = \L^T \calC~~, 
     ~~~~\calC\equiv \pmatrix{C^{\a\b} & O \cr O & C^{\Dot\a\Dot\b} \cr }
     {~~.~~~~~ ~~~~~}   
&(3.10{\rm c}) \cr } $$ 
   
The original fermionic lagrangian now is $~\Lag_{\rm F} = 
\sum_\m \Lag\dashs_{\!\chi,\xi,\m} + \Lag_\l$, and 
the total path-integral to be considered in the Euclidian space is 
$$ e^{-V E(\l)} = \int \[ d A\du a I\, \] 
       \[ d z\] \[ d z^*\] \[ d u\] \[ d u^*\]
       \, \, I_{\rm F} \, \, 
       e^{-\int d^4 x\, \big( \Lag_{\rm B} + i \l X\big) }~~,  
\eqno(3.11) $$
where the fermionic path-integral $~I_{\rm F}$~ is from (3.5) and (3.10) 
with $~\calM$~ replaced by $~\calM$: 
$$ \li{I_{\rm F} & = \int \[ d \chi\] \[ d \Bar\chi \] 
     \[ d \xi\] \[ d \Bar\xi\] \[ d \l \] \[ d \Bar\l\]  
     \exp\Big[ \, -\int d^4 x \,\Big( \sum_\m
      \Lag\dashs_{\!\chi,\xi,\m} + \Lag_\l \Big) \, \Big] \cr 
& = \bigg( \prod_\m \int \[ d\Tilde\chi_\m \] \[ d\Tilde{\Bar\chi}_\m \] 
     \[ d\Tilde\xi_\m \] \[ d\Tilde{\Bar\xi}_\m \] \,  
    e^{-\int d^4 x\, \Lag\dashs_{\!\chi,\xi,\m} } \bigg) 
      \int \[ d\l \]  \[ d\Bar\l \] \, 
      e^{-\int d^4 x\, \Lag_\l }  \cr
& = \big( \det m  \, \big)^{2n_0}  
     \bigg| \, \prod_{\m>0} \det\big( m+ i\m I_N \big) \, \bigg|^2 \,\,   
      \big[ \, \Det\big( i \calDsl
     + \calM \big) \, \big]^{1/2}  {~~. ~~~~~ ~~~ }
&(3.12) \cr } $$
The first two factors are from the $~\int{\scst\[ d
\Tilde\chi\] \[ d \Tilde{\Bar\chi} \] \[d\Tilde\xi \] \[
d\Tilde{\Bar\xi} \]}\-$integral as in the  non-supersymmetric case (2.14) now
with the shifted variables $~\Tilde\chi,~\Tilde{\Bar\chi},~\Tilde\xi,~
\Tilde{\Bar\xi}$, and the 
remaining factor is from the $~\int{\scst\[d \l\] \[ d\Bar\l\]}\-$integral.  
There is potential phase ambiguity \ref\wittensutwo{E.~Witten,
\pl{117}{82}{324}; S.D.H.~Hsu, \hepth{9704149}; R.~Narayana and P.~Vranas,
\np{506}{97}{373}.}\ref\alvarez{L.~Alvarez-Gaum\'e 
and Witten, \np{234}{83}{269}.} for taking the square root 
in the last factor in (3.12). 
However, we will shortly show that there is no problem with this ambiguity in
vector-like theories.  

