This is a set of macros for the paper. %%% %%% FTMACROS %%% Version JF 30/11/92 %%% \magnification1200 \tolerance=10000 \hsize=17truecm\vsize=23truecm \parindent=40pt \multiply\baselineskip by 15\divide \baselineskip by 10 \def\today{\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\day, \number\year} \def\header#1{\rm\nopagenumbers \hfil\underbar{#1} \hfil\today\bigskip \headline={\rm\ifnum\pageno>1 #1\hfil\today\hfil Page \folio\else\hfil\fi}} % % % \def\dst{\displaystyle} \def\sty{\scriptstyle} \def\sst{\scriptstyle} \def\tst{\textstyle} \def\leftdisplay#1{\leftline{$\displaystyle{#1}$}} \def\leftrightdisplay#1#2{\line{$\displaystyle{#1}$\hfil$\displaystyle{#2}$}} \def\frac#1#2{\dst {#1\over#2}} % fractions in displaystyle \def\sfrac#1#2{{\tst{#1\over#2}}} % fractions in textstyle \def\pmb#1{\setbox0=\hbox{#1} % generate bold face \kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0 \kern-.025em\box0}%Knuth puts in \raise.0433em before box0 \def\shbox#1{\vbox{\hrule\hbox{\vrule\vbox{#1}\vrule}\hrule}} % box it % % Greek letters % \def\al{\alpha} \def\be{\beta} \def\ga{\gamma} \def\de{\delta} \def\ep{\epsilon} \def\ze{\zeta} \def\et{\eta} \def\th{\theta} \def\ka{\kappa} \def\la{\lambda} \def\rh{\rho} \def\si{\sigma} \def\ta{\tau} \def\up{\upsilon} \def\ph{\phi} \def\ch{\chi} \def\ps{\psi} \def\om{\omega} \def\Al{\Alpha} \def\Be{\Beta} \def\Ga{\Gamma} \def\De{\Delta} \def\Ze{\Zeta} \def\Et{\Eta} \def\Th{\Theta} \def\Ka{\Kappa} \def\La{\Lambda} \def\Rh{\Rho} \def\Si{\Sigma} \def\Ta{\Tau} \def\Up{\Upsilon} \def\Ph{\Phi} \def\Ch{\Chi} \def\Ps{\Psi} \def\Om{\Omega} % % Characters % \def\0{{\bf 0}} \def\a{{\bf a}} \def\b{{\bf b}} \def\k{{\bf k}} \def\el{\hbox{\pmb{$\ell$}}} \def\m{{\bf m}} \def\t{{\bf t}} \def\s{{\bf s}} \def\S{{\cal S}} \def\q{{\bf q}} \def\x{{\bf x}} \def\y{{\bf y}} \def\r{{\bf r}} \def\s{{\bf s}} \def\L{{\bf L}} \def\p{{\bf p}} \def\w{{\bf w}} \def\z{{\bf z}} \def\F{{\cal F}} \def\bpsi{{\bf \Psi}} % % Vincent's capital roman double letters % \def\AA{{\it I}\hskip-3pt{\tt A}} \def\CC{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$} \hskip -0.1em C}} \def\FF{\hbox to 8.33887pt{\rm I\hskip-1.8pt F}} \def\NN{\hbox to 9.3111pt{\rm I\hskip-1.8pt N}} \def\PP{\hbox to 8.61664pt{\rm I\hskip-1.8pt P}} \def\QQ{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}} {\hskip -0.1em Q}} \def\RR{\hbox to 9.1722pt{\rm I\hskip-1.8pt R}} \def\ZZ{\hbox to 8.2222pt{\rm Z\hskip-4pt \rm Z}} %\def\ZZ{Z\!\!\!Z} % % % Blackboard characters % \def\bbbr{{\rm I\!R}} \def\bbbn{{\rm I\!N}} \def\bbbm{{\rm I\!M}} \def\bbbh{{\rm I\!H}} \def\bbbf{{\rm I\!F}} \def\bbbk{{\rm I\!K}} \def\bbbp{{\rm I\!P}} \font \tensans = cmss10 \font \fivesans = cmss10 at 5pt \font \sixsans = cmss10 at 6pt \font \sevensans = cmss10 at 7pt \font \ninesans = cmss10 at 9pt \font \twelvesans = cmss12 \newfam\sansfam \textfont\sansfam=\tensans\scriptfont\sansfam=\sevensans \scriptscriptfont\sansfam=\fivesans \def\sans{\fam\sansfam\tensans} \def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} %\bbbone {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}} \def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox %\bbbc to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}} \def\bbbe{{\mathchoice {\setbox0=\hbox{\smalletextfont e}\hbox{\raise %\bbbe 0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.3pt height0.7\ht0\hss}\box0}} {\setbox0=\hbox{\smalletextfont e}\hbox{\raise 0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.3pt height0.7\ht0\hss}\box0}} {\setbox0=\hbox{\smallescriptfont e}\hbox{\raise 0.1\ht0\hbox to0pt{\kern0.5\wd0\vrule width0.2pt height0.7\ht0\hss}\box0}} {\setbox0=\hbox{\smallescriptscriptfont e}\hbox{\raise 0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.2pt height0.7\ht0\hss}\box0}}}} \def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm %\bbbq Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}} \def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm %\bbbt T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}} \def\bbbs{{\mathchoice %\bbbs {\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}} \def\bbbz{{\mathchoice {\hbox{$\sans\textstyle Z\kern-0.4em Z$}} %\bbbz {\hbox{$\sans\textstyle Z\kern-0.4em Z$}} {\hbox{$\sans\scriptstyle Z\kern-0.3em Z$}} {\hbox{$\sans\scriptscriptstyle Z\kern-0.2em Z$}}}} % % Symbols % \def\const{{\rm const}\,} \def\sgn{{\rm sgn}} \def\slim{\mathop{\,\hbox{s-lim}\,}} \def\cartprod{\mathop{\lower2pt\hbox{{\twelvesans X}}}} \def\eqdef{{\buildrel \rm def \over =}} \def\v{\pmb{\big\vert}} \def\bbar#1{\setbox0=\hbox{$#1$}% \copy0 \raise.0433em\box0} \def\optbar#1{\vbox{\ialign{##\crcr\hfil${\scriptscriptstyle(}\mkern -1mu \vrule height 1.2pt width 3pt depth -.8pt {\scriptscriptstyle)}$\hfil\crcr \noalign{\kern-1pt\nointerlineskip}$\hfil\displaystyle{#1}\hfil$\crcr}}} % % Section heading % \font \tafontt = cmbx10 scaled\magstep2 \font \tbfontss = cmbx5 scaled\magstep1 \font \tbfonts = cmbx7 scaled\magstep1 \font \tbfontt = cmbx10 scaled\magstep1 \font \tasys = cmex10 scaled\magstep1 \font \tamsss = cmmib10 scaled 833 \font \tamss = cmmib10 \font \tams = cmmib10 scaled\magstep1 \font \tbst = cmsy10 scaled\magstep1 \font \tbsss = cmsy5 scaled\magstep1 \font \tbss = cmsy7 scaled\magstep1 \font \chfont = cmbx12 at 14pt \font \subchfont = cmbx12 at 12pt \def\titlea#1{\centerline{\tafontt #1 }\vskip.75truein} \def\titleb#1{\removelastskip\vskip.2truein\noindent{\tbfontt #1 }\vskip.4truein} \def\titlec#1{\removelastskip\vskip\baselineskip\noindent{\bf #1 } \vskip\baselineskip} \def\sec{\mathhexbox278} \def\section#1#2{% \vskip25pt plus 4pt minus4pt \bgroup \textfont0=\tbfontt \scriptfont0=\tbfonts \scriptscriptfont0=\tbfontss \textfont1=\tams \scriptfont1=\tamss \scriptscriptfont1=\tamsss \textfont2=\tbst \scriptfont2=\tbss \scriptscriptfont2=\tbsss \textfont3=\tasys \scriptfont3=\tenex \scriptscriptfont3=\tenex \baselineskip=16pt \lineskip=0pt \rightskip 0pt plus 6em \pretolerance=10000 \tbfontt \setbox0=\vbox{\vskip25pt \noindent \if!#1!\ignorespaces#2 \else\ignorespaces#1\unskip\enspace\ignorespaces#2\fi \vskip15pt}% \dimen0=\pagetotal \ifdim\dimen0<\pagegoal \dimen0=\ht0\advance\dimen0 by\dp0\advance\dimen0 by 4\normalbaselineskip \advance\dimen0 by\pagetotal \advance\dimen0 by-\pageshrink \ifdim\dimen0>\pagegoal\vfill\eject\fi\fi \noindent \if!#1!\ignorespaces#2 \else\ignorespaces#1\unskip\enspace\ignorespaces#2\fi \egroup \vskip12.5pt plus4pt minus4pt \ignorespaces} % % Theorem etc. % \def\newenvironment#1#2#3#4{\long\def#1##1##2{\removelastskip \vskip\baselineskip\noindent{#3#2\if!##1!.\else\unskip\ \ignorespaces ##1\unskip\fi\ }{#4\ignorespaces##2\vskip\baselineskip}}} % \newenvironment\lemma{Lemma}{\bf}{\it} \newenvironment\proposition{Proposition}{\bf}{\it} \newenvironment\theorem{Theorem}{\bf}{\it} \newenvironment\corollary{Corollary}{\bf}{\it} \newenvironment\example{Example}{\it}{\rm} \newenvironment\problem{Problem}{\bf}{\rm} \newenvironment\definition{Definition}{\bf}{\rm} \newenvironment\remark{Remark}{\it}{\rm} \def \thm#1{\vskip.25truein\noindent{\bf Theorem#1}\par} \def \lem#1{\vskip.25truein\noindent{\bf Lemma#1}\par} % % Proof, qed, equation % \long\def\proof#1{\removelastskip\vskip\baselineskip\noindent{\bf Proof\if!#1!\else\ \ignorespaces#1\fi.\ }\ \ \ignorespaces} \def\sq{\hbox{\rlap{$\sqcap$}$\sqcup$}} \def\qed{\ifmmode\sq\else{\unskip\nobreak\hfil \penalty50\hskip1em\null\nobreak\hfil\sq \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi} \def \prf{\vskip.25truein\noindent{\bf Proof\ }:\ \ \ } %\def\endproof{\hfil \break \line{ \hfill\vrule height .75em width .75em depth 0pt}} \def\endproof{\hfill\vrule height .6em width .6em depth 0pt\vskip.25in } \def\eqn#1{\eqno{({\rm #1})}} % ******** FIG.TEX ************************ % (c) R. A. Adams and R. B. Israel % Dept. of Mathematics % The University of British Columbia % Vancouver, B.C., Canada V6T 1Y4 % % November, 1990. Version Feldman 09/11/92 % ********************************************** % % being macros suitable for the inclusion of figures created by % "MG" into TeX documents. % %%%%%%% %%%%%%% SELECT YOUR SYSTEM NOW! \newcount\system %\global\system=1 % for textures %\global\system=2 % for msdos \global\system=3 % for unix(dvips)vincent % % In vertical mode use: % % \cfig{}{} % % to include a centred figure whose PostScript description and % associated TeX labels, both produced by "MG", are in the % files .PS and .LBL respectively. % The first line of the .LBL file is the MG version number. The % second and third lines are the width and height of the figure % (in points). The macro reads them there to provide % enough vertical space. % % example: % % \cfig{myplot}{Figure 3} % % Similarly, two side-by-side graphs can be inserted by % % \twofigs{}{}{}{} % % A graph on the right side of some text can be produced by % \rfig{}{}{} % % % Labels on figures will print in the default TeX families % (typically 10 point Computer Modern or whatever families % are installed in your version of TeX.) % If you want to change this use, for example, % % \def\setlabelsize{\ninepoint} % % within your TeX document. To reinstate the defaults use % % \def\setlabelsize{} % % macros \ds, \ss, \ts defined below can also be used to control % size of labels % % \ninepoint and \eightpoint are defined hereunder for CM fonts. % If you want any other size or family you will have to define % your own. % \font\ninerm=cmr9 \font\sixrm=cmr6 \font\eightrm=cmr8 \font\ninei=cmmi9 \font\sixi=cmmi6 \font\eighti=cmmi8 \font\ninesy=cmsy9 \font\sixsy=cmsy6 \font\eightsy=cmsy8 \font\ninebf=cmbx9 \font\sixbf=cmbx6 \font\eightbf=cmbx8 \font\nineit=cmti9 \font\eightit=cmti8 \font\ninesl=cmsl9 \font\eightsl=cmsl8 \font\ninett=cmtt9 \font\eighttt=cmtt8 \font\twelverm=cmr12 \font\twelvei=cmmi12 \font\twelvesy=cmsy10 at 12pt \font\twelvebf=cmbx12 \font\twelveit=cmti12 \font\twelvesl=cmsl12 \font\twelvett=cmtt12 % \def\twelvepoint{\def\rm{\fam0\twelverm}% switch to 12-point type \textfont0=\twelverm \scriptfont0=\eightrm \scriptscriptfont0=\sixrm \textfont1=\twelvei \scriptfont1=\eighti \scriptscriptfont1=\sixi \textfont2=\twelvesy \scriptfont2=\eightsy \scriptscriptfont2=\sixsy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\twelveit \def\it{\fam\itfam\twelveit}% \textfont\slfam=\twelvesl \def\sl{\fam\slfam\twelvesl}% \textfont\ttfam=\twelvett \def\tt{\fam\ttfam\twelvett}% \textfont\bffam=\twelvebf \scriptfont\bffam=\eightbf \scriptscriptfont\bffam=\sixbf \def\bf{\fam\bffam\twelvebf}% \normalbaselineskip=14pt \setbox\strutbox=\hbox{\vrule height10pt depth4pt width0pt}% \let\sc=\tenrm \let\big=\twelvebig \normalbaselines\rm} % \def\ninepoint{\def\rm{\fam0\ninerm}% switch to 9-point type \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\nineit \def\it{\fam\itfam\nineit}% \textfont\slfam=\ninesl \def\sl{\fam\slfam\ninesl}% \textfont\ttfam=\ninett \def\tt{\fam\ttfam\ninett}% \textfont\bffam=\ninebf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ninebf}% \normalbaselineskip=11pt \setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt}% \let\sc=\sevenrm \let\big=\ninebig \normalbaselines\rm} % \def\eightpoint{\def\rm{\fam0\eightrm}% switch to 8-point type \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm} % % The next five \TeX\ macros can be used in labels to simplify % some size choices. (If they interfere with other macros in % your system, comment them out.) % % % use the current tex font family for labels \def\setlabelsize{} % % by default, boxes will not be drawn around figures \newif\ifboxfigures \boxfiguresfalse % you can declare \boxfigurestrue in your TeX source if you want % the boxes drawn \newdimen\boxrulewidth \newdimen\boxborderwidth \boxrulewidth=1pt \boxborderwidth=2pt % boxes drawn around figures will have rules 1 point wide % surrounding a 2 point wide border by default. % You can change these measurements by redefining \boxrulewidth % and \boxborderwidth in your TeX source. % % macro for drawing boxes % \def\boxit#1{\vtop{\hrule height\boxrulewidth% \hbox{\vrule width\boxrulewidth\kern\boxborderwidth% \vbox{\kern\boxborderwidth#1\kern\boxborderwidth}\kern\boxborderwidth% \vrule