\begin{filecontents}{xspace.sty} %% %% This is file `xspace.sty', %% generated with the docstrip utility. %% %% The original source files were: %% %% xspace.dtx (with options: `package') %% %% IMPORTANT NOTICE: %% %% This file is a generated file from the sources of the `tools' bundle %% in the LaTeX2e distribution. %% %% For the copyright notice see the source file(s). %% %% You are not allowed to modify this file. %% %% You are allowed to distribute this file if and only if %% it is distributed with the corresponding source files in %% the `tools' bundle. %% %% For the copying and distribution conditions of the source files, %% see the file readme.txt distributed with the tools bundle. %% %% File: xspace.dtx Copyright (C) 1991-1996 David Carlisle \NeedsTeXFormat{LaTeX2e} \ProvidesPackage{xspace} [1996/05/17 v1.04 Space after command names (DPC)] \DeclareRobustCommand\xspace{\futurelet\@let@token\@xspace} \def\@xspace{% \ifx\@let@token\bgroup\else \ifx\@let@token\egroup\else \ifx\@let@token\/\else \ifx\@let@token\ \else \ifx\@let@token~\else \ifx\@let@token.\else \ifx\@let@token!\else \ifx\@let@token,\else \ifx\@let@token:\else \ifx\@let@token;\else \ifx\@let@token?\else \ifx\@let@token/\else \ifx\@let@token'\else \ifx\@let@token)\else \ifx\@let@token-\else \space \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi} \endinput %% %% End of file `xspace.sty'. \end{filecontents} \begin{filecontents}{lutypaper.sty} % ----------------------------------------------------------------------------- % Page style settings % ----------------------------------------------------------------------------- \setlength{\textwidth}{6in} \setlength{\oddsidemargin}{0.25in} \setlength{\evensidemargin}{0.25in} \setlength{\topmargin}{-0.6in}%-0.40in \setlength{\headheight}{0.3in} \setlength{\headsep}{0.4in} \setlength{\textheight}{8.4in}%8.4in %\setlength{\footheight}{0.2in} \setlength{\footskip}{0.7in} \renewcommand{\baselinestretch}{1.1} % line spacing \setlength{\footnotesep}{.15in} \setlength{\parskip}{3pt plus 0.2pt} % ---------------------------------------------------------------------------- % Draft mode % ---------------------------------------------------------------------------- \newcounter{papermode} \newcommand{\draft}{\setcounter{papermode}{1}} % ----------------------------------------------------------------------------- % Title page % ----------------------------------------------------------------------------- \renewenvironment{titlepage}{% \setcounter{page}{0}% reset page # \pagestyle{empty}% no page number \renewcommand{\thefootnote}{\fnsymbol{footnote}}}% use symbols % {\newpage% send it out \renewcommand{\thefootnote}{\arabic{footnote})}% use numbers \setcounter{footnote}{0}% reset footnote # \setcounter{page}{1}}% reset page # % preprint numbers \newcommand{\preprint}[1]{% \begin{flushright}% #1% \end{flushright}} % title \renewcommand{\title}[1]{% \bigskip% \begin{center}% \Large\bf #1% \end{center}% \vskip .2in} % author \renewcommand{\author}[1]{% {\begin{center} #1 \end{center}}} % address \newcommand{\address}[1]{\vspace{-1.7em}\vspace{0pt} {\begin{center} \it #1 \end{center}}} \renewenvironment{abstract}{% \noindent% \begin{center}% {\bf