Even though we can not diagonalize $~\calDsl$~ and $~\calM$~
simultaneously,  we still can use the eigenstate $~|\n\rangle$~ for the
eigenvalue $~\n\in\IR$~ of the operator $~\calDsl$:
$$ \calDsl \ranu = + \n \ranu~~, ~~~~{\it i.e.,}~~~~
       \calDsl \L_\n = \calDsl \pmatrix{\l_\n \cr \Bar\l_\n \cr } 
     = + \n \pmatrix{\l_\n \cr \Bar\l_\n \cr} \equiv \n \L_\n ~~, 
\eqno(3.13) $$
where the subscript $~_\n$~ on $~\L_\n$~ {\it etc.}~denotes the eigenvalue 
for the four-component spinor $~\L$, with the adjoint index $~^I$~ omitted.  As
usual, we can define  
$$ \li{ & \G_5 \equiv I_2\otimes \pmatrix{I_g & O \cr O & - I_g \cr } ~~, 
&(3.14) \cr } $$
satisfying $~\{ \G_5, \calDsl\} = 0$, 
so that an eigenstate $~\raminu$~ of $~\calDsl$~ can be constructed by   
$$ \calDsl \big[\, \G_5 \ranu\,\big] =
     = - \G_5 \calDsl\ranu = (-\n) \big[\, \G_5 \ranu \,\big] 
      ~~~~\Longrightarrow ~~~~\G_5 | \n \rangle = \raminu ~~. 
\eqno(3.15) $$ 
Therefore any eigenstate for $~\forall\n>0$~ is always paired up with 
an eigenstate $~-\n<0$.  We now see that a conjugate state 
$~\lanu$~ is related to $~\ranu$~ as follows:  Consider 
$$ \li{ & \L^\dagger = \pmatrix {\l_\a\cr \Bar\l_{\Dot\a} \cr}^\dagger 
    = \big( \! -\Bar\l_{\Dot\a}~, ~ - \l_\a \big) = \big( \l,\Bar\l \big) 
     \pmatrix{O & - I_g\cr - I_g & O \cr} 
     = \L^T \calF =\Bar\L \calC^{-1} \calF {~~, ~~~~~ ~~~~~} 
&(3.16{\rm a}) \cr 
&\calF\equiv I_2 \otimes \pmatrix{O & -I_g \cr -I_g & O \cr} ~~, ~~~~
    \calF^{-1} = \calF ~~, ~~~~\[\calF, \calC \] = 0 ~~, 
     ~~~~\[ \calF, \calM \] =0 ~~.  
&(3.16{\rm b}) \cr } $$
Here $~\Bar\L$~ is the usual Dirac conjugate of $~\L$, and $~\calF$~ is 
needed for complex-conjugation.  Therefore 
$$\li{& \ranu^\dagger = \lanu\calC^{-1} \calF^{-1} ~~, ~~~~ 
      \lanu^\dagger = \calF\calC\ranu ~~. 
&(3.17) \cr } $$ 
Accordingly, as in (3.15) we can confirm that  
$$ \li{ & \laminu = \lanu \G_5 ~~. 
&(3.18) \cr } $$ 

Other important relations needed are  
$$ \li{ &\calF \calM \calF^{-1} = \calM^\dagger ~~, 
      ~~~~\calC \calM \calC^{-1} = \calM  ~~, ~~~~
      \[ \G_5, \calM\] = 0 ~~, ~~~~  
&(3.19) \cr 
&\calF \calDsl \calF^{-1} = + \Hat\calDsl 
      \equiv \pmatrix{ 0 & \Dsl\ud\b{\Dot\a} \cr
                        \Dsl\du\a{\Dot\b} & 0 \cr} ~~, ~~~~        
      \calC \calDsl \calC^{-1} = - \Hat\calDsl ~~, ~~~~
       \calDsl^\dagger = - \Hat\calDsl~~, ~~~~~
&(3.20) \cr } $$ 
as easily confirmed.  Using these as well as (3.16), we get 
$$ \li{ \G_5 \big( i\calDsl + \calM \big)\G_5 = & - i\calDsl + \calM \cr 
= & + \calC^{-1} \calF^{-1} \big( i\Hat\calDsl + \calM^\dagger\big) 
        \calF\calC 
        = \calC^{-1} \calF^{-1} \big( - i \calDsl^\dagger + \calM^\dagger\big)
        \calF\calC \cr 
= & + \calC^{-1} \calF^{-1} \big(i \calDsl + \calM \big)^\dagger \calF\calC~~.
&(3.21) \cr } $$
 