width\boxrulewidth}% \hrule height\boxrulewidth}} % % Now for the definitions and main macro for figure inclusion. % \newcount\mgversion \newdimen\pswidth \newdimen\xleft \newdimen\psheight \newdimen\ytop \newdimen\ybot \newcount\justx \newcount\justy \global\justx=0 \global\justy=0 \newdimen\vpos \newtoks\label \newread\labelfile \newdimen\xcoord \newdimen\ycoord \newif\ifdoit \newbox\labox % \newcount\temp \def\readdim#1{\global\read\labelfile to \temp \global #1=\temp pt} % \def\figinsert#1{\par% #1=filename \openin\labelfile=#1.lbl \global\read\labelfile to\mgversion\message{#1} \readdim{\pswidth} \readdim{\psheight} \ifboxfigures\boxit{\fi\vbox to\psheight {\vfill %%% %%% NOTE: next lines may have to be changed for your DVIPS driver %%% \ifnum\system=1\special{postscript grestore newpath gsave}\fi%%textures \ifnum\system=2\special{postscript grestore newpath gsave}\fi%%msdos %\ifnum\system=3\special{" grestore newpath gsave}\fi %%unix:dvips \ifnum\system=1\special{postscriptfile #1.PS }\fi %%textures \ifnum\system=2\special{ps: plotfile #1.PS} \fi %%msdos \ifnum\system=3\special{psfile=#1.ps hscale=120 vscale=120} \fi %%unix:dvips %%% \vskip-\psheight \setlabelsize% \hbox to\pswidth {\hss}% \parindent=0pt\offinterlineskip \vpos=0 pt% \loop\readdim{\xcoord} \ifnum \temp < -999 \doitfalse\else\doittrue\fi \ifdoit \readdim{\ycoord} \global\read\labelfile to\justx \global\read\labelfile to\justy \global\read\labelfile to\label \global\setbox\labox=\hbox{\label\hskip-0.3em}% \advance\vpos by-\ycoord \vskip-\vpos \vpos=\ycoord \hbox to\pswidth{\hskip\xcoord % \hbox to 0pt{\ifnum\justx>0\hss\fi% \vbox to0pt{% \ifnum\justy<2\vss\fi% \copy\labox\kern0pt% \ifnum\justy>0\vss\fi}% \ifnum\justx<2\hss\fi}% \hss}% \repeat% \advance\vpos by-\psheight% \vskip-\vpos % }\ifboxfigures}\fi\closein\labelfile} % % % figcrop{} treats the first two labels as marking % the upper left and lower right corners of the figure. This is for % positioning purposes only. The figure may extend beyond the corners. % The corner markers are not printed. % % \def\figcrop#1{\par% #1=filename \openin\labelfile=#1.lbl \global\read\labelfile to\mgversion\message{#1} \global\read\labelfile to\temp%read overall dimensions \readdim{\ybot} \readdim{\xleft}% read upper left point \readdim{\ytop} \global\read\labelfile to\justx%ignore \global\read\labelfile to\justy%ignore \global\read\labelfile to\label%ignore \readdim{\pswidth}% read lower right point \global\advance\pswidth by -\xleft \readdim{\psheight} \global\advance\ybot by -\psheight \global\advance\psheight by -\ytop \global\read\labelfile to\justx%ignore \global\read\labelfile to\justy%ignore \global\read\labelfile to\label%ignore \ifboxfigures\boxit{\fi\vbox to\psheight{\vfill %%% %%% NOTE: next line may have to be changed for your DVIPS driver %%% \ifnum\system=1\special{postscript grestore newpath gsave}\fi %%textures \ifnum\system=2\special{postscript grestore newpath gsave}\fi %%msdos %\ifnum\system=3\special{" grestore newpath gsave}\fi %%unix:dvips \ifnum\system=1 \hbox to \pswidth{\kern-\xleft\special{postscriptfile #1.PS }\hfil}\fi%textures \ifnum\system=2 \hbox to \pswidth{\kern-\xleft\special{ps: plotfile #1.PS}\hfil}\fi%mdos \ifnum\system=3 \hbox to \pswidth{\kern-\xleft\special{psfile=#1.ps hscale=120 vscale=120}\hfil}\fi %unix:dvips %%% \vskip -\baselineskip \vskip -\ybot \vskip-\psheight \setlabelsize% \hbox to\pswidth {\hss}% \parindent=0pt\offinterlineskip \vpos=0 pt% \loop\readdim{\xcoord} \ifnum \temp < -999 \doitfalse\else\doittrue\fi \ifdoit \advance \xcoord by -\xleft \readdim{\ycoord} \advance \ycoord by -\ytop \global\read\labelfile to\justx \global\read\labelfile to\justy \global\read\labelfile to\label \global\setbox\labox=\hbox{\label\hskip-0.3em}% \advance\vpos by-\ycoord \vskip-\vpos \vpos=\ycoord \hbox to\pswidth{\hskip\xcoord % \hbox to 0pt{\ifnum\justx>0\hss\fi% \vbox to0pt{% \ifnum\justy<2\vss\fi% \copy\labox\kern0pt% \ifnum\justy>0\vss\fi}% \ifnum\justx<2\hss\fi}% \hss}% \repeat% \advance\vpos by-\psheight% \vskip-\vpos % }\ifboxfigures}\fi\closein\labelfile} % % % \figplace{}{}{} % moves to the right by and down by % and then applies \figcrop % \def\figplace#1#2#3{ \openin\labelfile=#1.lbl \ifeof\labelfile\immediate\write16{Can't find #1.lbl; I quit!}\end\fi \closein\labelfile \vbox{\vskip #3 \hbox{\hskip#2 \figcrop{#1}}}} % % % define a font for figure captions in the following macros % one of the next two lines should be commented out %\font\figfont=psmhlvb at10pt % e.g. Arbortext's mapped helvetica bold \def\figfont{\bf} % boldface from your default family. % % You may wish to modify the following high-level figure inclusion macros % or create new ones of your own. % % Macro for centred figure, captioned below. % \def\cfig#1#2{\par\smallskip \openin\labelfile=#1.lbl \ifeof\labelfile\immediate\write16{Can't find #1.lbl; I quit!}\end\fi \closein\labelfile \vbox{\centerline{\figinsert{#1}}\smallskip \centerline{\figfont#2}}\smallskip} % % Macro for text on left side of page and figure on right side. % Figure is captioned below. % \newdimen\textsize \long\def\rfig#1#2#3{\par\smallskip \openin\labelfile=#2.lbl \ifeof\labelfile\immediate\write16{Can't find #2.lbl; I quit!}\end\fi \global\read\labelfile to\mgversion \readdim{\pswidth} \closein\labelfile \textsize=\hsize \advance\textsize by-\pswidth \advance\textsize by-1pc \line{\vtop{\hsize=\textsize{#1}\vfil} \hfill\vtop{\hsize=\pswidth \vskip0pt \figinsert{#2}\smallskip \line{\hfil\figfont#3\hfil}\smallskip}}} % % Macro for two side-by-side figures, captioned below. % \newdimen\rightfigwidth \newdimen\leftfigwidth \def\twofigs#1#2#3#4{\par\smallskip \openin\labelfile=#1.lbl \ifeof\labelfile\immediate\write16{Can't find #1.lbl; I quit!}\end\fi \global\read\labelfile to\mgversion \readdim{\leftfigwidth} \closein\labelfile \openin\labelfile=#3.lbl \ifeof\labelfile\immediate\write16{Can't find #3.lbl; I quit!}\end\fi \global\read\labelfile to\mgversion \readdim{\rightfigwidth} \closein\labelfile \vbox{\line{\hfill\figinsert{#1}\hfill\hfill\figinsert{#3}\hfill} \smallskip \line{\hfill\hbox to\leftfigwidth {\hfil\figfont#2\hfil} \hfill\hfill\hbox to\rightfigwidth {\hfil\figfont#4\hfil}\hfill}}\smallskip} % % %\input ../ftmacros \input ftmacros %\input ../myfig \input myfig \def\<{\left <} \def\>{\right >} \def\norm{|\!|\!|} \multiply\baselineskip by 10\divide \baselineskip by15 \null\vskip2.5truecm % % (1) Put title in next two lines % %\centerline{The Pauli Exclusion Principle in Many Body Theory} %\centerline{A Convergent Linked Cluster Expansion for Many Fermion Models} %\centerline{A Convergent Cluster-Like Expansion for Many Fermion Green's %Functions} \centerline{\subchfont An Infinite Volume Expansion for Many Fermion Green's Functions} %\centerline{} \vskip20pt \centerline{Joel Feldman\footnote{$^{*}$}{ Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Schweizerischer Nationalfonds zur Forderung der wissenschaftlichen Forschung}$^{\dagger}$} \centerline{Department of Mathematics} \centerline{University of British Columbia} \centerline{Vancouver, B.C. } \centerline{CANADA\ \ V6T 1Z2}\vskip.5truecm \centerline{Jacques Magnen$^{\dagger}$, Vincent Rivasseau\footnote{$^{\dagger}$} { Research supported in part by the Forschungsinstitut f\"ur Mathematik, Z\"urich}} \centerline{Centre de Physique Th\'eorique} \centerline{Ecole Polytechnique} \centerline{F-91128 Palaiseau Cedex} \centerline{FRANCE}\vskip.5truecm \centerline{Eugene Trubowitz} \centerline{Mathematik} \centerline{ETH-Zentrum} \centerline{CH-8092 Z$\ddot{\rm u}$rich} \centerline{SWITZERLAND}\vskip2truecm % % (2) Dedication (optional) % %\vskip1truecm \noindent \underbar{Abstract}\ \ \ % % (3) Put abstract below here We prove the convergence of a simplified cluster/Mayer expansion at one energy scale for three space-time dimensional many Fermion systems. The bounds are uniform in the scale. We iterate them to show that the sum of all diagrams that contain no two or four-legged subdiagrams converges. Our results are suited to a multiscale construction of the full system. \vfill \eject \multiply\baselineskip by 15\divide \baselineskip by10 \noindent{\subchfont \sec I Introduction} \vskip.25truein In this paper we consider many Fermion systems formally characterized by the effective potential $$ {\cal G}(\psi^e,\bar \psi^e) =\log\,{1\over Z}\int e^{-\lambda {\cal V}(\psi+\psi^e,\bar\psi+\bar\psi^e)} d\mu_{C}(\psi,\bar\psi) $$ for the external fields $\psi^e,\bar\psi^e$. Here, $d\mu_C(\psi,\bar\psi)$ is the fermionic Gaussian measure in the Grassmann variables $\{\psi(\xi),\bar\psi(\xi)\ |\ \xi\in \bbbr\times\bbbr^d\times\{\uparrow,\downarrow\}\}$ with propagator $$ C(\xi,\bar\xi)=\delta_{\sigma,\bar\sigma}\int {d^{d+1}p\over(2\pi)^{d+1}} {e^{i_-} \over ip_0-e(\p)} $$ where $$ _-=\p\cdot\x-p_0\tau $$ and $$ e(\p)={\p^2\over2m}-\mu. $$ The variable $\xi=(\tau,\x,\sigma)$ consists of time, space and spin components and the $(d+1)$-momentum $p=(p_0,\p)$. The interaction is given by $$ {\cal V}={1\over 2} \int \prod_{i=1}^4 d\xi_i\ \ V\!\left(\xi_1,\xi_2,\xi_3,\xi_4\right) \bar\psi(\xi_1)\bar\psi(\xi_2)\psi(\xi_4)\psi(\xi_3) $$ where the kernel $V\!\left(\xi_1,\xi_2,\xi_3,\xi_4\right)$ is translation invariant with $V\!\left(0,\xi_2,\xi_3,\xi_4\right)$ integrable and $$ \int d\xi=\sum_{\sigma\in\{\uparrow,\downarrow\}} \int_\bbbr d\tau\int_{\bbbr^d} d\x. $$ The partition function $$ Z=\int e^{-\lambda {\cal V}(\psi,\bar\psi)}d\mu_{C}(\psi,\bar\psi) $$ so that ${\cal G}(0,0)=0.$ The Euclidean Green's functions $$ G_p(\xi_1,\bar\xi_1,\dots,\xi_p,\bar\xi_p)=\prod_{i=1}^p {\delta^2\hfill\over \delta\psi^e(\xi_i)\delta\bar\psi^e(\bar\xi_i)}{\cal G} $$ generated by the effective potential are the connected Green's functions amputated by the free propagator. By definition, ${\cal G}$ exists when the norm $$ \norm G_p\norm=\max_{j}\ \sup_{\xi_j}\ \int\prod_{i\ne j}d\xi_i\ |G_p(\xi_1,\cdots,\xi_{2p})| $$ of each of its moments, $G_p,\ p\ge 1$, is finite. Intuitively, $\norm G_p\norm$ is the supremum in momentum space of $G_p$. In fact, the supremum in momentum space was used as the standard norm on vertices in [FT2]. Our long term goal is to give a rigorous proof that the standard model for an interacting system of electrons and phonons has a superconducting ground state at sufficiently low temperature. Perturbation theory and, in particular, the renormalization of the two point function was controlled in [FT1]. (See [BG] for related results.) A renormalization group flow for the four point function was defined and analyzed in [FT2]. Two additional ingredients are required to complete this program. First an infinite volume expansion that combines power counting at fixed energy with the exclusion principle and second, control of the Goldstone boson. This paper provides the first ingredient, in three space-time dimensions. With the exception of Lemma 3 all components of the expansion apply in all dimensions. We restrict to $d=2$ only when it is necessary to do so. As in [FT1,2], the model is sliced into energy regimes by decomposing momentum space into shells around the Fermi surface. The $j^{\rm th}$ slice has covariance $$ C^{(j)}(\xi,\bar\xi)=\delta_{\sigma,\bar\sigma}\int {d^{d+1}p\over(2\pi)^{d+1}} {e^{i_-} \over ip_0-e(\p)}f_j(p), $$ where $$ f_j(p)=f\left(M^{-2j}\left(p_0^2+e(\p)^2\right)\right) $$ effectively forces $|ip_0-e(\p)|\sim M^j$. The function $f\in C^\infty_0([1,M^4])$. The parameter $M$ is strictly bigger than one so that the scales near the Fermi surface have $j$ near $-\infty$. The model is defined in finite volume and at fixed scale by the following lemma. However, the radius of convergence depends on volume and scale in a patently unsatisfactory way.$$ \int d\xi=\sum_{\sigma\in\{\uparrow,\downarrow\}} \int_\bbbr d\tau\int_{\bbbr^d} d\x. $$\vfill\eject \lem{\ 1}{\it $$ {\cal G}_\Lambda^{(j)}(\psi^e,\bar\psi^e) =\log\,{1\over Z_\Lambda^{(j)}}\int e^{-\lambda {\cal V}_\Lambda(\psi+\psi^e,\bar\psi+\bar\psi^e)} d\mu_{C^{(j)}}(\psi,\bar\psi) $$ where $$ {\cal V}_\Lambda={1\over 2} \int_{\Lambda^4} \prod_{i=1}^4 d\xi_i\ \ V\!