Abstract}\\\medskip% \begin{minipage}[t]{5.3in}}% % {\end{minipage}% \end{center}} % pacs numbers \newcommand{\pacs}[1]{% {\rightline{PACS #1}}} % date \renewcommand{\date}[1]{% \begin{center}% #1% \end{center}} % ----------------------------------------------------------------------------- % Sectioning style % ----------------------------------------------------------------------------- % Reduce space in section headings \renewcommand{\@seccntformat}[1]{% {\csname the#1\endcsname}\ \ } % Section \renewcommand{\section}{% \@startsection{section}{1}{\z@}% {-3.5ex plus -1ex minus -.2ex}% {2.3ex plus.2ex}% {\centering\normalsize\bfseries}} %\renewcommand{\sec}[2][]{% optional argument is label % \ifempty#1% no optional argumet % \section{#2}% just start section % \else\section{#2\label{sec:#1}}}% label section \newcommand{\Sec}[1]{{Sec.~\ref{sec:#1}}} % Subsection \renewcommand{\subsection}{\@startsection{subsection}{2}{0pt}% {-3.25ex plus -1ex minus -.2ex}% {1.5ex plus .2ex}% {\centering\normalsize\itshape}} % Start appendices % % Example: % % \startappendices % \section{Appendix A: ...} % ... % \section{Appendix B: ...} % ... % etc. % \newcommand{\startappendices}{% \setcounter{equation}{0}% \setcounter{section}{1}% \setcounter{subsection}{1}% \renewcommand{\thesection}{\Alph{section}}} \newcommand\fakesection{\@startsection {section}{1}{\z@}% {-3.5ex \@plus -1ex \@minus -.2ex}% {2.3ex \@plus.2ex}% {\centering\normalsize\bfseries}} % Appendix % % Do *not* use with \startappendices! \newcounter{appendixcount} \renewcommand{\appendix}[1]{% \ifnum\value{appendixcount} = 0% first appendix %\newpage% \setcounter{equation}{0}% \setcounter{section}{1}% \setcounter{subsection}{1}% \renewcommand{\thesection}{\Alph{section}}% \fi% \stepcounter{appendixcount} \section*{#1}} % ----------------------------------------------------------------------------- % References % ----------------------------------------------------------------------------- \newenvironment{references}{% %\newpage% start new page \begin{thebibliography}{99}% \frenchspacing}% use tighter spacing % {\end{thebibliography}} % ----------------------------------------------------------------------------- % Equation numbering % ----------------------------------------------------------------------------- % Turn on equation numbering by section \makeatletter% allow access to internal LaTeX commands \renewcommand{\theequation}{\thesection.\arabic{equation}}% \@addtoreset{equation}{section}% \makeatother% turn off access to internal LaTeX commands % ----------------------------------------------------------------------------- % Citation abbreviations % ----------------------------------------------------------------------------- \newcommand{\cites}[2]{[#1--#2]} % ----------------------------------------------------------------------------- % Draft mode stuff % ----------------------------------------------------------------------------- \def\endignore{} \def\ignore #1\endignore{} \end{filecontents} \begin{filecontents}{mydefs.sty} % -------------------------------------------------------------------------- % Text abbreviations % -------------------------------------------------------------------------- \input{xspace.sty} \input{equations.sty}% by Charles Karney % General \newcommand{\lhs}{left-hand\ side\xspace} \newcommand{\rhs}{right-hand\ side\xspace} \newcommand{\ie}{\textit{i.e.