The determinant in the square root in
the last factor in (3.12) can be re-expressed 
as the usual definition of the determinant 
in terms of exponential, trace and logarithmic
functions:
$$ \Det\big( i\calDsl + \calM \big) 
     = \big( \det M \big)^{\Tilde n_0} \big( \det\Bar M \big)^{\Tilde n_0}
       \prod_{\n\neq0} \, \exp\big[ \, \lanu \ln\big( i \calDsl 
      + \calM \big) \ranu \, \big]  ~~.  
\eqno(3.22) $$ 
As in the case of quarks, we consider only the 
instanton number zero background, so that the number of left- and right-handed
zero-modes are the same: $~\Tilde n\low{{\rm L}0} = \Tilde n\low{{\rm
R}0} \equiv\Tilde n_0$, whose contributions in (3.22) can be computed
separately, as  
$$ \li{& \big( \det M \big)^{\Tilde n_0} 
      \big( \det\Bar M \big)^{\Tilde n_0}    
      = \big( \det M\big)^{\Tilde n_0} \big(\det M\big)^*{}^{\Tilde n_0} 
       = \big|\, \det M \, \big|^{2\Tilde n_0} ~~.  
&(3.23) \cr} $$
As for the $~\n\neq 0$~ contributions, due to the pairing between the
$~\ranu$~ and $~\raminu$, (3.22) is rewritten as 
$$ \li{ &\Det\big( i \calDsl + \calM \big) \cr 
&~ = \big|\, \det M \, \big|^{2\Tilde n_0} 
      \left[\, \prod_{\n>0} \exp \big[ \, \lanu  
      \ln\big( i \calDsl + \calM \big) \ranu \, \big]\, \right] \,  
       \left[\, \prod_{\n<0} \, \exp \big[ \, \lanu 
       \ln\big( i \calDsl + \calM \big) \ranu\, \big] \, \right] \cr 
&~ = \big|\, \det M \, \big|^{2\Tilde n_0} 
     \prod_{\n>0} \exp \big[ \, \lanu 
      \ln\big(i \calDsl+\calM \big)\ranu \, \big]
     \,\exp \big[\,\laminu \ln\big( i \calDsl +\calM
      \big)\raminu\,\big] {~~, ~~~~~ ~~~~~}
&(3.24) \cr } $$
where the exponent in the last factor is simplified by the aid of 
(3.17) and (3.21) as 
$$ \li{ &~ \laminu \ln\big(i\calDsl+ \calM \big) \raminu  
     =\lanu \G_5\ln\big( i\calDsl +\calM \big) \G_5\ranu \cr      
& = \lanu \calC^{-1} \calF^{-1}
      \big\{ \ln \big( i\calDsl +\calM \big) \big\}^\dagger  
      \calF\calC \ranu 
      = \big[\, \lanu \ln \big( i\calDsl +\calM \big)\ranu \,\big]^\dagger\cr
& = \big[\, \lanu \ln \big( i\calDsl +\calM \big)
      \ranu \,\big]^*  ~~. 
&(3.25) \cr } $$
Therefore (3.24) is semi-positive definite:
$$\li{ & \Det\big( i \calDsl + \calM \big) 
     = \big|\, \det M \, \big|^{2\Tilde n_0} 
      \prod_{\n>0} \Big|\, \exp\lanu \ln\big(i\calDsl + \calM  \big) \ranu \,
      \Big|^2 \ge 0 ~~.  
&(3.26) \cr } $$
Combining this with (3.12), we get the semi-positive definiteness of 
the fermionic determinant:
$$ \li{&I_{\rm F} = \big|\, \det m \,\big|^{2n_0} \, 
     \big|\, \det M \, \big|^{\Tilde n_0} \, 
     \bigg| \, \prod_{\m>0} \det\big( m+ i\m I_N \big)\,\bigg|^2 \, 
     \prod_{\n>0} \bigg|  \, 
         \exp \lanu \ln\big( i \calDsl + \calM \big)\ranu\, \bigg| \ge 0 
     {~.~~~~~ ~~~~~}  
&(3.27) \cr} $$
 
Notice that the usual phase ambiguity when taking the square root
\wittensutwo\alvarez\ does not arise 
here, because of the semi-positive definite expression of (3.26), as 
conbributions from $~\ranu$~ and $~\raminu$~ always in pairs.  The main
ingredient in this proof is the usage of the eigenstate $~\ranu$~ with   the
properties of $~\G_5$~ and complex conjugations, which do not require the
diagonalization of $~\calM$, or even its commutator with $~\calDsl$.  The
crucial procedure we have relied on is the expression of the determinant in
terms of  exponential, trace and logarithmic functions, which is to be the
universal definition for a determinant.   