\left(\xi_1,\xi_2,\xi_3,\xi_4\right) \bar\psi(\xi_1)\bar\psi(\xi_2)\psi(\xi_4)\psi(\xi_3) $$ and $$ Z_\Lambda^{(j)}=\int e^{-\lambda {\cal V}_\Lambda(\psi,\bar\psi)} d\mu_{C^{(j)}}(\psi,\bar\psi) $$ is analytic in $\lambda$ in a neighborhood of the origin that includes at least the disk of radius $\const\left(M^{2j}|\Lambda|\right)^{-1}.$} \prf Expand the exponential $$\eqalign{ Z_\Lambda^{(j)}&=\int e^{-\lambda {\cal V}_\Lambda(\psi,\bar\psi)} d\mu_{C^{(j)}}(\psi,\bar\psi)\cr &=\sum_{n=0}^\infty {(-\lambda)^n\over n!}\int {\cal V}_\Lambda(\psi,\bar\psi)^n d\mu_{C^{(j)}}(\psi,\bar\psi)\cr &=\sum_{n=0}^\infty {(-\lambda)^n\over 2^n n!}\int_{\Lambda^{4n}} \left(\prod_{j=1}^n\prod_{i=1}^2 d\xi_{j,i}d\bar\xi_{j,i}\right) \left(\prod_{j=1}^nV(\bar\xi_{j,1},\bar\xi_{j,2},\xi_{j,1},\xi_{j,2}) \right)\det\left[C^{(j)}(\xi_k,\bar\xi_\ell)\right] }$$ where the indices $k$ and $\ell$ run over $\{(j,i)\,|\,1\le j\le n\,,\,1\le i\le 2\}$. The $(k,\ell)$ matrix element $$\eqalign{ C^{(j)}(\xi_k,\bar\xi_\ell) &=\delta_{\sigma_k,\bar\sigma_\ell} \int {d^{d+1}p\over(2\pi)^{d+1}} {e^{i_-} \over ip_0-e(\p)}f_j(p)\cr &=\left }$$ where $$ A_k(p,\alpha):= \delta_{\alpha,\sigma_k}{e^{i_-} \over\left[ip_0-e(\p)\right]^{1\over 2}} f_j(p)^{1/2} $$ and $$ \bar A_\ell(p,\bar \alpha) := \delta_{\bar\alpha,\bar\sigma_\ell} \left\{{e^{-i_-}\over {\left[ip_0-e(\p)\right]^{1\over 2}}}\right\}^* f_j(p)^{1/2} $$ are in $L^2\left(\bbbr^{d+1}\times\{\uparrow,\downarrow\}\right).$ Here, any branch of the square root will do, since\ $ip_0-e(\p)$\ does not vanish on the domain of integration. Consequently, by Gram's inequality, $$\eqalign{ \left|\det\left[C^{(j)}(\xi_k,\bar\xi_\ell)\right]\right| &\le\prod_{k} \|A_k\|_2\ \|\bar A_k\|_2\cr &\le {\rm const}^n\ M^{2jn} }$$ since $$\eqalign{ \|\bar A_k\|_2^2=\|A_k\|_2^2 &=\int{d^{d+1}\p\over(2\pi)^{d+1}}\left|ip_0-e(\p)\right|^{-1}f_j(p)\cr &\le {\rm const\ }M^{j}. }$$ Substituting, $$\eqalign{ \left|Z_\Lambda^{(j)}\right| &\le\sum_{n=0}^\infty {|\lambda|^n\over 2^n n!}\int_{\Lambda^{4n}} \left(\prod_{j=1}^n\prod_{i=1}^2 d\xi_{j,i}d\bar\xi_{j,i}\right) \left|\prod_{j=1}^nV(\bar\xi_{j,1},\bar\xi_{j,2},\xi_{j,1},\xi_{j,2}) \right| \left|\det\left[C^{(j)}(\xi_k,\bar\xi_\ell)\right]\right|\cr &\le\sum_{n=0}^\infty {|\const\,\lambda|^n\over n!}\int_{\Lambda^{4n}} \left(\prod_{j=1}^n\prod_{i=1}^2 d\xi_{j,i}d\bar\xi_{j,i}\right) \left|\prod_{j=1}^nV(\bar\xi_{j,1},\bar\xi_{j,2},\xi_{j,1},\xi_{j,2}) \right| M^{2jn}\cr &\le\sum_{n=0}^\infty {|\const\,\lambda|^n\over n!} \left(|\Lambda|M^{2j}\norm V\norm\right)^n\cr &=\exp\left(\const\,|\lambda||\Lambda|M^{2j} \norm V\norm\right). }$$ It follows that the partition function is an entire function of $\lambda$. Each Taylor coefficient of the expansion of the numerator $$ \int e^{-\lambda {\cal V}_\Lambda(\psi+\psi^e,\bar\psi+\bar\psi^e)} d\mu_{C^{(j)}}(\psi,\bar\psi) $$ in powers of the external fields is of the form $$ \int P(\psi,\bar\psi)e^{-\lambda {\cal V}_\Lambda(\psi,\bar\psi)} d\mu_{C^{(j)}}(\psi,\bar\psi) $$ for some polynomial $P$. It is estimated in exactly the same way and is also an entire function of $\lambda$. The Green's functions of ${\cal G}$ are finite sums of finite products of such numerators divided by a power of $Z_\Lambda^{(j)}$. Thus they are meromorphic with poles at the zeroes of the partition function. Therefore the radius of convergence is at least the absolute value of the smallest root of $Z_\Lambda^{(j)}$. When $\lambda=0$, the partition function is one. As above $$ \left|{d\hfill\over d\lambda}Z_\Lambda^{(j)}\right| \le \const\,|\Lambda|M^{2j}\norm V\norm \exp\left(\const\,|\lambda||\Lambda|M^{2j} \norm V\norm\right) $$ so that $$ \left|Z_\Lambda^{(j)}(\lambda)-1\right|\le \const\,|\lambda||\Lambda|M^{2j}\norm V\norm\exp\left(\const\,|\lambda| |\Lambda|M^{2j}\norm V\norm\right). $$ Finally, for $|\lambda||\Lambda|M^{2j}\norm V\norm\le\const$, $$ \const\,|\lambda||\Lambda|M^{2j}\norm V\norm\, \exp\left(\const\,|\lambda||\Lambda|M^{2j} \norm V\norm\right)\ <\ 1\ . $$ \endproof Let $G^{(j,\Lambda)}_p$ be the $p$-point Green's function generated by ${\cal G}^{(j)}_\Lambda$. By the Lemma above the Taylor series $$ G^{(j,\Lambda)}_p=\sum_{n=0}^\infty g_n(p,j,\Lambda)\lambda^n $$ has a strictly positive, though possibly $j$ and $\Lambda$ dependent, radius of convergence. The main result of this paper is \thm{\ 1}{\it Let $d=2$. There exists a $\ \const\ $, independent of $j$ and $\Lambda$, such that $$ \norm g_n(p,j,\Lambda)\norm \le\const^{n+p}M^{(2-5p/2)j}\norm V\norm^n $$ where $$ \norm g_n(p,j,\Lambda)\norm= \max_{k}\ \sup_{\xi_k}\ \int\prod_{i\ne k}d\xi_i\ |g_n(p,j,\Lambda)(\xi_1,\cdots,\xi_{2p})|. $$ Furthermore the limits $$ g_n(p,j)=\lim_{\Lambda\rightarrow \bbbr^3}g_n(p,j,\Lambda) $$ exist and the infinite volume Green's functions at scale $j$ $$ G^{(j)}_p=\sum_{n=0}^\infty g_n(p,j)\lambda^n $$ are analytic in $|\lambda|_-} \over ip_0-e(\p)}f_j(p)\,\eta_m(\p) $$ and of the fields $$ \psi=\sum_{m=1}^{M^{-(d-1)j}}\psi^{(m)},\hskip.25truein \bar\psi=\sum_{m=1}^{M^{-(d-1)j}}\bar\psi^{(m)}. $$ The standard power counting bound for an individual graph is still easy to prove when there are sectors. First, one selects a spanning tree for the graph. To each line not in the tree there is a corresponding momentum loop, obtained by joining its ends through a path in the tree. This construction produces a complete set of independent loops. Ignoring unimportant constants, each propagator is bounded by its supremum $M^{-j}$. The volume of integration for each loop is now $M^{(d+1)j}$. A priori, there is one sector sum with $M^{-(d-1)j}$ terms for each line. But, by conservation of momentum, there is only one sector sum per loop. Thus, if there are $n$ vertices and $E$ external lines, the supremum in momentum space of the graph is bounded by $$\eqalign{ \prod_{\rm lines}M^{-j}\prod_{\rm loops}M^{(d+1)j}M^{-(d-1)j} &=M^{-j(4n-E)/2}M^{2j\left[(4n-E)/2-(n-1)\right]}\cr &=M^{{1\over 2}(4-E)j}. }$$ In the course of a non-perturbative construction, estimates cannot be made graph by graph because there are too many of them. Rather, the perturbation series must be blocked and the blocks estimated as units. The blocks are estimated using the exclusion principle to implement strong cancellations between the roughly $n!^2$ graphs of order $n$. However, once the series is blocked, momentum loops can't be defined and the argument leading to the estimate above cannot be made. Conservation of momentum has to be implemented at each vertex, rather than through loops. Even though the volume cutoff $\Lambda$ breaks exact conservation of momentum, many of the $M^{-2\ell(d-1)j}$ terms in the sector sums for a general $2\ell$-legged vertex must be zero. \lem{\ 2}{\it Fix $\ m\in {\bf Z}^{d+1}\ $ and $\ \ell\ \ge\ 2\ $. Then, the number of $2\ell$-tuples $$ \left\{S_1,\ \cdots\,S_{2\ell}\right\} $$ of sectors for which there exist $\ \k_i\in \bbbr^d,\ i=1,\ \cdots,\ 2\ell\ $ satisfying $$ \k_i^\prime\in S_i,\ \ |\k_i-\k_i^\prime|\le {\rm const}\ M^j \ , \ \ \ i=1,\ \cdots,\ 2\ell\ $$ and $$ |\k_1+\ \cdots\ +\k_{2\ell}|\le{\rm const}\left(1+|m|\right)M^j $$ is bounded by $$ \const^\ell(1+|m|)^dM^{-(2\ell-1)(d-1)j}M^j \left\{1+|j|\delta_{d,2}\delta_{\ell,2}\right\}. $$ In particular, for a four legged vertex, the number of $4$-tuples is at most $$ \const(1+|n|)^dM^{(-3d+4)j}\left\{1+|j|\delta_{d,2}\right\}. $$ Here, $\ \k^\prime = {\k\over |\k|}\ $ denotes the projection of $\ \k\ $ onto the Fermi surface.} \vskip.25truein Lemma 2 is proved in the next section. Specializing to two space dimensions, the number of active sector 4-tuples at a vertex is of order $\ |j|M^{-2j}\ $. Four planar vectors of equal length whose sum is zero form a parallelogram. The factor $\ M^{-2j}\ $ is natural since a parallelogram is determined by two of its sides. The logarithmic factor $\ |j|\ $ is not an artifact of our bounds. It arises from the degenerate situation in which all four vectors are roughly collinear. One of the main technical difficulties of the paper is to overcome the logarithm. Note that in three dimensions, the parallelogram is hinged and the logarithm $|j|$ jumps to the power $M^{-j/2}$. To circumvent the logarithm in two dimensions we divide the Fermi circle into sectors of length $M^{j/2}$ rather than $M^j$ through a smooth partition of unity $$ 1=\sum_{\ell=1}^{M^{-j/2}}\zeta_\ell(\p),\hskip .25truein \zeta_\ell(\p)=\zeta_\ell\left({\p\over|\p|}k_F\right) $$ where $\zeta_\ell$ is supported on the $\ell^{th}$ sector, $\Sigma_\ell$, and its 2 neighbors. We denote by $r_\ell$ the center of $\Sigma_\ell$. The new sectors are long and skinny since they are still $M^j$ thick. We shall show in a moment that the sector propagator $$ C^{(j,\ell)}(\xi,\bar \xi)=\delta_{\sigma,\bar\sigma} \int {d^{3}p\over(2\pi)^{d+1}} {e^{i_-} \over ip_0-e(\p)}f_j(p)\,\zeta_\ell(\p) $$ decays anisotropically. To accommodate the anisotropy, we will introduce a different lattice of dual boxes for each sector. The boxes will be short in the direction perpendicular to $r_\ell$ and long in the direction of $r_\ell$. The reason that approximately collinear configurations generate a logarithm for sectors of length $M^j$, but not for sectors of length $M^{j/2}$, may be seen in the proof (\sec II) of \lem{\ 3} {\it Let $d=2$ and divide the Fermi circle into sectors $\Sigma_\ell,\ \ell=1,\cdots,M^{-j/2}$ of width $M^{j/2}$. Fix $\ m\in {\bf Z}^{3}$ and any sector $\Sigma_{\ell_1}$. The number of sector quadruples $\ \{\Sigma_{\ell_1},\Sigma_{\ell_2},\Sigma_{\ell_3},\Sigma_{\ell_4}\}$ for which there exist $\ \k_i\in \bbbr^2,\ i=1,\ \cdots,\ 4\ $ satisfying $$ \k_i^\prime\in \Sigma_{\ell_i}, \ \ |\k_i-\k_i^\prime|\le{\rm const}\, M^j \ , \ \ \ i=1,\ \cdots,4\ $$ and $$ |\k_1+\ \cdots\ +\k_{4}|\le{\rm const}\,\left(1+|m|\right)M^j $$ is bounded by $$ \const\,(1+|m|^{2})M^{-j/2}. $$} The anisotropic decay of the covariance is given in \lem{\ 4} \noindent(a) Let $$ \rho^{(j,\ell)}(\xi,\bar\xi)\ \eqdef\ 1+M^{j}|\xi_{0} -\bar\xi_{0}| + M^{j}|\xi_{\|} -\bar\xi_{\|}| + M^{j/2} |\xi_{\bot} -\bar\xi_{\bot}| $$ {\it where $\xi_{\|}$ is the component of $\xi$ parallel to $r_{\ell}$ and $\xi_{\bot}$ is the component orthogonal to $r_{\ell}$. The same notation is used for $\bar \xi$. Then, for any $\gamma>0$,} $$ \vert C^{(j,\ell)} (\xi, \bar \xi) \vert \le \const\ M^{3j/2} \rho^{(j,\ell)}(\xi,\bar\xi)^{-\gamma}. $$ \noindent(b) $$ \left\vert D_0^{n_0}D_\|^{n_1}D_\bot^{n_2} \left(e^{-i<\r_\ell,\xi-\bar\xi>_-}C^{(j,\ell)} (\xi, \bar \xi)\right) \right\vert \le \const\ M^{({3\over 2}+n_0+n_1+{n_2\over 2})j} \rho^{(j,\ell)}(\xi,\bar\xi)^{-\gamma}. $$ {\it Here $D_0={\partial\hfill\over \partial\xi_0} , D_\|={\r_\ell\over|\r_\ell|}\cdot\nabla_\xi$ and $D_\bot=\hat\pi_\ell\cdot\nabla_\xi$ where $\hat\pi_\ell$ is any unit vector perpendicular to $\r_\ell$.} \prf (a) A pointwise bound on $$ C^{(j,\ell)}(\xi,\bar \xi)=\delta_{\sigma,\bar\sigma} \int {d^{3}p\over(2\pi)^{3}} {e^{i_-} \over ip_0-e(\p)}f_j(p)\,\zeta_\ell(\p) $$ is obtained by observing that the integrand is bounded by $M^{-j}$ and that the volume of integration is $M^{{5\over 2}j}$. Multiplying $C^{(j,\ell)}$ by $\rho^{(j,\ell)}(\xi,\bar\xi)^{\gamma}$ is converted, by integration by parts, into $p$-derivatives acting on $$ {1\over ip_0-e(\p)}f_j(p)\,\zeta_\ell(\p)\ \ . $$ A derivative acting on $\zeta^{(\ell)}$ produces an $M^{-{1\over 2}j}$ while one acting on $f$ produces an $M^{-j}$. However, the directional derivative $\hat \pi_\ell\cdot\nabla_\p$ acting on $f_j(p)=f\left(M^{-2j}\left(p_0^2+e(\p)^2\right)\right)$ produces $$ M^{-2j}2e(\p){1\over 2m}2\p\cdot\hat \pi_\ell $$ which is bounded by $\const\ M^{-2j}M^jM^{j/2}|\hat \pi_\ell|=\const\ M^{-j/2}$ on the support of the integrand. \noindent(b) Each derivative, with respect to $\xi$ or $\bar\xi$, of $$ e^{-i<\r_\ell,\xi-\bar\xi>_-}C^{(j,\ell)}(\xi,\bar \xi) =\delta_{\sigma,\bar\sigma} \int {d^{3}p\over(2\pi)^{3}} {e^{i_-} \over ip_0-e(\p)}f_j(p)\,\zeta_\ell(\p) $$ brings down a factor of $p-\r_\ell$. On the domain of integration, the components of this vector\hfill\break \hfil\figplace{lce1}{1.25in}{0in}\hfil\break in the time, $\r_\ell$ and $\hat\pi_\ell$ directions are bounded by $M^j,M^j$ and $M^{j/2}$ respectively. This is obvious except for the $\r_\ell$ component. For it we have $$\eqalign{ \left|(p-\r_\ell)_\|\right|&\le k_F-\big(k_F-O(M^{j})\big)\cos \theta\cr &\le k_F-\big(k_F-O(M^{j})\big)\big(1-O(M^{j/2})^2\big)\cr &\le O(M^j) }$$ since $\theta\le O(M^{j/2})$. \endproof We now give a rough description of the expansion. In perturbation theory, $g_n(p,j,\Lambda)$ is written as ${1\over n!