}\xspace} \newcommand{\eg}{\textit{e.g.}\xspace} \newcommand{\etc}{\textit{etc.}\xspace} \newcommand{\etal}{\textit{et.~al.}\xspace} \newcommand{\apriori}{\textit{\=B9a priori}\xspace} \newcommand{\naive}{na\"\i ve\xspace} \newcommand{\naively}{na\"\i vely\xspace} \newcommand{\Naive}{Na\"\i ve\xspace} \newcommand{\Naively}{Na\"\i vely\xspace} % Quantum mechanics \newcommand{\vev}{va\-cuum\ ex\-pec\-ta\-tion\ value\xspace} \newcommand{\vevs}{va\-cuum\ ex\-pec\-ta\-tion\ values\xspace} \newcommand{\Vev}{Va\-cuum\ ex\-pec\-ta\-tion\ value\xspace} \newcommand{\Vevs}{Va\-cuum\ ex\-pec\-ta\-tion\ values\xspace} \newcommand{\SE}{Schr\"o\-ding\-er\ equa\-tion\xspace} % Group theory \newcommand{\rep}{rep\-re\-sen\-ta\-tion\xspace} \newcommand{\reps}{rep\-re\-sen\-ta\-tions\xspace} \newcommand{\Rep}{Rep\-re\-sen\-ta\-tion\xspace} \newcommand{\Reps}{Rep\-re\-sen\-ta\-tions\xspace} \newcommand{\trans}{trans\-for\-ma\-tion\xspace} \newcommand{\transs}{trans\-for\-ma\-tions\xspace} \newcommand{\Trans}{Trans\-for\-ma\-tion\xspace} \newcommand{\Transs}{Trans\-for\-ma\-tions\xspace} % Effective Field Theory \newcommand{\cpt}{chi\-ral\ per\-tur\-ba\-tion\ theory\xspace} \newcommand{\lag}{la\-gran\-gian\xspace} % Standard Model \newcommand{\ew}{elec\-tro\-weak\xspace} \newcommand{\FCNC}{fla\-vor-chan\-ging neu\-tral cur\-rent\xspace} \newcommand{\FCNCs}{fla\-vor-chan\-ging neu\-tral cur\-rents\xspace} \newcommand{\tHooft}{'t~Hooft\xspace} % \newcommand{\CP}{{\it CP}\xspace} % \newcommand{\CPT}{{\it CPT}\xspace} \newcommand{\CP}{$C\!P$\xspace} \newcommand{\CPT}{$C\!P\!T$\xspace} % SUSY \newcommand{\Kahler}{K\"ah\-ler\xspace} \newcommand{\OR}{O'Rai\-fear\-taigh\xspace} \newcommand{\spot}{su\-per\-po\-ten\-tial\xspace} \newcommand{\susy}{su\-per\-sym\-me\-try\xspace} \newcommand{\Susy}{Su\-per\-sym\-me\-try\xspace} \newcommand{\susc}{su\-per\-sym\-me\-tric\xspace} \newcommand{\Susc}{Su\-per\-sym\-me\-tric\xspace} \newcommand{\MSSM}{mi\-ni\-mal \susc stan\-dard mo\-del\xspace} % -------------------------------------------------------------------------- % Hyphenation % -------------------------------------------------------------------------- \hyphenation{ba-ry-on ba-ry-ons} \hyphenation{la-gran-gi-an la-gran-gi-ans} \hyphenation{mo-del mo-dels} \hyphenation{mo-men-ta mo-men-tum} \hyphenation{par-ti-cle par-ti-cles} \hyphenation{phy-si-cal phy-sics} % use to "comment out" text \def\endignore{} \def\ignore #1\endignore{} % -------------------------------------------------------------------------- % Equation abbreviations % -------------------------------------------------------------------------- % reference equations \newcommand{\eq}[1]{(\ref{eq:#1})} % plain equation reference \newcommand{\Eq}[1]{Eq.~\eq{#1}} % always capitalize "Eq" \newcommand{\Eqs}[1]{Eqs.