Note that (3.27) implies only 
non-negativity of $~I_{\rm F}$~ which can still be zero.  This is because the
matrix $~\calM$~ can depend on the scalar coordinates $~Z\equiv (z,z^*, u,
u^*)$.  However, we can further 
show that there exists a measurable support (a connected domain with non-zero
measure) in the $~Z\-$space, on which $~I_{\rm F}(Z)>0$~ and non-zero.  
In fact, consider the particular point $~Z_0=0$~ on which 
$~\calM=O$~ due to (3.8), (3.9) and (3.10).  It follows that 
$$\li{&I_{\rm F}(0) = \big|\, \det m \,\big|^{2n_0} \,
     \big|\, \det M \, \big|^{\Tilde n_0}  
     \bigg|\, \prod_{\m>0} \det\big( m+ i\m I_N \big)\,\bigg|^2 \, 
     \prod_{\n>0} \bigg|  \, 
         \exp \lanu \ln\big( i\n \big)\ranu\, \bigg| > 0 
     {~~.~~~~~ ~~~~~}  
&(3.28) \cr} $$
Once we get $~I_{\rm F}(0) >0$, then relying on the smoothness of $~I_{\rm
F}(Z)$~ as a function of $~Z$, we can conclude that $~I_{\rm F}(Z) >0$~ 
on a measurable support including $~Z_0=0$.    
The existence of a measurable support for $~I_{\rm F}>0$~ leads us to the 
positivity of the path-integral measure $~\int {\scst \[d z \] \[d z^* \] 
\[d u \] \[d u^* \] } \,  I_{\rm F} > 0$, and we conclude that parity is
conserved in supersymmetric vector-like theories.    

In the above analysis, we have performed the most usual Wick rotation from the
Minkowskian metric $~(+,-,-,-)$~ into the Euclidian one $~(-,-,-,-)$~ by
replacing formally the coordinate $~x^0\rightarrow i x^4$.  However, this may
need more care, when it comes to the complex conjugation of spinors. 
Motivated by this, we have re-confirmed our result above by an
alternative Wick rotation into the metric $~(+,+,+,+)$.  Additionally, the
spinors in these Euclidian spaces are only `formally' defined, in such a way
that their complex conjugation rule is essentially the parallel to the
Minkowskian case, like the simple replacement $~x^0 \rightarrow i x^4$, and this
is the very reason why the Feynman rules in the usual Euclidian
path-integral is essentially the same as those in the Minkowskian. 
Rigorously speaking,  spinors in the Euclidian spaces can exist only as
$~\hbox{\it USp}(2)$~ spinors \ref\kt{T.~Kugo and P.K.~Townsend,
\np{211}{83}{357}.}, and moreover the dotted and undotted spinors as
eigenvectors of the $~\g_5\-$matrix are no longer related by complex
conjugations \kt.\footnotew{This situation is similar to what is called
Aiyah-Ward space-time with the signature $~(+,+,-,-)$~ studied in
\ref\gkn{S.J.~Gates, Jr., S.V.~Ketov and H.~Nishino,  \pl{307}{93}{323};
\pl{307}{93}{331}; \pl{297}{92}{99}; \np{393}{93}{149}.}.}  For this precaution,
we have also reformulated the Wick rotation, such that the dotted $~\l_\a$~ and
undotted $~\Bar\l_{\Dot\a}$~ spinors in the final Euclidian space are not
related to each other under complex conjugation, as they should be \kt. 
Interestingly, we have reached the same conclusion for the semi-positive
definiteness of the determinant (3.27), even though the meaning of the bra- and
cket-vectors are slightly modified, and all the pseudo-scalar Yukawa couplings
with $~\g_5$~ acquire an extra factor of $~i$~ like the $~\l X\-$term in (2.1). 
One  additional feature in this case we seem to rely on is that the
gluino zero-modes are unstable and disappear from the physical spectrum, 
based on the analysis in ref.~\ref\cs{A.~Casher and Y.~Shamir, \np{314}{89}{390};
Y.~Shamir, {\it private communication}; {\it See e.g.,} V.V.~Khoze, M.P.~Mattis,
M.J.~Slater, hep-th/9804009.}.  
Since the details of this formulation is rather technical leading essentially to
the same conclusion, we skip them in this paper.  