}$ times the sum of approximately $n!^2$ connected Feynman diagrams of order $n$. Every connected diagram is spanned by a connected tree. By Cayley's Theorem there are $n^{n-2}$ labeled trees of order $n$. The number of trees can be compensated for by the ${1\over n!}$. The expansion starts with trees and inductively builds graphs from them by joining the remaining $4n-2p-2(n-1)$ legs. However, for each fixed tree, these legs may be joined in $\sim n!$ ways to form connected graphs. As in the standard cluster expansion, part of this $n!$ will be cancelled using the decay of the propagator. We now review this procedure. Pretend that the propagator decays like $(1+M^j|\xi-\bar\xi|)^{-\gamma}$. Introduce a lattice, $K_j$, of cubes of side $M^{-j}$ that paves space-time. If we are in the midst of the inductive process, some, but not all, legs of the tree have been joined to form lines. At the next step we select any leg, say in $\Delta\in K_j$, that has not yet been contracted and sum over all possible ways of connecting it to another uncontracted leg $\tau$. Block the sum $$\eqalign{ \sum_{{\rm target\ legs\ }\tau} = \sum_{\Delta^\prime\in K_j} \sum_{{\rm target\ legs}\atop {\rm in\ }\Delta^\prime}\ \ . }$$ Ultimately we must estimate such blocked sums. This is done by \def\eqone{I.1} $$\eqalign{ \left|\sum_{\Delta^\prime\in K_j} \sum_{{\rm target\ legs}\atop {\rm in\ } \Delta^\prime} F(\tau)\right| &\le\left[\sum_{\Delta^\prime\in K_j} \big(1+M^j{\rm dist}(\Delta,\Delta^\prime)\big)^{-d-2}\right]\cr &\hskip.5truein\times\ \sup_{\Delta^\prime\in K_j} \left|\sum_{{\rm target\ legs}\atop {\rm in\ }\Delta^\prime} \big(1+M^j{\rm dist}(\Delta,\Delta^\prime)\big)^{d+2}F(\tau)\right|. }\eqn{\eqone}$$ The factor $\big(1+M^j{\rm dist}(\Delta,\Delta^\prime)\big)^{d+2}$ is balanced by the decay of the propagator. Now the number of terms in the sum $\sum_{{\rm target\ legs}\atop {\rm in\ }\Delta^\prime}$ is the number of target legs in $\Delta^\prime$ rather than the total number of target legs. When applied to all contractions, this technique converts ``global $n!$'s'' to ``local $n!$'s''. For a local $n!$ to be large, there must be many fields of the same momentum slice in a single dual cube. This is prevented by the Pauli exclusion principle. To see how the combinatorial analysis suggested in the last two paragraphs may be carried out in a manner suitable for the proof of Theorem 1, we develop a complete expansion for a toy model with a simplified propagator and a local interaction. All the complications due to the presence of the Fermi surface are removed, by hand. The proof of Theorem 1 for the true propagator and full interaction is presented in \sec III. Let the dimension $d$ be arbitrary and let $C^{(j)}$ be any propagator obeying the bounds \def\eqtwo{I.2} $$ \left\vert \nabla^n C^{(j)} (\xi, \bar \xi)\right\vert \le \const\ M^{({d+1\over 2}+n)j} \left(1+M^j(|\xi-\bar\xi|\right)^{-\gamma}. \eqn{\eqtwo}$$ for some large $\gamma$ and all $n\le N$, which are typical for strictly renormalizable field theories. For example the propagator for the Gross-Neveu model [FMRS], [GK] in two space-time dimensions is of this type. The interaction of the toy model is $$ {\cal V}_\Lambda={1\over 2}\sum_{\{\uparrow,\downarrow\}} \int_{\Lambda} d\tau d\x\ \ \bar\psi(\tau,\x,\sigma)\bar\psi(\tau,\x,\sigma^\prime) \psi(\tau,\x,\sigma^\prime)\psi(\tau,\x,\sigma). $$ We now expand $$ \<{\cal A}\>={1\over Z_\Lambda}\int{\cal A} e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)} d\mu_{C^{(j)}}(\psi,\bar\psi) $$ where $$ {\cal A}=\int \prod_{j=1}^{2p} d\xi_j\ A(\xi_1,\cdots,\xi_{2p}) \prod_{j=1}^{2p}\optbar\psi(\xi_j) $$ is an arbitrary monomial. Just as in the proof of Lemma 1, Gram's inequality (or, in the unlikely event that $C^{(j)}$ does not factorize suitably, Hadamard's inequality) implies that both the numerator and denominator of $\<{\cal A}\>$ are entire functions of $\lambda$. The denominator $Z$ can have many $j$ and $\Lambda$ dependent zeros. But when $\lambda=0$, $Z=1$ so that $\<{\cal A}\>$ is meromorphic on all of $\bbbc$ and analytic at zero. We shall develop a formal power series expansion for $\<{\cal A}\>$ with the property that for every $N$ $$ \<{\cal A}\>=\sum_{n=0}^N a_n(j,\Lambda)\lambda^n+ O\left(\lambda^{N+1}\right). $$ A priori we do not claim that the tail $O\left(\lambda^{N+1}\right)$ is uniform in $j$ or $\Lambda$. Nevertheless, since $\<{\cal A}\>$ is analytic at zero we must have \def\eqthr{I.3} $$ \<{\cal A}\>=\sum_{n=0}^\infty a_n(j,\Lambda)\lambda^n \eqn{\eqthr} $$ in some, possibly $j$ and $\Lambda$ dependent, neighborhood of zero. Observe that $a_n(j,\Lambda)$ must be the sum of all connected Feynman diagrams of order $n$ with $2p$ external legs, since (\eqthr) is an asymptotic expansion. We shall also show that there exists a $\ \const\ $, independent of $j$ and $\Lambda$, such that $$ |a_n(j,\Lambda)|\le\const^{n+p}M^{(d+1)pj/2}\|A\|_1. $$ As a consequence, equation (\eqthr) applies for all $|\lambda| =\sum_{n=0}^\infty a_n(j)\lambda^n $$ for all $|\lambda|=\sum_{T\in {\cal T}_s} {\int{\cal A}(T,s,\Lambda) e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}} \over \int e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}} }. \eqn{\eqfou} $$ Each ${\cal A}(T,s,\Lambda)$ is a monomial in the fields $\psi,\bar\psi$ of degree $2\pi(T,s)$. In the event that $\pi(T,s)=0$ the ratio of integrals simplifies to the number ${\cal A}(T,s,\Lambda)$. Thus \def\eqfiv{I.5} $$ \<{\cal A}\>=\sum_{T\in {\cal T}_s\atop \pi(T,s)=0}{\cal A}(T,s,\Lambda) +\sum_{T\in {\cal T}_s\atop \pi(T,s)\ne 0} {\int{\cal A}(T,s,\Lambda) e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}} \over \int e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}} }\ . \eqn{\eqfiv} $$ The numbers in the first sum are not touched in subsequent steps of the expansion. We shall show that each ratio in the second sum is $O\left(\lambda^s\right)$. During step $s+1$, two operations are performed. The first is integration by parts. We apply the integration by parts formulae \def\eqsix{I.6} $$\eqalign{ \int \psi(\xi) F(\psi,\bar\psi)d\mu_{C^{(j)}}(\psi,\bar\psi) &=\int d\bar\xi \ C^{(j)}(\xi,\bar\xi)\int {\delta\hfill\over\delta \bar\psi(\bar\xi)} F(\psi,\bar\psi)d\mu_{C^{(j)}}(\psi,\bar\psi)\cr \int \bar\psi(\bar\xi) F(\psi,\bar\psi)d\mu_{C^{(j)}}(\psi,\bar\psi) &=-\int d\xi \ C^{(j)}(\xi,\bar\xi)\int {\delta\hfill\over\delta \psi(\xi)} F(\psi,\bar\psi)d\mu_{C^{(j)}}(\psi,\bar\psi) }\eqn{\eqsix}$$ to each of the fields appearing in the monomial ${\cal A}(T,s,\Lambda)$. The order is chosen arbitrarily. Of course if a derivative ${\dst\delta\hfill\over\dst\delta \optbar\psi(\optbar\xi)}$ acts on a field in ${\cal A}(T,s,\Lambda)$, the field disappears. Thus we need not apply the integration by parts formulae to eliminate it. If no derivative acts on the exponential, all fields downstairs disappear, producing a term of degree zero. (We use the expression ``downstairs'' to identify fields multiplying the exponential.) Each time a derivative ${\dst\delta\hfill\over\dst\delta \optbar\psi(\optbar \xi)}$ acts on the exponential a new vertex and new fields are brought downstairs. Since the new vertex comes with a $\lambda$ we see that, by the end of step $s+1$ all terms with degree $2\pi(T,s,\Lambda)$ different from zero are $O\left(\lambda^{s+1}\right)$. We interpret this construction in terms of graphs. Each integration by parts adds a line, that is, a propagator to the graph. The lines produced by differentiating the exponential form a spanning tree. The terms ${\cal A}(T,s,\Lambda)$ of degree zero are values of completely formed graphs. The other terms are gestating and will at some later stage become the values of full graphs. Fix any $T\in {\cal T}_s$. When all fields of ${\cal A}(T,s,\Lambda)$ have been eliminated by partial integration, we apply the second operation. It implements the Pauli exclusion principle. At this point there are still fields downstairs that were generated by derivatives acting on the exponential. They will be eliminated by partial integration in step $s+2$. As in the appendix of [IM], we expand each of these fields in a Taylor polynomial of degree $t$ around $c_\delta$, both to be determined: $$\eqalign{ \optbar\psi(\xi)=\optbar\psi(c_\delta)+\ \cdots\ +&{1\over t!}\big((\xi-c_\delta)\cdot\nabla_\xi\big)^{t} \ \optbar\psi(c_\delta)\cr &+{1\over t!}\int_0^1dw (1-w)^{t} \big((\xi-c_\delta)\cdot\nabla_\xi\big)^{t+1}\ \optbar\psi(c_\delta+w(\xi-c_\delta)). }$$ When, in step $s+2$ a field is eliminated by integration by parts, we obtain a Taylor expansion of the corresponding propagator. Thus lines in graphs carry these more general propagators. Note that the Taylor expansion contains $$ 1+(d+1)+(d+1)^2+\cdots+(d+1)^{t}+(d+1)^{t+1}\le 2(d+1)^{t+1} $$ terms. We now describe how the (field dependent) expansion point $c_\delta$ is determined. Recall that $K_j$ is a paving of $\bbbr^{d+1}$ by cubes of side $M^{-j}$. Each interaction vertex downstairs is rewritten as the sum \def\eqsev{I.7} $$ {\cal V}_\Lambda={1\over 2}\sum_{\{\uparrow,\downarrow\}} \sum_{\Delta\in K_j}\int_{\Lambda\cap\Delta} d\tau d\x\ \ \bar\psi(\tau,\x,\sigma)\bar\psi(\tau,\x,\sigma^\prime) \psi(\tau,\x,\sigma^\prime)\psi(\tau,\x,\sigma). \eqn{\eqsev}$$ In the $s=0$ step the only fields downstairs belong to the monomial ${\cal A}$. They are also expanded in terms of the paving. Multiple applications of (\eqsix) and (\eqsev) have generated from ${\cal A}(T,s,\Lambda)$ a sum of terms. Pick a term. Each field of this term is localized in a cube $\Delta\in K_j$. Let $\pi(s+1,\Delta,\sigma,b)$ be the number of fields that have the specified values of $\Delta,\sigma$ and $b$. Here $s+1$ reminds us that we are in the midst of step $s+1$ and $b$ distinguishes between $\psi$'s and $\bar\psi$'s. For each $\sigma,b$ and $\Delta\in K_j$ we divide $\Delta$ into $\pi(s+1,\Delta,\sigma,b)^\epsilon$ identical cubes $\delta$ each of side $\pi(s+1,\Delta,\sigma,b)^{-{\epsilon\over d+1}}M^{-j}$. The value of $\epsilon$ will be picked later. The center of $\delta$ is called $c_\delta$ and the number of fields in $\delta$ with specified values of $\sigma$ and $b$ is called $\pi(s+1,\delta,\sigma,b)$. The $\delta$ in the above Taylor expansion is, of course, that containing $\xi$. In particular the Taylor expansion must be done inside the $\xi$ integrals. Note that, by the hypothesis (\eqtwo) on the behavior of the propagator, each $(\xi-c_\delta)\cdot\nabla_\xi$ that acts on a $\psi$ produces a factor of \def\eqeig{I.8} $$ \pi(s+1,\Delta,\sigma,b)^{-{\epsilon\over d+1}}M^{-j}\cdot M^j. \eqn{\eqeig}$$ Since the fields anticommute, any nonzero integral may contain at most one field having any given value of $\sigma,b$ at any $c_\delta$. The same is true for each derivative of the fields. Thus, for each $\sigma,b,\delta$, all but $2(d+1)^{t}$ of the fields having this $\sigma,b$ and located in $\delta$ must be Taylor remainders. That is, there are at least $\pi(s+1,\delta,\sigma,b)-2(d+1)^{t}$ Taylor remainders. This completes the description of the expansion. We now prove \lem{5} {\it Let the propagator $C^{(j)}$ obey {\rm (\eqtwo)}. Then there exists a $\ \const\ $, independent of $j$ and $\Lambda$, such that $$ |a_n(j,\Lambda)|\le\const^{n+p}M^{(d+1)pj/2}\|A\|_1. $$ Furthermore the limits $$ a_n(j)=\lim_{\Lambda\rightarrow \bbbr^{d+1}} a_n(j,\Lambda)$$ exist and obey the same bounds. } \prf We use the ``method of combinatorial factors'' to keep track of the many sums in the expansion. This technique uses the elementary estimate \def\eqnin{I.9} $$ \kappa_i>0\ ,\ \sum_i \kappa_i^{-1}\le 1\ \ \ \Rightarrow\ \ \ \left|\sum_i U_i\right|\le\sup_i \left|\kappa_i U_i\right|. \eqn{\eqnin}$$ to replace each sum by a supremum. To help remember the combinatorial factor $\kappa_i$ multiplying the value $U_i$ of a given diagram, the factor is assigned to a specific line or vertex of the diagram. Here are the combinatorial factors used to control each of the operations. \vskip.25truein \noindent(1){\it Integration by parts.