~\eq{#1}} % label equation \newcommand{\eql}[1]{\label{eq:#1}} % inline equation \newcommand{\beq} {\begin{eqalignno}% works even if equation has no & } \newcommand{\eeq}{\end{eqalignno}} \newcommand{\bal}{\begin{eqalign}} \newcommand{\eal}{\end{eqalign}} % begin unnumbered equation (or array) \newcommand{\beqn}{\begin{eqalignno*}} \newcommand{\eeqn}{\end{eqalignno*}} % eqalign abbreviation \newcommand{\bea}{\begin{eqalign}} \newcommand{\eea}{\end{eqalign}} % -------------------------------------------------------------------------- % Citation abbreviations % -------------------------------------------------------------------------- % reference \newcommand{\Ref}[1]{{Ref.~\cite{#1}}} \newcommand{\Refs}[1]{{Refs.~\cite{#1}}} % -------------------------------------------------------------------------- % Figures % -------------------------------------------------------------------------- % change figure heading style (by J. Shultis) % \long\def\@makecaption#1#2{\vskip 10\p@ % \setbox\@tempboxa\hbox{{\bf #1.} #2}% bold and "." instead of ":" % \ifdim \wd\@tempboxa >\hsize % {\bf #1.} #2\par% bold and "." instead of ":" % \else % \hbox to\hsize{\hfil\box\@tempboxa\hfil}% % \fi} % % % change "Figure" to "Fig" % \newcommand{\thefigurename}{Fig.} % \def\fnum@figure{\thefigurename\ \thefigure} % % % caption % \newcommand{\capt}[1]{% % \begin{minipage}[t]{5in}% % \caption{#1}% % \end{minipage}% % } % % % include eps figure (first optional argument is label} % \newcommand{\epsfig}[3][]{% % \begin{figure}[htbp]% % \begin{center}% % \centerline{\epsfbox{#2}}% % \capt{#3}% % \end{center}% % \end{figure}% % \ifempty#1\else% optional argument present % \label{fig:#1}\fi% % } % % % reference figures % % reference equations % \newcommand{\fig}[1]{\ref{fig:#1}} % plain figure reference % \newcommand{\Fig}[1]{Fig.~\fig{#1}} % always capitalize "Fig" % \newcommand{\Figs}[1]{Figs.~\fig{#1}} % -------------------------------------------------------------------------- % Math relations % -------------------------------------------------------------------------- \newcommand{\too}{\longrightarrow} \newcommand{\mapstoo}{\longmapsto} \newcommand{\df}{\mathrel{:=3D}} \newcommand{\fd}{\mathrel{=3D:}} % -------------------------------------------------------------------------- % Math functions % -------------------------------------------------------------------------- \newcommand{\Prob}{\mathop{\rm Prob}} \newcommand{\tr}{\mathop{\rm tr}} \newcommand{\Tr}{\mathop{\rm Tr}} \newcommand{\Det}{\mathop{\rm Det}} \newcommand{\Pf}{\mathop{\rm Pf}} \newcommand{\sgn}{\mathop{\rm sgn}} \newcommand{\diag}{\mathop{\rm diag}} \renewcommand{\Re}{\mathop{\rm Re}} \renewcommand{\Im}{\mathop{\rm Im}} % -------------------------------------------------------------------------- % Groups % -------------------------------------------------------------------------- \newcommand{\Group}[2]{{\hbox{{\itshape{#1}}($#2$)}}} \newcommand{\U}[1]{\Group{U\kern0.05em}{#1}} \newcommand{\SU}[1]{\Group{SU\kern0.1em}{#1}} \newcommand{\SL}[1]{\Group{SL\kern0.05em}{#1}} \newcommand{\Sp}[1]{\Group{Sp\kern0.05em}{#1}} \newcommand{\SO}[1]{\Group{SO\kern0.1em}{#1}} % -------------------------------------------------------------------------- % Math accents % -------------------------------------------------------------------------- \newcommand{\scr}[1]{{\mathcal{#1}}} \newcommand{\sub}[1]{^{\vphantom{\dagger}}_{#1}} \newcommand{\rsub}[1]{\mathstrut_{\hbox{\scriptsize #1}}} \newcommand{\rsup}[1]{\mathstrut^{\hbox{\scriptsize #1}}} \newcommand{\twi}{\widetilde} \newcommand{\mybar}[1]% {{\kern 0.8pt\overline{\kern -0.8pt#1\kern -0.8pt}\kern 0.8pt}} \newcommand{\sla}[1]% {{\raise.15ex\hbox{$/$}\kern-.57em #1}}% Feynman slash \newcommand{\roughly}[1]% {{\mathrel{\raise.3ex\hbox{$#1$\kern-.75em\lower1ex\hbox{$\sim$}}}}} % -------------------------------------------------------------------------- % Other math structures % -------------------------------------------------------------------------- % bras, kets, etc. \newcommand{\bra}[1]{\langle #1 |} \newcommand{\ket}[1]{| #1 \rangle} \newcommand{\braket}[2]{\langle #1 | #2 \rangle} \newcommand{\Bra}[1]{\left\langle #1 \left|} \newcommand{\Ket}[1]{\right| #1 \right\rangle} \newcommand{\Tvev}[1]{\langle 0 | T #1 | 0 \rangle} \newcommand{\avg}[1]{\langle #1 \rangle} \newcommand{\Avg}[1]{\left\langle #1 \right\rangle} \newcommand{\nop}[1]{:\kern-.3em#1\kern-.3em:} \newcommand{\vacbra}{{\bra 0}} \newcommand{\vac}{{\ket 0}} % rough comparisons \newcommand{\lsim}{\mathrel{\roughly<}} \newcommand{\gsim}{\mathrel{\roughly>}} % integrals \newcommand{\myint}{\int\mkern-5mu} \newcommand{\ddx}[2]{d^{#1}#2\,} \newcommand{\ddp}[2]{\frac{d^{#1}#2}{(2\pi)^{#1}}\,} % derivatives \newcommand{\del}{\partial} \newcommand{\ddel}[2]{\frac{\partial{#1}}{\partial{#2}}} % Math constructs ---------------------------------------------------------- \newcommand{\sfrac}[2]{{\textstyle\frac{#1}{#2}}} \newcommand{\hc}{\hbox{\rm h.c.}} % Abbreviations ------------------------------------------------------------ % Greek letters (first two letters of name) \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\epsilon} \newcommand{\ze}{\zeta} \newcommand{\et}{\eta} \renewcommand{\th}{\theta} \newcommand{\Th}{\Theta} \newcommand{\vth}{\vartheta} \newcommand{\ka}{\kappa} \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} \newcommand{\rh}{\rho} \newcommand{\si}{\sigma} \newcommand{\Si}{\Sigma} \newcommand{\ta}{\tau} \newcommand{\Up}{\Upsilon} \newcommand{\ph}{\phi} \newcommand{\vph}{\varphi} \newcommand{\Ph}{\Phi} \newcommand{\ps}{\psi} \newcommand{\Ps}{\Psi} \newcommand{\om}{\omega} \newcommand{\Om}{\Omega} % Squares ------------------------------------------------------------------ % draw box with width #1 pt and line thickness #2 pt \newcommand{\drawsquare}[2]{\hbox{% \rule{#2pt}{#1pt}\hskip-#2pt% left vertical \rule{#1pt}{#2pt}\hskip-#1pt% lower horizontal \rule[#1pt]{#1pt}{#2pt}}\rule[#1pt]{#2pt}{#2pt}\hskip-#2pt% upper horizontal \rule{#2pt}{#1pt}}% right vertical % Young tableaux \newcommand{\Yfund}{\drawsquare{7}{0.6}}% fundamental \newcommand{\Ysymm}{\drawsquare{7}{0.6}\hskip-0.6pt% \drawsquare{7}{0.6}}% symmetric second rank tensor \newcommand{\Yasymm}{\drawsquare{7}{0.6}\hskip-7.6pt% \raisebox{7pt}{\drawsquare{7}{0.6}}}% antisymmetric second rank % small Young Tableaux \newcommand{\smYfund}{\drawsquare{4}{0.6}}% fundamental \newcommand{\smYsymm}{\drawsquare{4}{0.6}\hskip-0.6pt% \drawsquare{4}{0.6}}% symmetric second rank tensor \newcommand{\smYasymm}{\drawsquare{4}{0.6}\hskip-4.6pt% \raisebox{4pt}{\drawsquare{4}{0.6}}}% small