   
\bigskip\bigskip\bigskip

\vbox{\centerline{\bf 4.~Vector-Like Theory with Broken Supersymmetry}

We mention the possibility of adding some gluino mass terms, which may be 
caused by some spontaneous, explicit, or non-perturbative breaking of 
supersymmetry.  This is easily considered, by adding the gluino mass terms} 
$$ \Lag_{m\dashs\l^2} \equiv m\dashs{}^{I J} \big(\l^{\a I} \l\du{\a}J \big) + 
      m\dashs{}^{I J} \big({\Bar\l}{}^{\Dot\a I} \Bar\l\du{\Dot\a}J \big) ~~, 
\eqno(4.1) $$
to our original lagrangian (3.1).  Here $~m\dashs\equiv \big(m\dashs{}^{I J} \big)$~ is
real and symmetric.  Accordingly, (3.10a) is now 
$$ \Lag\dashs_\l \equiv - \big( \l^\a, \Bar\l{}^{\Dot\a} \big) 
       \pmatrix{ I_2 \otimes (M + m\dashs)  & i \Dsl\otimes I_g \cr 
                i \Dsl^T\otimes I_g  & I_2 \otimes (\Bar M + m\dashs) \cr } 
     \pmatrix{\l_\b \cr \Bar\l_{\Dot\b} \cr} ~~.  
\eqno(4.2) $$
This implies that the matrix $~M$~ is replaced by $~M+m\dashs$~ and 
$~\Bar M$~ by $~\Bar M+m\dashs$.  Under this shift,   
the relations such as $~M^\dagger = \Bar M$~ are intact.  
Eventually (3.27) is now replaced by 
$$ \li{& I\dashs_{\!\rm F} =  
      \big| \,\det m \,\big|^{2n_0} \big|\, \det M \, \big|^{\Tilde n_0} \, 
     \bigg| \, \prod_{\m>0} \det\big( m+ i\m I_N \big)\,\bigg|^2 
         \, \bigg|\, \prod_{\n>0}   
         \exp \lanu \ln\big( i \calDsl + \calM\dashs \big)\ranu \,\bigg| 
        \ge 0 {~,~~~~~ ~~~~~}  
&(4.3) \cr} $$
where $~\calM\dashs$~ is a $~4g\times 4g$~ matrix similar to $~\calM$~ 
defined by 
$$ \calM\dashs \equiv \calM + I_2\otimes \pmatrix{m\dashs & O \cr  O & m\dashs
\cr } ~~,  ~~~~ \calM'^{\,\dagger} = \calM\dashs{}^{\,*} ~~.    
\eqno(4.4) $$
Hence the presence of $~m\dashs$~ does not affect the semi-positive definiteness
of the fermionic determinant.  Accordingly, we can also show
that $~I\dashs_{\!\rm F} > 0 $~ on a measurable support in the
$~Z\-$space,  and therefore we conclude that parity is conserved also in broken
supersymmetric vector-like theories with non-zero gluino masses.      
 


\bigskip\bigskip\bigskip

\centerline{\bf 5.~Concluding Remarks} 

In this paper we have shown the conservation of parity in 
supersymmetric vector-like theories.  
The main body of our proof is the confirmation that the determinant as  
the fermionic path-integral $~I_{\rm F}$~ is
positive and non-zero on a measurable support under the bosonic integral 
$~\int {\scst \[ d z \] \[ d z^* \] \[ d u \] \[ d u^* \]}$. 
We have also seen that the quark-gluino-squark mixing  
terms in the supersymmetric theory pose no problem.  
The supersymmetric
vector-like theory seems to avoid the problem with these mixing terms thanks to
parings between the eigenstates of the Dirac operator, 
despite of the complication caused by the mixing with gluini.  

In our analysis of the fermionic path-integral, we first integrated over
the quark fields $~\chi,~\Bar\chi, ~\xi,~\Bar\xi$, making the computation more
organized, instead of integrating over the gluino field first as in \cvetic.  
By so doing, we have seen that the final gluino path-integral is less involved 
and more controllable, in 
particular when we need to consider the Majorana gluino determinant
which used to have subtlety with $~\g_5\-$pseudo-scalar couplings.  We have
understood that the usual phase ambiguity in the
square root of the determinants for a 2-component spinor
\wittensutwo\alvarez\ does not arise in a vector-like theory, due to
the pairing between the integrals over dotted and undotted spinors, combined
with the pairing between the positive and negative eigenstates of the Dirac
operator, including the zero-modes.     
 