} \item{-}A factor of two, assigned to the leg initiating the integration by parts suffices to decide whether or not the leg brings down a new vertex from the exponent. \noindent When it does, \item{-}a factor of two, assigned to the target leg, counts the number of possible target legs. \noindent When it doesn't, we need to work harder. Vertices are continuously being added downstairs from the exponent during the $(s+1)^{\rm st}$ step. Thus the set of possible target legs both grows and shrinks as the stage progresses. On the other hand, all source legs must come from the set of legs downstairs at the beginning of the stage. So it is easier to count the number of contractions by labeling each target leg with the name of the source leg that contracted to it rather than use the usual procedure, which is to label each source leg with the name of the target leg to which it contracts. \item{-}A factor of two per leg suffices to decide whether or not the leg is a target leg. \noindent For each target leg, we organize the sum over source legs according first, to the cube $\Delta\in K_j$ of the source and second, to which leg in $\Delta$ is the source. As in (\eqone) \item{-}a factor of $\ \const\left(1+M^j{\rm dist}(\Delta,\Delta^\prime)\right)^{d+2}\ $, assigned to the line generated, suffices to control the sum over $\Delta$. \item{-}A factor $\pi(s+1,\Delta,\sigma,b)$, assigned to the source leg, counts the number of possible source legs within $\Delta$. \noindent(2){\it Implementation of the Pauli exclusion principle.} \item{-} a factor of $\ \const\left(1+M^j{\rm dist}(\Delta,\Delta^\prime)\right)^{d+2}\ $ assigned to the propagator that brought a vertex down from the exponent, will control the sum over localization cubes of the vertex. \noindent The sum over localization cubes for the fields of ${\cal A}$ is not controlled using combinatorial factors. It will shortly be performed explicitly. \item{-}The Taylor expansion splits each leg into at most $\ 2(d+1)^{t+1}\ $ pieces. \vskip .25truein It remains only to bound an integral. The integration variables are the positions of the fields of ${\cal A}$. The integrand is the supremum, over positions and diagrams, of the product of \item{-} the above combinatorial factors \item{-}$\const\left(1+M^j{\rm dist} (\Delta,\Delta^\prime)\right)^{-\gamma}$ per line of the diagram \item{-}$|\lambda|\int_{\Delta}d^{d+1}\xi\ 1\ \le |\lambda| M^{-j(d+1)}$ per vertex of the diagram \item{-}$|A(\xi_1,\cdots,\xi_{2p})|$ \item{-}and the factors that come from the Taylor expansions used to implement the Pauli exclusion principle. \vfill\eject \noindent The latter are \def\eqten{I.10} $$\eqalign{ &\prod_s\prod_{\sigma,b}\ \prod_{\Delta\in K_j}\prod_{\delta\in \Delta} \left[\pi(s,\Delta,\sigma,b)^{-{\epsilon\over d+1}}\right]^ {(t+1)\big(\pi(s,\delta,\sigma,b)-2(d+1)^{t}\big)}\cr &\le \prod_s\prod_{\sigma,b}\ \prod_{\Delta\in K_j} \left[\pi(s,\Delta,\sigma,b)^{-{\epsilon\over d+1}}\right]^ {(t+1)\big(\pi(s,\Delta,\sigma,b)-2(d+1)^{t} \pi(s,\Delta,\sigma,b)^\epsilon\big)}.\cr }\eqn{\eqten}$$ Pick any $\zeta>0$. It is possible to choose $\epsilon$ and $t$, depending only on $d$ and $\zeta$ so that $$ (\eqten)\le\prod_s\prod_{\sigma,b}\ \prod_{\Delta\in K_j} \const \pi(s,\Delta,\sigma,b)^{-\zeta \pi(s,\Delta,\sigma,b)}. $$ Altogether $$\eqalign{ \left|a_n(j,\Lambda)\right|&\le \const^{n+p} M^{{d+1\over 2}j{4n+2p\over 2}}M^{-(d+1)jn} \|A\|_1\cr &=\const^{n+p}M^{{d+1\over 2}pj}\|A\|_1 }$$ The first power of $M^j$ came from covariance bounds and the second from integration over the positions of the interaction vertices. When the $\Lambda$ constraint is removed, $a_n$ is expressed as an absolutely convergent series obeying the same bound.\endproof Our discussion of the toy model is now complete. The rest of the paper is devoted to the full model. %\vfill\eject \vskip.25truein \noindent{\subchfont \sec II Sector Counting Lemmas} \vskip.25truein In this section we prove Lemmas 2 and 3 formulated in the introduction. \lem{\ 2} {\it Fix $\ m\in \bbbz^{d+1}\ $ and $\ \ell\ge 2\ $. Then, the number of $2\ell$-tuples $$ \left\{S_1,\ \cdots\,S_{2\ell}\right\} $$ of sectors of side $M^j$ on the Fermi sphere for which there exist $\ \k_i\in \bbbr^d,\ i=1,\ \cdots,\ 2\ell\ $ satisfying $$ \k_i^\prime\in S_i,\ \ |\k_i-\k_i^\prime|\le{\rm const}\,M^j \ , \ \ \ i=1,\ \cdots,\ 2\ell\ $$ and $$ |\k_1+\ \cdots\ +\k_{2\ell}|\le{\rm const}\left(1+|m|\right)M^j $$ is bounded by $$ \const^\ell(1+|m|)^dM^{-(2\ell-1)(d-1)j}M^j \left\{1+|j|\delta_{d,2}\delta_{\ell,2}\right\}. $$ Here, $\ \k^\prime = {\k\over |\k|}\ $ denotes the projection of $\ \k\ $ onto the Fermi surface.} \prf For any fixed $\ \k_i,\ i=1,\ \cdots,\ 2\ell-1\ $ there are at most $\ O(1+|m|)^{d-1}\ $ sectors within $\ O\left((1+|m|)M^j\right)\ $ of $\ \k_1+\ \cdots\ +\k_{2\ell-1}\ $. Thus, the problem is reduced to determining the number of $(2\ell-1)$-tuples $\ \left\{S_1,\ \cdots\ ,S_{2\ell-1}\right\}\ $ of sectors for which there exist momenta $\ \k_i\in \bbbr^d,\ i=1,\ \cdots,\ 2\ell-1\ $ with $\ \k_i^\prime\in S_i,\ \ |\k_i-\k_i^\prime|\le {\rm const}\ M^j \ $, such that $\ \k_1+\ \cdots\ +\k_{2\ell-1}\ $ is within $\ O\left((1+|m|)M^j\right)\ $ of the Fermi surface. Since $\ |\k_i-\k_i^\prime|\le {\rm const}\,M^j\ $ there must exist $\k^\prime_i\in S_i$ such that $\ \k_1^\prime+\ \cdots\ +\k_{2\ell-1}^\prime\ $ is within $\ O\left((1+|m|+\ell)M^j\right)\ $ of the Fermi surface. As the projections $\ \k_i^\prime\ ,\ 1\le i\le 2\ell-1\ $, vary independently over their sectors, the sum $\ \k_1^\prime+\ \cdots\ +\k_{2\ell-1}^\prime\ $ varies by $\ O(\ell M^j)\ $. Thus, the problem is further reduced to counting the number $\ N_\ell\ $ of $(2\ell-1)$-tuples $\ \left\{S_1,\ \cdots\ ,S_{2\ell-1}\right\}\ $ of sectors such that for all $\ \k_i^\prime\in S_i\ $ the sum $\ \k_1^\prime+\ \cdots\ +\k_{2\ell-1}^\prime\ $ is within $\ O\left((1+|m|+\ell)M^j\right)\ $ of the Fermi surface. Observe that, if the volume of every sector is at least $\ \const\,M^{(d-1)j}$, $$ N_\ell\ \le\ \const^{-(2\ell-1)}M^{-(2\ell-1)(d-1)j}\ \prod_{i=1}^{2\ell-1}\left(\int_{k_FS^{d-1}}d\k_i^\prime\right) f\left(\k_1^\prime+\ \cdots\ +\k_{2\ell-1}^\prime\right)\ $$ where $\ f\ $ is a smooth function that is one when $\ \k_1^\prime+\ \cdots\ +\k_{2\ell-1}^\prime\ $ is within $\ O\left((1+|m|+\ell)M^j\right)\ $ of the Fermi surface and is zero when it is at least a distance $\ 2O\left((1+|m|+\ell)M^j\right)\ $ from the Fermi surface. Rewriting, and recalling that the Fourier transform of the $(d-1)$-sphere $\ \delta(|\k|-k_F)\ $ is $$\ {\rm const}\ \left(k_F|\p|\right)^{1-{d\over 2}} J_{{d\over 2}-1}(k_F|\p|) $$ we have $$\leftdisplay{\prod_{i=1}^{2\ell-1} \left(\int_{k_FS^{d-1}}d\k_i^\prime\right) f\left(\k_1^\prime+\ \cdots\ +\k_{2\ell-1}^\prime\right)}$$ $$\leftdisplay{\hskip.5truein= {\rm const}^{2\ell-1}\ \int\ \prod_{i=1}^{2\ell-1}d^d\k_i \ \prod_{i=1}^{2\ell-1}\ \delta(|\k_i|-k_F) \ f\left(\k_1+\ \cdots\ +\k_{2\ell-1}\right)}$$ $$\leftdisplay{\hskip.5truein={\rm const}^{2\ell-1}\ \int\ d^d\t\ \prod_{i=1}^{2\ell-1}d^d\k_i\ \prod_{i=1}^{2\ell-1}d^d\p_i\ \prod_{i=1}^{2\ell-1}\ \left(k_F|\p_i|\right)^{1-{d\over 2}} J_{{d\over 2}-1}(k_F|\p_i|)}$$ $$\leftdisplay{\hskip 2.1truein\ \prod_{i=1}^{2\ell-1}\ e^{i<\k_i,\p_i>} \ {\widehat f}(\t) \ e^{i<\t,\k_1+\ \cdots\ +\k_{2\ell-1}>}}$$ $$\leftdisplay{\hskip.5truein={\rm const}^{2\ell-1}\ \int\ d^d\t\ \prod_{i=1}^{2\ell-1}d^d\p_i \ \prod_{i=1}^{2\ell-1}\ \left(k_F|\p_i|\right)^{1-{d\over 2}} J_{{d\over 2}-1}(k_F|\p_i|) \ \prod_{i=1}^{2\ell-1}\ \delta(\p_i+\t)\ {\widehat f}(\t)}$$ $$\leftdisplay{\hskip.5truein={\rm const}^{2\ell-1}\ \int\ \ d^d\t\ \ \left\{\left(k_F|\t|\right)^{1-{d\over 2}} J_{{d\over 2}-1}(k_F|\t|)\right\}^{2\ell-1} \ {\widehat f}(\t)\ .} $$ \vskip .25truein The classical estimates %(Stein and Weiss P.158) $$J_\alpha(r)\sim \const\,r^\alpha\ ,\ \ r\rightarrow 0$$ $$J_\alpha(r)=O\left(r^{-{1\over 2}}\right)\ ,\ \ r\rightarrow \ \infty\ $$ imply that $$ \left(k_F|\t|\right)^{1-{d\over 2}} J_{{d\over 2}-1}(k_F|\t|) =\cases{O(1) & ${\rm for\ small}\ |\t| $\cr O\left( |\t|^{{1-d\over 2}}\right) & $ {\rm for\ large}\ |\t| $\cr} $$ Consequently, $${\widehat f}(\t)={\rm const}\ \int d^d\p\ e^{-i<\p,\t>}\ f(|\p|)\ =\ {\rm const}\ \int_0^\infty dr\ r^{d-1} \int_{S^{d-1}}d\p^\prime\ e^{-i}\ f(r)$$ $$ \leftdisplay{\hskip .55truein={\rm const}\ \int_0^\infty dr \ r^{d-1}\ \left(r|\t|\right)^{1-{d\over 2}}J_{{d\over 2}-1}(r|\t|)\ f(r)} $$ $$ \leftdisplay{\hskip .55truein={\rm const} \ (1+|m|+\ell)\ M^j\ \int_0^\infty dr\ r^{d-1} \ \left(r|\t|\right)^{1-{d\over 2}} J_{{d\over 2}-1}(r|\t|)\ {f(r)\over \|f\|_1}} $$ $$\leftdisplay{\hskip.55truein= \cases{O\left((1+|m|+\ell)M^j|\t|^{{1-d\over 2}}\right) & ${\rm for}\ |\t|\ \ge\ 1 $\cr O\left((1+|m|+\ell)M^j \right) & ${\rm for}\ |\t|\ \le\ 1$\cr}}$$ We now obtain $$\eqalign{N_\ell\ &\le\ {\rm const}^\ell\ M^{-(2\ell-1)(d-1)j}\ \int d^d\t\ \ \left\{\left(k_F|\t|\right)^{1-{d\over 2}} J_{{d\over 2}-1}(k_F|\t|)\right\}^{2\ell-1} \ {\widehat f}(\t)\cr &\le\ {\rm const}^\ell\ (1+|m|)\ M^{-(2\ell-1)(d-1)j}M^j \ \left\{\int_{|\t|\ \ge\ 1}d^d\t\ |\t|^{-\ell(d-1)}\ + \ \int_{|\t|\ \le\ 1}d^d\t\right\}\cr &\le\ \cases{ {\rm const}^\ell\ (1+|m|)\ M^{-(2\ell-1)(d-1)j}M^j & ${\rm for}\ \ell(d-1)\ >\ d $\cr {\rm logarithmically\ divergent}&${\rm for}\ \ell(d-1)= d $\cr}}$$ \vskip .25truein If $\ \ell(d-1)= d\ $, then $\ \ell=d=2\ $ and we consider the integral $$\int d^d\t\ \left\{\left(k_F|\t|\right)^{1-{d\over 2}} J_{{d\over 2}-1}(k_F|\t|)\right\}^3 \ {\widehat f}(\t)$$ Of course $f(t)$ is Schwartz class and hence so is $\widehat f$. The convergence of the integral is not in question. However each derivative of $\ f\ $ produces a factor of $\ M^{-j}\ $. Thus, we may gain a power of $\ |\t|^{-2}\ $ in our estimate of $\ {\widehat f}(\t)\ $ only at the cost of one $\ M^{-2j}\ $. Taking the geometric mean between $$|{\widehat f}(\t)|\le{\rm const}\, (1+|m|)\ M^j|\t|^{-{1\over 2}}$$ and $$|{\widehat f}(\t)|\le{\rm const}\, (1+|m|)\ M^{-j}|\t|^{-{5\over 2}}$$ one obtains for every $\ \epsilon>0\ $ $$|{\widehat f}(\t)|\le{\rm const}\, (1+|m|)\ M^{(1-2\epsilon)j} |\t|^{-({1\over 2}+2\epsilon)}\ .$$ The constant in the last line is independent of $\ \epsilon\ $. Finally, for $\ d=2\ $ $$N_2\le{\rm const}\ (1+|m|)\ M^{-3j}\ M^j \ \left\{M^{(-2\epsilon)j}\ \int_{|\t|\ \ge\ 1}d^2\t \ |\t|^{-(2+2\epsilon)}\ +\ 1\right\}$$ $$=\ {\rm const}\, (1+|m|)\ M^{-3j}\ M^j \ \left\{M^{(-2\epsilon)j}\ {\rm const}\ \epsilon^{-1} +\ 1\right\}\ .