antisymmetric second rank % Delambertian \newcommand{\Dlamb}{{\mathop{\drawsquare{7}{0.6}}}} % -------------------------------------------------------------------------- % Units % -------------------------------------------------------------------------- \newcommand{\eV}{{\hbox{~eV}}} \newcommand{\keV}{{\hbox{~keV}}} \newcommand{\MeV}{{\hbox{~MeV}}} \newcommand{\GeV}{{\hbox{~GeV}}} \newcommand{\TeV}{{\hbox{~TeV}}} \newcommand{\cm}{{\hbox{~cm}}} \newcommand{\degree}{^{\circ}} % -------------------------------------------------------------------------- % Journals % -------------------------------------------------------------------------- \newcommand{\Journal}[4]{\textit{#1}\ \textbf{#2}, #3 (#4)}% APS style \newcommand{\AP}[3]{\Journal{Ann.\ Phys.}{#1}{#2}{#3}} \newcommand{\CMP}[3]{\Journal{Comm.\ Math.\ Phys.}{#1}{#2}{#3}} \newcommand{\ARNPS}[3]{\Journal{Ann.\ Rev.\ Nucl.\ Part.\ Sci.}{#1}{#2}{#3}} \newcommand{\HPA}[3]{\Journal{Helv.\ Phys.\ Act.}{#1}{#2}{#3}} \newcommand{\IJTP}[3]{\Journal{Int.\ Jour.\ Theor.\ Phys.}{#1}{#2}{#3}} \newcommand{\JETP}[3]{\Journal{JETP}{#1}{#2}{#3}} \newcommand{\JETPL}[3]{\Journal{JETP Lett.}{#1}{#2}{#3}} \newcommand{\JMP}[3]{\Journal{Jour.\ Math.\ Phys.}{#1}{#2}{#3}} \newcommand{\JPA}[3]{\Journal{Jour.\ Phys.}{A#1}{#2}{#3}} \newcommand{\MPLA}[3]{\Journal{Mod.\ Phys.\ Lett.}{A#1}{#2}{#3}} \newcommand{\NC}[3]{\Journal{Nuovo Cim.}{#1}{#2}{#3}} \newcommand{\NPB}[3]{\Journal{Nucl.\ Phys.}{B#1}{#2}{#3}} \newcommand{\NPPS}[3]{\Journal{Nucl.\ Phys.\ Proc.\ Suppl.}{#1}{#2}{#3}} \newcommand{\PA}[3]{\Journal{Physica}{#1A}{#2}{#3}} \newcommand{\PLB}[3]{\Journal{Phys.\ Lett.}{#1B}{#2}{#3}} \newcommand{\PR}[3]{\Journal{Phys.\ Rep.}{#1}{#2}{#3}} \newcommand{\PRD}[3]{\Journal{Phys.\ Rev.}{D#1}{#2}{#3}} \newcommand{\PRC}[3]{\Journal{Phys.\ Rev.}{C#1}{#2}{#3}} \newcommand{\PRL}[3]{\Journal{Phys.\ Rev.\ Lett.}{#1}{#2}{#3}} \newcommand{\PRSL}[3]{\Journal{Proc. Roy. Soc. (London)}{#1}{#2}{#3}} \newcommand{\PRV}[3]{\Journal{Phys.\ Rev.}{#1}{#2}{#3}} \newcommand{\PTP}[3]{\Journal{Prog.\ Theor.\ Phys.}{#1}{#2}{#3}} \newcommand{\RMP}[3]{\Journal{Rev.\ Mod.\ Phys.}{#1}{#2}{#3}} \newcommand{\SJNP}[3]{\Journal{Sov.\ J.\ Nucl.\ Phys.}{#1}{#2}{#3}} \newcommand{\ZETFP}[3] {\Journal{Zh.\ Eksp.\ Teor.\ Fiz.\ Pis'ma}{#1}{#2}{#3}} \newcommand{\ZPC}[3]{\Journal{Zeit.\ Phys.}{C#1}{#2}{#3}} % e-print archives \newcommand{\hepph}[1]{{hep-ph/#1}} \newcommand{\hepth}[1]{{hep-th/#1}} \newcommand{\nuclth}[1]{{nucl-th/#1}} \end{filecontents} \begin{filecontents}{equations.sty} % This file, equations.sty (July 1990), contains a collection of macros % to aid in constructing displayed equations in LaTeX. Written by % % Charles Karney % Plasma Physics Laboratory E-mail: Karney@Princeton.EDU % Princeton University Phone: +1 609 243 2607 % Princeton, NJ 08543-0451 FAX: +1 609 243 2662 % with some ideas and macros borrowed from John Hobby and Stephen Gildea. \def\fileversion{2.0} \def\filedate{93/12/17} \def\docdate {93/08/30} % 93/12/17 -- take out the \showthe % \changes{v2.0}{93/08/30}{Added optional arguments to \yesnumber and % \begin{equation}} % % --------------------USER DOCUMENTATION-------------------- % Use