We have also studied the effect of the gluino masses, as a result of either  
spontaneous, explicit, or non-perturbative breaking of supersymmetry, and 
reached the conclusion that parity is also 
conserved in these cases with broken supersymmetry.  In principle, we can also
consider the squark masses caused by the supersymmetry breaking, but these
terms are purely bosonic affecting only $~\Lag_{\rm B}$, so that they 
are not expected to change our analysis or result in this paper.     

Note that our result relies on the original method in \vw, namely we deal only
with bosonic parity-breaking observables in path-integrals.  Therefore our
result does not cover the parity-breaking {\it via} fermionic observables
\ag\ which do not acquire the imaginary unit $~i$~ under the Wick-rotation.   

Our result here seems to contradict with ref.~\ads, which shows that the
vacuum structure is disturbed, when there are more colors than 
flavors: $~N_f< N$.  However, there is actually no conflict,
because we interpret this as the result of the masslessness of quarks treated in
ref.~\ads.  In our system, due to the massive quarks from the outset, the Witten
index $~\Tr(-1)^F$~ is well-defined and non-zero \wittenindex, {\it e.g.},
$~\Tr(-1)^F = N$~ for the $~SU(N)$~ gauge group.  Hence we expect no breaking 
of supersymmetry even at the non-perturbative level, which would have ruined the
foundation of our proof.  Since the topological stability due to the
well-defined $~\Tr (-1)^F$~ is reliable for massive quarks, it is quite natural
that chiral symmetry or parity symmetry is also conserved.      
Additionally, our parity-conservation is also consistent with the result of
\ks\ about the phase with unbroken discrete axial symmetry.  
Some subtlety arises, when the masses of the quarks become zero, 
because in such a case the
Witten index  $~\Tr(-1)^F$~ is no longer well-defined
\wittenindex\ref\shamir{Y.~Shamir, \prl{66}{91}{3101}.}, and
therefore the vacuum loses its stability against chiral or parity breakings
as in \ads.  From this viewpoint, we see no contradiction 
of our result with refs.~\ads\shamir, in which the masses of quarks are zero 
from the outset.     

We have seen that the supersymmetry breaking, if its only effect on fermions
is the gluino masses, does not alter the conservation of parity.  Even though
this statement seems contradictory with the previous paragraph, we understand
that the breaking of supersymmetry lifts the vacuum energy higher than the 
original supersymmetric and parity-conserving vacuum with $~E(0)=0$.  

\doit0{
We can try to generalize our result to include appropriate Yukawa
couplings {\it via} the usual cubic interactions among chiral multiplets.  
However, it seems impossible to require a generalized mass matrix 
with the Yukawa-couplings  
is positive definite.  This is because the potential $~\big|W(z,u)\big|^2$~ 
is supposed to have both negative and positive 
curvature, depending on the $~(z,z^*,u,u^*)\-$coordinates.  
Therefore its second derivative with respect to
the chiral fields has no definite signature, ruining the foundation
of our proof.   
}

We can try to apply our method to other arbitrary global or discrete 
symmetries, such as baryon number, in a supersymmetric vector-like theory,    
using the prescription using upper bounds for fermionic propagators in
ref.~\ref\weinberg{S.~Weinberg, {\it `The Quantum Theory of Fields'}, Vol.~II,
page 238, Cambridge University Press, 1996.}.  However, there seems to be an
obstruction caused by the  zero-ness of mass eigenvalues, {\it e.g.}, our matrix
$~\calM$~ in (3.10) hitting zeros, that upsets the upper bound for
fermionic propagators, undermining the foundation for the stability of
fermion-anti-fermions Greens functions against symmetry-breaking parameters
\vwnp\weinberg.  In other words, our method using the semi-positive definite 
fermionic determinant is powerful only for parity symmetry, or other symmetries
based only on the vacuum to vacuum amplitude.    

\bigskip\bigskip

\vbox{
We are grateful for J.C.~Pati for bringing about the problem with parity 
conservation in supersymmetric QCD, and for other helpful discussions.  We are
also indebted to M.~Luty, R.~Mohapatra, Y.~Shamir, and E.~Witten for important
suggestions  and comments.  Additional acknowledgement is due to M.~Cveti\v c  
for a copy of reference \cvetic, and other important discussions.  Special 
acknowledgement is for C.~Vafa for pointing out mistakes in an
earlier  version of the manuscript.  
}
 
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