$$ Taking the limit $\ \epsilon \rightarrow 0\ $ gives the desired result. \endproof We now turn to the proof of Lemma 3. Actually we shall prove the stronger \lem{\ $3^\prime$} {\it Let $d=2$ and $\ell \ge 4$. Let $\Omega_2,\dots,\Omega_\ell$ be intervals on the Fermi circle each of length $\Omega\ge M^{j/2}$. Let $\Sigma_1$ be a fixed sector. The number of $(\ell-1)$-tuples of sectors $\{\Sigma_{2},\cdots,\Sigma_{\ell}\}$ for which there exist $\ \k_i\in\bbbr^2,\ i=1,\cdots,\ell\ $ satisfying $$ \k_{i}^\prime\in \Sigma_{i}\cap\Omega_i, \ \ |\k_i-\k_i^\prime|\le{\rm const}\, M^j \ , \ \ \ i=1,\cdots,\ell\ $$ and $$ |\k_1+\ \cdots\ +\k_{2\ell}|\le{\rm const}\,\left(1+|m|\right)M^j $$ is bounded by $$ \const^{\ell}\,(1+|m|)^2\left(\Omega M^{-j/2}\right)^{\ell-3}\ . $$ } \prf Denote by $\r_n$ the center of the sector containing $\k_n$. Renumber $\k_2,\cdots,\k_\ell$ so that $|\r_\ell\cdot\r_{\ell-1}|$ is minimal amongst all $\left\{|\r_n\cdot\r_p|\ \big|\ n,p\ne 1\right\}$. In other words $\phi=\angle(\k^\prime_{\ell-1},\k^\prime_\ell)$ is as close to $\pi/2$ as possible. All other $\angle(\k^\prime_{n},\k^\prime_{p})$'s with $\ n,p\ge 2\ $ must be within $\phi+O(M^{j/2})$ of either 0 or $\pi$. When $M^i\le\phi\le M^{i+1}$ or $M^i\le\pi-\phi\le M^{i+1}$ the number of accessible $(\ell-1)$-tuples of sectors is bounded by $$ N_\ell\prod_{i=2}^{\ell-2}\min\left\{\Omega,M^i\right\}M^{-j/2} $$ where $N_\ell$ is the number of sectors accessible to the last two $\k$'s once the sectors for $\k_2,\cdots,\k_{\ell-2}$ have been fixed. We shall shortly show that $N_\ell\le \const^\ell\,(1+|m|)^2$. The sum over those $i$'s with $M^i\le\Omega$ is bounded by $$ \const^\ell\,\sum_{i\atop M^i\le\Omega} (1+|m|)^2M^{i(\ell-3)} M^{-j(\ell-3)/2} \le \const^\ell\,(1+|m|)^2 \left(\Omega M^{-j/2}\right)^{\ell-3} $$ provided $\ell\ge 4$. Now consider $M^i>\Omega$. Once the sectors of all the $\k_i$'s except the $\ell^{\rm th}$ have been selected there can be at most one $i$ consistent with $\k_\ell$ falling in $\Omega_\ell$. For this one value of $i$ $$\eqalign{ N_\ell\prod_{i=2}^{\ell-2}\min\left\{ \Omega ,M^i\right\}M^{-j/2} &\le \const^\ell\,(1+|m|)^2 \left(\Omega M^{-j/2}\right)^{\ell-3} }$$ It suffices to consider $i$ obeying $\ M^i >\const^\ell(1+|m|)M^{j/2}\ $ so fix any such $i$ and $\ \Sigma_1,\cdots,\Sigma_{\ell-2}$. We now compute $\a$ and $\epsilon$, defined by $$\eqalign{ \a&=-\r_1-\,\cdots\,-\r_{\ell-2}\cr \a+\epsilon&=\k_{\ell-1}^\prime+\k_\ell^\prime\cr &=-\k_1-\,\cdots\,-\k_{\ell-2}+O\!\big((1+|m|)M^j\big)\ . }$$ Chose a coordinate system in which $\ \r_2=(k_F,0)\ $. Then, since $\ \theta_n=\angle(\r_2,\r_n),\ $ is within $\ 2M^{i+1}\ $ of 0 or $\pi$ the $x$ and $y$ coordinates of every $\r_n$ and $\k_n$, $\ \ 2\le n\le\ell,\ $ obey $$\eqalign{ x_n&=\pm[k_F+O\!\big(M^j\big)]\cos O\!\big(M^i\big) =\pm k_F+O\!\big(M^{2i}\big)\cr y_n&=[k_F+O\!\big(M^j\big)]\sin O\!\big(M^i\big)=O\!\big(M^i\big) }$$ respectively. Consequently $\ \k_1=-\k_2-\,\cdots\,-\k_{\ell}\ $ and hence $\r_1$ obey $$\eqalign{ x_1&=\pm k_F+O\!\big((|m|+\ell)M^{2i}\big)\cr y_1&=O\!\big((|m|+\ell)M^i\big) }$$ We need to know with greater precision how much $\k_n$ can wiggle. $$\eqalign{ \k_n-\r_n&=\k_n^\prime-\r_n+O\!\big(M^j\big)\cr &=k_F\Big(\cos\big(\theta_n+O\!(M^{j/2})\big), \sin\big(\theta_n+O\!(M^{j/2})\big)\Big)-k_F\Big(\cos\theta_n, \sin\theta_n\Big)+O\!\big(M^j\big)\cr &=O\!\big((M^iM^{j/2},M^{j/2})\big)\ . }$$ Summing the individual wiggles yields $$\eqalign{ \epsilon&=\sum_{n=1}^{\ell-2}(\r_n-\k_n)+O\!\big((1+|m|)M^j\big)\cr &=(|m|+\ell)\,O\!\big((M^iM^{j/2},M^{j/2})\big) }$$ and $$\eqalign{ \a&=-\r_1-\,\cdots\,-\r_{\ell-2}\cr &=\k_{\ell-1}+\k_\ell-\sum_{n=1}^{\ell}\k_n +\sum_{n=1}^{\ell-2}(\k_n-\r_n)\cr &=N(2k_F,0)+O\!\big((M^{2i},M^{i})\big) +O\!\big((1+|m|)M^{j})\big)+\ell\, O\!\big((M^iM^{j/2},M^{j/2})\big)\cr &=N(2k_F,0)+O\!\big((M^{2i},M^{i})\big) }$$ where $\ N\in\{1,0,-1\}\ $. We are now in a position to bound $N_\ell$. There are two cases to be considered. First, suppose $|N|=1$. Rotate the coordinate system by $\ \pi\delta_{N,-1}+O\!\big(M^{i}\big)\ $ to make $\a$ run along the positive $x$ axis. In the new coordinates, $$ \epsilon=(|m|+\ell)\,O\!\big((M^iM^{j/2},M^{j/2})\big)\ . $$ is still obeyed.\hfill\break \figplace{lce3}{1.25in}{-.35in} \vskip.25truein Then the two components of $$ k_F\big(\cos\alpha,\sin\alpha\big) +k_F\big(\cos(\phi-\alpha),-\sin(\phi-\alpha)\big)=\a+\epsilon $$ are $$\eqalign{ \cos\alpha+\cos(\phi-\alpha)&={|\a|\over k_F}+(|m|+\ell)\, O\!\big(M^iM^{j/2}\big)\cr \sin\alpha-\sin(\phi-\alpha)&=(|m|+\ell)\,O\!\big(M^{j/2}\big)\ .\cr }$$ The $y$ component implies that $$ |2\alpha-\phi|=(|m|+\ell)\,O\!\big(M^{j/2}\big) $$ so that there are $\ \const\,(|m|+\ell)\ $ sectors accessible to $\k_\ell$ (once $\r_{\ell-1}$ has been fixed) and $$ \alpha={1\over 2}\phi+(|m|+\ell)\,O\!\big(M^{j/2}\big)\ . $$ is bounded above and below by $\ \const\,M^i$. Since $$\eqalign{ \cos\alpha+\cos(\phi-\alpha) &=\cos\alpha+\cos\alpha\cos(\phi-2\alpha) -\sin\alpha\sin(\phi-2\alpha)\cr &=\left[2+(|m|+\ell)^2\,O\!\big(M^{j}\big)\right]\cos\alpha +(|m|+\ell)\,O\!\big(M^iM^{j/2}\big)\cr }$$ the $x$ component gives $$ \alpha=\cos^{-1}\left({|\a|\over 2k_F}\right) +(|m|+\ell)\,O\!\left({M^iM^{j/2}\over M^i}\right) $$ Thus, there are $\ \const\,(|m|+\ell)\ $ sectors accessible to $\k_{\ell-1}$ as well. Finally, suppose that $N=0$. This time rotate the coordinate system by $\ O\!\big(M^{i}\big)\ $ or $\ \pi+O\!\big(M^{i}\big)\ $so that $\k_{\ell-1}$ runs along the negative $x$ axis. \hfill\break \figplace{lce4}{1.5in}{-.1in}\hfill\break \noindent The angle $\phi$ is determined by $$\eqalign{ \sin\left({\pi-\phi\over 2}\right) &={\left|\a+\epsilon\right|\over 2k_F}\cr &={|\a|\over 2k_F}+(|m|+\ell)\,O\!\big(M^{j/2}\big)\ . }$$ Thus $$ \phi=\pi-2\sin^{-1}\left({|\a|\over 2k_F}\right) +(|m|+\ell)\,O\!\big(M^{j/2}\big) $$ and $\k_\ell$ has access to $O\!\big(|m|+\ell\big)$ sectors when $\r_{\ell-1}$ is held fixed. The angle $\theta$ is determined by $$\eqalign{ \left|\sin\left(\theta-{\phi\over 2}\right)\right| &={\left|\epsilon\cdot\left(\cos\left({\pi-\phi\over 2}\right) ,\sin\left({\pi-\phi\over 2}\right)\right)\right| \over|\a|}\cr &\le{(|m|+\ell)\,O\!\big(M^iM^{j/2}\big) \over M^i-(|m|+\ell)\,O\!\big(M^{j/2}\big)}\cr &\le(|m|+\ell)\,O\!\big(M^{j/2}\big)\ . }$$ In the second last line we used the hypothesis that $\ M^i\le\phi\le M^{i+1}\ $. This forces $$\eqalign{ \left|\a+\epsilon\right| &=2k_F\sin\left({\pi-\phi\over 2}\right)\cr &\ge \const\,M^i\ . }$$ Once again there are at most $O\!\big(|m|+\ell\big)$ sectors accessible to $\k_{\ell-1}$. \endproof %\vfill\eject %\input myfig \def\eqtwo{I.2} \def\eqsix{I.6} \def\eqsev{I.7} \vskip.25truein \noindent{\subchfont \sec III The Full Expansion} \vskip.25truein In this Section we elaborate on the expansion presented in the introduction to prove a lemma and the two theorems. All are of the form of Lemma 5, but treat the true propagator and interaction. They are restricted to three space-time dimensions. In Lemma 6, we bound Schwinger functions. Theorem 1 constructs the effective potential. Finally in Theorem 2, we generalize the expansion to accommodate $\nu$-body interactions, $\nu\ge 2$, of the type generated by the renormalization group flow of [FT2]. Throughout this section we expand the true propagator $$ C^{(j)}=\sum_{\ell=1}^{M^{-j/2}} C^{(j,\ell)} $$ where $$ C^{(j,\ell)}(\xi,\bar \xi)=\delta_{\sigma,\bar\sigma} \int {d^{3}p\over(2\pi)^{d+1}} {e^{i_-} \over ip_0-e(\p)}f_j(p)\,\zeta_\ell(\p)\ . $$ The partition of unity $$ 1=\sum_{\ell=1}^{M^{-j/2}}\zeta_\ell(\p),\hskip .25truein \zeta_\ell(\p)=\zeta_\ell\left({\p\over|\p|}k_F\right) $$ divides the Fermi circle into long sectors $\Sigma_\ell$. We denote by $r_\ell$ the center of $\Sigma_\ell$. Recall that $$ _-=\p\cdot\x-p_0\tau\ , $$ $$ e(\p)={\p^2\over 2m}-\mu. $$ and $$ f_j(p)=f\left(M^{-2j}\left(p_0^2+e(\p)^2\right)\right). $$ The corresponding decomposition of the fields is $$ \psi(\xi)=\sum_{\ell=1}^{M^{-j/2}} \psi^{(\ell)}(\xi)\hskip.5in \bar\psi(\xi)=\sum_{\ell=1}^{M^{-j/2}} \bar\psi^{(\ell)}(\xi)\ . $$ For each sector $\Sigma_\ell$ we introduce a lattice ${\bf D}_{\ell}$ of rectangular parallelepipeds (called boxes for short) that pave $\bbbr^{3}$. These boxes have three axes. One axis is the fixed time direction and the length of any box in that direction is $M^{-j}$. In the orthogonal $\bbbr^{2}$ plane, one of the axes is $r_{\ell}$, the center of $\Sigma_\ell$, and the length in this direction is $M^{-j}$. The third axis is orthogonal to $r_{\ell}$ and has length $M^{-j/2}$. Finally, the two-body interaction in volume $\Lambda\subset \bbbr^3$ is $$ {\cal V}_\Lambda={1\over 2}\hskip-1.5pt \int \hskip-1.5pt d^3\xi\prod_{i=1}^4d^3\eta_i \ \chi_\Lambda(\xi) V(\eta_1,\eta_2,\eta_3,\eta_4) \delta(\eta_1\!+\!\hskip1pt\eta_2\!+\!\hskip1pt\eta_3\!+\hskip1pt\!\eta_4) {\bar\psi}(\xi+\eta_1)\psi(\xi+\eta_3) {\bar\psi}(\xi+\eta_2)\psi(\xi+\eta_4). $$ It is convenient for us to restrict only the center of mass $\xi$ to $\Lambda$. \lem{\ 6} {\it Let $$ {\cal A}=\int \prod_{j=1}^{2p} d\xi_j\ A(\xi_1,\cdots,\xi_{2p}) \prod_{j=1}^{2p}\optbar\psi(\xi_j) $$ be a monomial of degree $2p$. Define $$ \<{\cal A}\>={1\over Z_\Lambda}\int{\cal A} e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)} d\mu_{C^{(j)}}(\psi,\bar\psi) $$ where $$ Z_\Lambda=\int e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)} d\mu_{C^{(j)}}(\psi,\bar\psi)\ . $$ Then, there exists a $\ \const\ $, independent of $j$ and $\Lambda$, such that the perturbation series $$ \<{\cal A}\>=\sum_{n=0}^\infty a_n(j,\Lambda)\lambda^n $$ converges for all $|\lambda|&=\sum_{T\in {\cal T}_s} {\int{\cal A}(T,s,\Lambda) e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}} \over \int e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}} }\cr &=\sum_{T\in {\cal T}_s\atop \pi(T,s)=0}{\cal A}(T,s,\Lambda) +\sum_{T\in {\cal T}_s\atop \pi(T,s)\ne 0} {\int{\cal A}(T,s,\Lambda) e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}} \over \int e^{-\lambda{\cal V}_\Lambda(\psi,\bar\psi)}d\mu_{C^{(j)}} }\ \ \ . }\eqn{\eqIIIone}$$ Recall that each ${\cal A}(T,s,\Lambda)$ is a monomial in the fields $\psi,\bar\psi$ of degree $2\pi(T,s)$. We used the fact that the ratio of integrals simplifies to the number ${\cal A}(T,s,\Lambda)$ when $\pi(T,s)=0$. Each ratio in the second sum will be $O\left(\lambda^s\right)$. We now outline the $(s+1)^{\rm th}$ step in the induction. The details will be presented shortly. First pick any field of ${\cal A}(T,s,\Lambda)$. Apply the integration by parts formula (\eqsix). In the event that the derivative brings a new vertex down from the exponent, expand the new vertex as a sum over momentum sectors $\ell$ and position space boxes $\Delta\in {\bf D}_\ell$. Here, this step is considerably more complicated than (\eqsev). Next, pick any other field of the original monomial ${\cal A}(T,s,\Lambda)$ and repeat the construction. When all the fields of ${\cal A}(T,s,\Lambda)$ have been exhausted, substitute Taylor expansions much as in \sec I. The sum over sectors and conjugate boxes is complicated because conservation of momentum must be exploited to restrict the sums over sectors at every vertex. We now consider this in more detail. The volume cutoff interaction is $$\eqalign{ {\cal V}_\Lambda&={1\over 2} \int d^3\xi\prod_{i=1}^4d^3\eta_i \ \chi_\Lambda(\xi)V(\eta_1,\eta_2,\eta_3,\eta_4) \delta(\Sigma \eta_i) {\bar\psi}(\xi\!+\!\eta_1)\psi(\xi\!+\!\eta_3) {\bar\psi}(\xi\!+\!\eta_2)\psi(\xi\!+\!\eta_4)\cr &={1\over 2}\int d^3\xi\prod_{i=1}^4\left(d^3\eta_i{d^3k_i\over (2\pi)^3}\right) \ \ \chi_\Lambda(\xi) V(\eta_1,\eta_2,\eta_3,\eta_4)\delta(\Sigma \eta_i) {\bar\psi}(k_1)\psi(k_3){\bar\psi}(k_2)\psi(k_4)\cr &\hskip2in e^{-i_-}e^{i_-}e^{-i_-} e^{i_-}\cr &={1\over 2}\int \prod_{i=1}^4{d^3k_i\over (2\pi)^3}\ {\widetilde \chi}_\Lambda(-k_1-k_2+k_3+k_4) \bar\psi(k_1)\psi(k_3)\bar\psi(k_2)\psi(k_4)\cr &={1\over 2}\sum_{m\in {\bf Z}^3}\int\prod_{i=1}^4{d^3k_i\over (2\pi)^3}\ \chi(k_1+k_2-k_3-k_4+M^jm) {\widetilde \chi}_\Lambda(-k_1-k_2+k_3+k_4)\cr &\hskip2in \bar\psi(k_1)\psi(k_3){\bar\psi}(k_2)\psi(k_4)\ . }$$ Here $$ 1=\sum_{m\in {\bf Z}^3}\chi(M^{-j}k+m) $$ is a partition of momentum space by $C^\infty_0$ functions supported on cubes of side $M^j$ and $$\eqalign{ &=\int\prod_{i=1}^4d^3\eta_i V(\eta_1,\eta_2,\eta_3,\eta_4)\delta(\eta_1+\eta_2+\eta_3+\eta_4)\cr &\hskip.5in e^{-i_-}e^{i_-}e^{-i_-} e^{i_-}. }$$ Reversing the calculation above expresses \def\eqIIItwo{III.2} $$\eqalign{ {\cal V}=&\sum_{m\in {\bf Z}^3} {1\over 2}\int d^3\xi\prod_{i=1}^4d^3\eta_i \ \chi_{m,\Lambda}(\xi)V(\eta_1,\eta_2,\eta_3,\eta_4) \delta(\Sigma \eta_i){\bar\psi}(\xi\!+\!\eta_1)\psi(\xi\!+\!\eta_3) {\bar\psi}(\xi\!+\!\eta_2)\psi(\xi\!+\!\eta_4)\cr &=\sum_{m\in {\bf Z}^3} {1\over 2}\int \prod_{i=1}^4d^3\xi_i \ \chi_{m,\Lambda}\left({\xi_1+\xi_2+\xi_3+\xi_4\over 4}\right) V(\xi_1,\xi_2,\xi_3,\xi_4) {\bar\psi}(\xi_1)\psi(\xi_3){\bar\psi}(\xi_2)\psi(\xi_4) }\eqn{\eqIIItwo}$$ where $\chi_{m,\Lambda}$ is the inverse Fourier transform of $\chi(M^{-j}k+m){\widetilde \chi}_\Lambda(k)$. Even if the interaction $V$ is of compact support, with respect to the center of mass, $\chi_{m,\Lambda}V$ is no longer supported in a compact neighborhood of $\Lambda$ because $\chi$ is of compact support in momentum space. This is no problem. The vertex is already connected by propagators to the external generalized vertex ${\cal A}$, so integration over the center of mass variable is taken care of. Only two properties of $\chi_{m,\Lambda}$ are required. The first is \def\eqIIIthr{III.3} $$\eqalign{ \sup_{\xi\in \bbbr^3} \left|\chi_{m,\Lambda}(\xi)\right| &\le\sup_\xi \left|M^{3j}\int d^3\eta\ e^{i<\eta,M^jm>_-} {\widetilde\chi}(M^j\eta) {\chi}_\Lambda(\xi-\eta)\right|\cr &\le {\rm const}_N\left(1+|m|\right)^{-N} } \eqn{\eqIIIthr}$$ independent of $j$, provided derivatives acting on $\chi_\Lambda$ produce, at worst, $M^j$'s. In other words $\chi_\Lambda$ must decay from 1 down to zero in a distance $O(M^{-j})$. This is a consequence of the standard integration by parts trick, as in the proof of Lemma 4, and $$ \left|{\nabla_\eta}^b {\widetilde\chi}(\eta)\right| \le \const^b\ {1\over (1+|\eta|)^{4}}\ . $$ Once conservation of momentum at a vertex has been implemented by (\eqIIItwo), each field at the vertex is expanded in sectors. Recall that the vertex was produced by an application of the integration by parts formulae (\eqsix) to a source field. So, one of its legs inherits its sector number from this source field. A priori, decomposing each of the three remaining fields into a sum over sectors could produce $M^{-3j/2}$ terms. The second property of $\chi_{m,\Lambda}$ is that there are only $O(|m|^{2} M^{-j/2})$ nonzero terms because of the constraints imposed by conservation of momentum and the fact that we are in two space dimensions. See Lemma 3. Finally, each field $\optbar\psi^{(\ell)}$ is expanded in boxes $\Delta\in{\bf D}_\ell$. Altogether, the ``sum over sectors and conjugate boxes'' operation replaces (\eqsev) by \def\eqIIIfou{III.4} $$\eqalign{ {\cal V} &=\sum_{m\in {\bf Z}^3}\sum_{\ell_1,\ell_2,\ell_3,\ell_4} {1\over 2}\prod_{i=1}^4 \left(\sum_{\Delta_i\in{\bf D}_{\ell_i}}\int_{\Delta_i} d^3\xi_i\right) \ \chi_{m,\Lambda}\left({\xi_1+\xi_2+\xi_3+\xi_4\over 4}\right)\cr &\hskip2in \times V(\xi_1,\xi_2,\xi_3,\xi_4) {\bar\psi}^{(\ell_1)}(\xi_1)\psi^{(\ell_3)}(\xi_3) {\bar\psi}^{(\ell_2)}(\xi_2)\psi^{(\ell_4)}(\xi_4) }\eqn{\eqIIIfou}$$ In step $s=0$ the only fields downstairs are those belonging to the monomial ${\cal A}$. One decomposes each field in sectors $\Sigma_\ell$ and then in conjugate boxes $\Delta\in {\bf D}_\ell$. There is no need for the sum over $m$ associated with conservation of momentum. We describe the Taylor expansions in more detail. Now there are sectors to be taken into account. Let $\pi(s+1,\ell,\Delta,\sigma,b)$ be the number of fields with the specified values of $\sigma, b, \ell$ and $\Delta\in{\bf D}_\ell$. For each $\ell,\sigma,b$ and $\Delta\in {\bf D}_\ell$ we divide $\Delta$ into $\pi(s+1,\ell,\Delta,\sigma,b)^\epsilon$ identical sub-boxes (really rectangular parallelepipeds) $\delta$ each similar to $\Delta$. Thus each little box has dimensions $$ \pi(s+1,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}M^{-j}\times \pi(s+1,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}M^{-j}\times \pi(s+1,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}M^{-j/2}. $$The value of $\epsilon$ will be picked later. As in \sec I the center of $\delta$ is called $c_\delta$ and the number of fields in $\delta$ with specified values of $\ell,\sigma$ and $b$ is called $\pi(s+1,\ell,\delta,\sigma,b)$. The action of derivatives, given by Lemma 4.b, differs from (\eqtwo) in several respects. In particular the appearance of the phase $e^{-i _-}$ forces us to modify the Taylor expansion. Based on this remark we define $$\eqalign{ \Psi^{(\ell)}(\xi)&=e^{-i<\r_\ell,\xi>}\psi^{(\ell)}(\xi)\cr \bar\Psi^{(\ell)}(\xi)&=e^{i<\r_\ell,\xi>}\bar\psi^{(\ell)}(\xi). }$$ We expand each $\Psi^{(\ell)}(\xi),\bar\Psi^{(\ell)}(\xi)$ downstairs in a Taylor polynomial \def\eqIIIfiv{III.5} $$\eqalign{ \optbar\Psi^{(\ell)}(\xi)=\optbar\Psi^{(\ell)}(c_\delta)+\ \cdots\ +&{1\over t!}\big((\xi-c_\delta)\cdot\nabla_\xi\big)^{t}\ \optbar\Psi^{(\ell)}(c_\delta)\cr &+{1\over t!}\int_0^1dw (1-w)^{t} \big((\xi-c_\delta)\cdot\nabla_\xi\big)^{t+1}\ \optbar\Psi^{(\ell)}(c_\delta+w(\xi-c_\delta)) }\eqn{\eqIIIfiv}$$ with $\delta$ being the box that contains $\xi$. As before, the Taylor expansion is done inside the $\xi$ integrals. One might worry that the restriction to a very small box $\delta$ weakens conservation of momentum and consequently increases the number of nonzero terms in the sector sums. However, the sector sums have already been cut down, so there is no problem. Each Taylor expansion contains $$ 1+3+3^2+\cdots+3^t+3^{t+1}\le 2\times3^{t+1} $$ terms. Note that, by Lemma 4.b, each $(\xi-c_\delta)\cdot\nabla_\xi$ that acts on a $\optbar\Psi^{(\ell)}$ produces a factor of $$ \pi(s+1,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}} \left[M^{-j}M^{j}+M^{-j}M^{j}+M^{-j/2}M^{j/2}\right] =3\pi(s+1,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}}. $$ Since the fields anticommute, any nonzero integral may contain at most one field having any given value of $\ell,\sigma,b$ at any $c_\delta$. The same is true for each derivative of the fields. Thus, for each $\ell,\sigma,b,\delta$, all but $2\times 3^{t}$ of the fields having this $\ell,\sigma,b$ and located in $\delta$ must be Taylor remainders. That is, there are at least $\pi(s+1,\ell,\delta,\sigma,b)-2\times 3^{t}$ Taylor remainders. This completes the description of the expansion. We use the same strategy for performing the estimates as in Lemma 5. Here are the combinatorial factors used to control each of the three operations.\vskip.25truein \noindent(1){\it Integration by parts.} \item{-}A factor of two, assigned to the leg initiating the integration by parts suffices to decide whether or not the leg brings down a new vertex from the exponent. \noindent When it does, \item{-}a factor of two, assigned to the target leg, counts the number of possible target legs. \noindent When it doesn't, we count the number of possible source legs for each target leg by applying the rules of Lemma 5 in each sector $\ell$. Precisely, for each target leg, we organize the sum over source legs according first, to the cube $\Delta\in {\bf D}_\ell$ of the source and second, to which leg in $\Delta$ is the source. \item{-}A factor of two per leg suffices to decide whether or not the leg is a target leg. \item{-}A factor of $\ \const\rho^{(j,\ell)}(\Delta,\Delta^\prime)^{4}\ $, assigned to the line generated, suffices to control the sum over $\Delta$. \item{-}A factor $\pi(s+1,\ell,\Delta,\sigma,b)$, assigned to the source leg, counts the number of possible source legs within $\Delta$. \noindent(2){\it Sums over sectors and conjugate boxes.}\ \ The sum $\ \sum_{m\in \bbbz^3}\ \cdots\ \chi_{m,\Lambda}\ $ is controlled by \item{-} a factor $\ \const|m|^{4}\ $ assigned to the vertex. \noindent When a vertex first moves downstairs from the exponent the sector of one of its legs is fixed by the source leg that initiates the process. By Lemma 3 the number of sectors accessible to the remaining legs is bounded by \item{-} a factor $\ \const(1+|m|^{4/3})M^{-j/2}$ assigned to the vertex. \item{-} If the external vertex does not impose any constraint on the sector sums of its $2p$ legs then the number of sectors accessible is $M^{-pj}$. \noindent The sums $\sum_{\Delta_i\in D_{\ell}}$ are not controlled by combinatorial factors. They will be performed explicitly. \noindent(3){\it Pauli exclusion principle.} \item{-}The Taylor expansion splits each leg into at most $\ 2\times 3^{t+1}\ $ pieces. \vskip .25truein It remains only to bound an integral. The integration variables are the positions of the fields of ${\cal A}$ and the interaction vertices ${\cal V}$ that have been brought down from the exponent. The integrand is the supremum, over positions and diagrams, of the product of \item{-} the above combinatorial factors \item{-}$\const\rho^{(j,\ell)}(\Delta,\Delta^\prime)^{-\gamma}$ per line of the diagram \item{-}$|\lambda| \left(\sup_{\xi\in \bbbr^3} \left|\chi_{m,\Lambda}(\xi)\right|\right) |V(\xi_1,\xi_2,\xi_3,\xi_4)| \le {\rm const}|\lambda|\left(1+|m|\right)^{-N}|V(\xi_1,\xi_2,\xi_3,\xi_4)| $ per vertex of the diagram (see (\eqIIIthr)) \item{-}$|A(\xi_1,\cdots,\xi_{2p})|$ \item{-}and the factors that come from the Taylor expansions used to implement the Pauli exclusion principle. \noindent The latter are \def\eqIIIsix{\rm III.6} $$\eqalign{ &\prod_s\prod_{\ell,\sigma,b}\ \prod_{\Delta\in {\bf D}_\ell} \prod_{\delta\in \Delta} \left[3\pi(s,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}} \right]^{(t+1)\big(\pi(s,\ell,\delta,\sigma,b)-2\times 3^{t}\big)}\cr &\le \prod_s\prod_{\ell,\sigma,b}\ \prod_{\Delta\in {\bf D}_\ell} \left[3\pi(s,\ell,\Delta,\sigma,b)^{-{\epsilon\over 3}} \right]^{(t+1)\big(\pi(s,\ell,\Delta,\sigma,b)-2\times 3^{t}\pi(s,\ell,\Delta,\sigma,b)^\epsilon\big)}.\cr }\eqn{\eqIIIsix}$$ Pick any $\zeta>0$. It is possible to choose $\epsilon$ and $t$, depending only on $\zeta$ so that $$ (\eqIIIsix)\le\prod_s\prod_{\ell,\sigma,b}\ \prod_{\Delta\in{\bf D}_\ell} \const \pi(s,\ell,\Delta,\sigma,b)^{-\zeta \pi(s,\ell,\Delta,\sigma,b)}. $$ Altogether \def\eqIIIsev{III.7} $$\eqalign{ \left|a_n(j,\Lambda)\right| &\le \const^{n+p}\! \int \prod_i d\xi_i |A|\! \left(\prod |V|\right)\! \left(\prod \rho^{(j,\ell)}(\Delta,\Delta^\prime)^{-\gamma+4}\! \right)\! M^{{3\over 2}j{4n+2p\over 2}}M^{-jn/2}M^{-pj}\cr }\eqn{\eqIIIsev}$$ where the second product is over vertices and the third is over lines. The first power of $M^j$ came from covariance bounds and the second and third from sector sums for fields at interaction vertices and ${\cal A}$ respectively. After discarding some of the decay factors $\rho^{(j,\ell)}$, we can view the integrand as a connected tree with generalized vertices $V$ and $A$ and lines $\rho^{(j,\ell)}$. Integrate starting at the extremities of the tree and working towards the root $A$. Suppose that the extremal vertex $V$ is connected to the tree by the argument $\xi_1$. The integral over $\xi_2,\xi_3,\xi_4$ with $\xi_1$ held fixed produces a $\norm V\norm $. A decay factor $\rho^{(j,\ell)}$ is enough to fix the box in which $\xi_1$ lives. The integral over $\xi_1$ within that box costs $M^{-5j/2}$, the volume of the box. Repeat for all the other $V$'s. Finally, the integral over the arguments of ${\cal A}$ gives $\|A\|_1$. In conclusion $$\eqalign{ \left|a_n(j,\Lambda)\right|&\le \const^{n+p} \|A\|_1\norm V \norm^n M^{-5jn/2} M^{3j(n+p/2)}M^{-jn/2}M^{-pj}\cr &=\const^{n+p} \|A\|_1\norm V\norm^n M^{-pj/2}\ . }$$ When the finite volume, i.e. $\Lambda$, constraint is removed, $a_n$ is expressed as an absolutely convergent series obeying the same bound.\endproof The next order of business is the effective potential $$ {\cal G}_\Lambda^{(j)}(\psi^e,\bar\psi^e) =\log\,{1\over Z_\Lambda^{(j)}}\int e^{-\lambda {\cal V}_\Lambda(\psi+\psi^e,\bar\psi+\bar\psi^e)} d\mu_{C^{(j)}}(\psi,\bar\psi). $$ Let $G^{(j,\Lambda)}_p$ be the $p$-point Green's function generated by ${\cal G}^{(j)}_\Lambda$. We now prove the \thm{\ 1}{\it Let $d=2$. There exists a $\ \const\ $, independent of $j$ and $\Lambda$, such that $$ \norm g_n(p,j,\Lambda)\norm \le\const^{n+p}M^{(2-5p/2)j}\norm V\norm^n $$ where $$ \norm g_n(p,j,\Lambda)\norm= \max_{k}\ \sup_{\xi_k}\ \int\prod_{i\ne k}d\xi_i\ |g_n(p,j,\Lambda)(\xi_1,\cdots,\xi_{2p})|. $$ Furthermore the limits $$ g_n(p,j)=\lim_{\Lambda\rightarrow \bbbr^3}g_n(p,j,\Lambda) $$ exist and the infinite volume Green's functions at scale $j$ $$ G^{(j)}_p=\sum_{n=0}^\infty g_n(p,j)\lambda^n $$ are analytic in $|\lambda|j}\sum_{q=1}^\infty \int\prod_{i=1}^qd\xi_id\bar\xi_i\ V_{\Lambda,q}^{(j^\prime)}(\xi_1,\bar\xi_1,\cdots,\xi_q,\bar\xi_q) \prod_{i=1}^q\bar\psi(\bar\xi_i)\psi(\xi_i) \eqn{\eqIIItena}$$ when evaluating the effective potential with covariance of scale $j$. The term ${\cal V}^{(0)}_{\Lambda , 2}$ in this sum is the ordinary vertex (\eqIIItwo) with four fields. Each kernel is expressed as a sum \def\eqIIItenb{III.10b} $$ V_{\Lambda,q}^{(j^\prime)}=\sum_{m\in{\cal M}_{q}} V_{\Lambda,q,m}^{(j^\prime)}\ . \eqn{\eqIIItenb}$$ The index $j^\prime$ gives the scale at which the graphs contributing to $V_{\Lambda,q,m}^{(j^\prime)}$ were formed. You should think of $m$ as measuring the extent to which exact conservation of momentum fails as well as giving the sectors of some internal lines. In particular, \def\eqIIItenc{III.10c} \item{-}there is an $E^{(j^\prime)}_{q,m}\in\bbbz$ such that $V_{\Lambda,q,m}^{(j^\prime)}$ is zero unless the sum of the momenta flowing into it is bounded by $E^{(j^\prime)}_{q,m}M^J$\hfill (\eqIIItenc) \noindent We assume that there is an $0<\alpha<1$ such that, when all the arguments of $V_{\Lambda,q}^{(j^\prime)}$ have momentum within $\ \const M^j\ $ of the Fermi surface and when one argument lies in a fixed sector of scale $j$, \def\eqIIItend{III.10d} \item{-}the remaining $2q-1$ legs of each nonzero $V_{\Lambda,q,m}^{(j^\prime)}$ are supported in \hfill\break $C_1^{q} \exp\left\{\left(E^{(j^\prime)}_{q,m}M^{J-j}\right)^\alpha\right\} M^{(2q-3)(j'-j)/2} K_{q,m}^{(j^\prime)}$ sector $(2q-1)$-tuples of scale $j$.\hfill\break\null\hfill (\eqIIItend) \noindent That is, $K_{q,m}^{(j^\prime)}$ is the number of sector $(2q-1)$-tuples at scale $j'$ when the kernel was first generated. However, a sector that is introduced with width $M^{j^\prime/2}$ is subdivided into $M^{(j^\prime-j)/2}$ sectors at scale $j$. The number of accessible sectors of scale $j$ for the $2q-1$ remaining external legs given their sectors at scale $j'$ is bounded by $const^{q}(1+E^{(j^\prime)}_{q,m}M^{J-j})^2 M^{(2q-3)(j'-j)/2}$ in Lemma 3. It is convenient for us to use the multiplicative property of the exponential so we bound $(1+EM^{J-j})^2\le\const\exp\left\{\left(EM^{J-j}\right)^\alpha\right\}$. We further assume that the kernel is analytic in the region $|\lambda|< R$ and obeys the bound \def\eqIIItene{III.10e} $$ \sum_{m\in{\cal M}_q} \exp\left\{\sum_{\imath=J}^{j^\prime-1} \left(E^{(j^\prime)}_{q,m}M^{J-\imath}\right)^\alpha\right\} K_{q,m}^{(j^\prime)}\norm V_{\Lambda,q,m}^{(j^\prime)}\norm \le K_1|\lambda|^{q/2}M^{{1\over 2}(5-3q)j^\prime}\ \ . \eqn{\eqIIItene}$$ throughout that region. Note that $\lambda$ times the vertex (\eqIIItwo) is of this form, with $$\eqalign{ V_{\Lambda,q}^{(j^\prime)}&=0 {\rm \ for\ all\ } q\ne 2,\ j^\prime\ne 0\cr {\cal M}_2&=\bbbz^3\cr V_{\Lambda,2,m}^{(0)}&={1\over 2}\chi_{m,\Lambda}V\cr K_{2,m}^{(0)}&=\const\cr E^{(0)}_{2,m}&=|m_1|+|m_2|+|m_3|\cr \norm V_{\Lambda,2,m}^{(0)}\norm&\le |\lambda|\const\,e^{-\const\,|m|^{1/2}}\norm V\norm\cr \alpha&={1\over 3}\cr R&=\infty\cr }$$ We are making a somewhat stronger hypothesis on the cutoff function $\chi_{m,\Lambda}$ here than in (\eqIIIthr). The standard $C^\infty$ compactly supported functions obey this stronger hypothesis. To be precise let $$\eqalign{ \chi_1(x)&=\cases{0&$x\le 0$\cr e^{-x^{-2}}e^{-(x-1)^{-2}}&$03/2$. Choosing $n$ to be the even integer nearest $|k|^{1/\beta}$ yields $$\eqalign{ \left|\widetilde \chi_3(k)\right| &\le\const^{\!n}\,{n^{3n/2}\over n^{\beta n}}\cr &\le e^{-(\beta-{3\over2})n\ln n+n\ln \const}\cr &\le \const e^{-(\beta-{3\over2})n}\cr &\le \const e^{-\const |k|^{1/\beta}}\ . }$$ We know from the perturbative analysis of [FT2] that two and four point interaction vertices which have internal scales $j'$ higher than the external scale $j$ have to be renormalized. This problem will be treated in a later paper. But we can already state a rigorous result if we limit ourselves to the part of the theory containing only convergent graphs. This part of the model is the sum of all graphs that have no two or four point subgraphs with the scales assigned to their internal lines higher than the scales assigned to their external lines. This was called completely positive power counting case in [Ri]. To isolate this part of the model it suffices to require that, with the exception of $q=2,\ j'=0$, all $V_{\Lambda,q}^{(j^\prime)}$ with $q\le 2$, $j'>j$ be zero.\vfill\eject \thm{\ 2} {\it Let ${\cal V}$ obey {\rm (III.10)} and let $V_{\Lambda,q}^{(j^\prime)}=0$ for $q=1,\ j'>j$ and $q=2,\ 0>j'>j$. Then, if $M$ is large enough, there is an $R_0(M,K_1)>0$ such that the effective potential ${\cal G}$ obeys {\rm (III.10)} for all $|\lambda|<\min(R,R_0)$.} \prf The expansion and bounds are the same as for Theorem 1, with the exception that the vertices are more complicated and that we leave the sums associated with external vertices explicit. We use a tilde to designate index sets and constants that refer to the effective potential. There is, of course the trivial graph in which a $V_{\Lambda,q}^{(j^\prime)}$ gets fed directly into the effective potential as a single vertex graph. All nontrivial contributions get put into $\widetilde V^{(j)}_{\Lambda,p,m}$'s. The new index set $$ \widetilde{\cal M}_{p}=\bbbz^{>0}\cartprod_{v=1}^{2p}\ {\tst\bigcup\limits_{j_v>j}} \ {\tst\bigcup\limits_{q_v=0}^\infty} \ {\tst\bigcup\limits_{m_v\in{\cal M}_{q_v}\atop I\subset\{1,\cdots,2q_v\}}} \{{\rm sector\ assignments\ to \ the\ lines\ }I{\rm\ of\ } V^{(j_v)}_{\Lambda,q_v,m_v}\} $$ Here $v$ labels the external vertices of terms contributing to $\widetilde V^{(j)}_{\Lambda,p,m}\,$. In the event that there are fewer than $2p$ external vertices, the extra $q_v$'s are set to zero. The set $I$ selects the legs of the vertex $v$ that will be internal to the $2p$-point function. We shall denote by $\imath_v$ and $e_v$ the number of legs of the vertex $v$ that end up being internal and external legs, respectively, of $\widetilde V^{(j)}_{\Lambda,p,m}.$ Suppose, for example, that the first seven vertices brought down from the exponent end up being internal, but that the eighth ends up being a $V_{\Lambda,q_1,m_1}^{(j_1)}$ with the legs in $I_1$, a proper subset of $\{1,\cdots,2q_1\}$, being internal and of sectors $\ell_1,\cdots,\ell_{|I_1|}$. Then this term will contribute to a $\widetilde V_{\Lambda,p,m}^{(j)}$ with the second component (out of $2p+1$ components) of $m$ being $(j_1,q_1,m_1,I_1,\ell_1,\cdots,\ell_{|I_1|})$. The new conservation of momentum index is $$ E^{(j)}_{p,m}=E+\sum_v E^{(j_v)}_{q_v,m_v} $$ where $E$, the first of the $2p+1$ components of $m$, is the sum of the conservation of momentum indices of all internal vertices. \vfill\eject The main bound (\eqIIInin) of Theorem 1 is replaced by \def\eqIIIninp{III.9'} $$\eqalign{ &\norm \widetilde V_{\Lambda,p,m}^{(j)}\norm \!\le\! \prod_{\rm internal\atop vertices}\!\!\left( \sum_{j_v>j}\sum_{q_v}\sum_{m_v\in {\cal M}_{q_v}} \hskip-5.5pt C_1^{q}\!\exp\! \left\{E^{(j_v)^\alpha}_{q_v,m_v}M^{(J-j)\alpha}\!\right\} M^{(q_v-{3\over 2})(j_v-j)}K_{q_v,m_v}^{(j_v)} \norm V_{\Lambda,q_v,m_v}^{(j_v)}\norm\!\right)\cr &\hskip1.5truein\left(\prod_{\rm external\atop vertices} \norm V_{\Lambda,q_v,m_v}^{(j_v)}\norm\right) M^{-5j(n-1)/2}M^{{3\over 2}j(\Sigma 2q_v-2p)/2}K_2^{\Sigma 2q_v-2p} K_3^{\Sigma 2q_v}\cr }\eqn{\eqIIIninp}$$ The constant $K_2$ includes the constants arising from bounding the propagator and summing or taking the supremum of $\rho^{(j,\ell)}(\Delta,\Delta^\prime)^{-\gamma+4}$. The constant $K_3$ includes the combinatorial factors associated with propagators and external legs. The latter include a factor of two for deciding whether or not a leg contracted to the exponent, a factor of two to decide which leg was the target leg, in the event that there was contraction to the exponent (note that $q\le 2^q$) and a factor of $(2\times 3^{t+1})^2$ from the Taylor expansion. The differences between $K_2$ and $K_3$ are that the former applies only to propagators, i.e. internal legs, while the latter is independent of $M$. Moving around the powers of $M$ $$\eqalign{ &M^{-5j(n-1)/2}M^{{3\over 2}j(\Sigma 2q_v-2p)/2} \prod_{\rm internal\atop vertices}M^{(q_v-{3\over 2})(j_v-j)}\cr &=M^{{1\over 2}(5-3p)j} \prod_{\rm internal\atop vertices}M^{(q_v-{3\over 2})(j_v-j)} M^{-{1\over 2}(5-3q_v)j} \prod_{\rm external\atop vertices}M^{-{1\over 2}(5-3q_v)j}\cr &=M^{{1\over 2}(5-3p)j} \prod_{\rm internal\atop vertices}M^{{1\over 2}(2-q_v)(j_v-j)} M^{-{1\over 2}(5-3q_v)j_v} \prod_{\rm external\atop vertices}M^{-{1\over 2}(5-3q_v)j}\cr }$$ we end up with $$\eqalign{ \norm \widetilde V_{\Lambda,p,m}^{(j)}\norm &\le \prod_{\rm internal\atop vertices} \Bigg(\sum_{j_v>j}\sum_{q_v}\sum_{m_v\in {\cal M}_{q_v}} M^{{1\over 2}(2-q_v)(j_v-j)}M^{-{1\over 2}(5-3q_v)j_v} \exp\! \left\{E^{(j_v)^\alpha}_{q_v,m_v}M^{(J-j)\alpha}\!\right\}\cr &\hskip2.5truein (C_1K_2^2K_3^2)^{q_v} K_{q_v,m_v}^{(j_v)}\norm V_{\Lambda,q_v,m_v}^{(j_v)}\norm\Bigg)\cr &\hskip.5truein\left(\prod_{\rm external\atop vertices} M^{-{1\over 2}(5-3q_v)j}K_2^{\imath_v}K_3^{2q_v}\norm V_{\Lambda,q_v,m_v}^{(j_v)}\norm\right) M^{{1\over 2}(5-3p)j}\cr }$$ Now summing over $m\in\widetilde{\cal M}_{p}$ entails, for each external vertex, a factor of two per line (to decide whether it is internal or not), a sum over $j_v$ and $m_v$ and a sum over sector assignments to the lines of $V_{\Lambda,q_v,m_v}^{(j_v)}$ that ended up being internal to $\widetilde V_{\Lambda,p,m}^{(j)}$. Including a factor $\widetilde K_{p,m}^{(j)}$ accounts for a sum over sector assignments to the lines that ended up external. So there is a sum over assignments to all legs and, since $$ E^{(j)^\alpha}_{q,m} \le\sum_{{\rm all \atop vertices}}E^{(j_v)^\alpha}_{q_v,m_v}\ , $$ the left hand side of (\eqIIItena) for $\widetilde V_{\Lambda,p,m}^{(j)}$ is bounded by $$\eqalign{ &M^{-{1\over2}(5-3p)j}\sum_{m\in\widetilde{\cal M}_{p}} \exp\left\{\sum_{\imath=J}^{j-1} \left(E^{(j)^\alpha}_{q,m}M^{J-\imath}\right)\right\} \widetilde K_{p,m}^{(j)} \norm \widetilde V_{\Lambda,p,m}^{(j)}\norm\cr &\hskip.25truein\le \prod_{\rm all\atop vertices}\Bigg(\sum_{j_v>j}\sum_{q_v}\sum_{m_v\in {\cal M}_{q_v}} M^{{1\over 2}(2-q_v)(j_v-j)}M^{-{1\over 2}(5-3q_v)j_v} \exp\left\{\sum_{\imath=J}^{j} \left(E^{(j_v)^\alpha}_{q_v,m_v}M^{J-\imath}\right)\right\}\cr &\hskip3.5truein K_2^{\imath_v}{(2C_1K_3^2)}^{q_v}K_{q_v,m_v}^{(j_v)} \norm V_{\Lambda,q_v,m_v}^{(j_v)}\norm\Bigg)\cr &\hskip.25truein\le \prod_{\rm all\atop vertices} \left(\sum_{j_v>j}\sum_{q_v} M^{{1\over 2}(2-q_v)(j_v-j)}K_1K_2^{\imath_v} {(2C_1K_3^2)}^{q_v}|\lambda|^{q_v/2}\right)\cr &\hskip.25truein\le |\lambda|^{p/2}\prod_{\rm all\atop vertices} \left(\sum_{j_v>j}\sum_{q_v}M^{-{1\over 4}(j_v-j)} M^{{1\over 4}(2-q_v)}K_1K_2^{\imath_v}{(2C_1K_3^2)}^{q_v} |\lambda|^{\imath_v/4}\right)\cr &\hskip.25truein\le |\lambda|^{p/2}\prod_{\rm all\atop vertices} K_1M^{1/2}|\lambda|^{1/8}\left(\sum_{j_v>j}\sum_{q_v} M^{-{1\over 4}(j_v-j)}M^{-q_v/4}K_2^{\imath_v}{(2C_1K_3^2)}^{q_v} |\lambda|^{\imath_v/8}\right)\cr }$$ For the second inequality we used $j_v>j$. For the third we also used $q_v\ge 3$. Since $C_1$ and $K_3$ are independent of $M$ we can choose $M$ so that $ 2C_1K_3^2\le M^{1/8} $. Then, if $|\lambda|^{1/8}\le K_2^{-1}$, $$\eqalign{ M^{-{1\over 2}(5-3p)j}\sum_{m\in\widetilde{\cal M}_{p}} &\exp\left\{\sum_{\imath=J}^{j-1} \left(E^{(j)^\alpha}_{q,m}M^{J-\imath}\right)\right\} \widetilde K_{p,m}^{(j)} \norm \widetilde V_{\Lambda,p,m}^{(j)}\norm \cr &\le |\lambda|^{p/2}\prod_{\rm all\atop vertices} K_1M^{1/2}|\lambda|^{1/8}\left(\sum_{j_v>j}\sum_{q_v} M^{-{1\over 4}(j_v-j)}M^{-q_v/8}\right)\cr &\le |\lambda|^{p/2}\prod_{\rm all\atop vertices}K_1M^{1/2}\, {M^{-{1\over 4}}\over 1-M^{-{1\over 4}}}\, {M^{-{5\over 8}}\over 1-M^{-{1\over 8}}}\, |\lambda|^{1/8}\cr &\le |\lambda|^{p/2} }$$ provided $|\lambda|$ is small enough. \endproof Using Theorem 2 inductively, we conclude that the sum over all scales of the completely convergent graphs contributing to any given Green's function is analytic in $\lambda$ at $\lambda=0$. %\vfill\eject \vskip.25truein \noindent{\subchfont References} \vskip.1truein \item{[BG]} G. Benfatto and G. 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