this as a style option, e.g., % \documentstyle[equations]{article} % These probably don't work in conjunction with the leqno option. % This implements the following: % (1) \yesnumber turns on an equation number in an eqnarray* environment % (just as \nonumber turns it off in an eqnarray environment). E.g., % \begin{eqnarray*} % a &=& b \\ % & & + c \\ % & & + d \\ % & & + e \\ % & & + f \\ % & & + g \yesnumber % \end{eqnarray*} % % \yesnumber takes an optional argument that will be used for the % equation number. The equation counter will not be incremented. % \begin{eqnarray*} % & & + f \\ % & & + g \yesnumber[**] % \end{eqnarray*} % % (2) The eqalign environment is just like Plain TeX's \eqalign. E.g., % \begin{equation} % \begin{eqalign} % a &= b, \\ % c &= d. % \end{eqalign} % \end{equation} % Note that \begin{equation} \end{equation} or equivalent is needed. % (3) The eqalignno environment is just like Plain TeX's \eqalignno. E.g., % \begin{eqalignno} % a &= b, \label{foo}\\ % c &= d. \label{bar} % \end{eqalignno} % Note the absence of \begin{equation} \end{equation}. \nonumber can be used % to suppress the equation number. eqalignno* is the same except that the % equation numbers are suppressed (unless a \yesnumber appears). % (4) The eqaligntwo environment is a two-equation per line equivalent of % eqalignno. E.g., % \begin{eqaligntwo} % a &= b, & x &= y, \label{foo} \\ % c &= d, & z &= w. \label{bar} % \end{eqaligntwo} % eqaligntwo* is defined similarly. % (5) The cases environment is just like Plain TeX's \cases. E.g., % \begin{equation} % u(x) = % \begin{cases} % 0, & for $x < 0$, \\ % 1, & for $x \ge 0$. % \end{cases} % \end{equation} % Note the first column is treated as math, the second column as text. % (6) I've borrowed John Hobby's modifications to the eqnarray environment % (to fix up the spacing around the operator), and to the \big, \bigg, etc. % operators (to make them scale with the point size). % (7) I've borrowed Stephen Gildea's subequations environment, and fixed it % so that it ignores spaces after the environment and so that you can refer % both to the overall set of equations and to individual subequations. E.g., % \begin{subequations} \label{foo} % \begin{eqalignno} % a &= b, \label{foo-a} \\ % c &= d, \label{foo-b} % \end{eqalignno} % text text text text ... % \begin{equation} % e = f. \label{foo-c} % \end{equation} % \end{subequations} % Now \ref{foo}, \ref{foo-a}, \ref{foo-b}, \ref{foo-c}, produce 1, 1a, 1b, % 1c. % (8) The equation number is always printed in text mode. Normally an % equation environment would print it in math mode (with the \rm family). % This only makes a difference if \rm in text and math modes gives you % different fonts. % % (8) \begin{equation} can take an optional argument that will be used for the % equation number. The equation counter will not be incremented. % \begin{equation}[$G'$] % 2 + 2 = 4 % \end{equation} % --------------------END OF USER DOCUMENTATION-------------------- % Fix up eqnarray* so that \yesnumber and \nonumber do the obvious things % Incorporate fixes of Peter Vanroose (peter@dit.lth.se) to make \yesnumber % actually work. \newif\if@defeqnsw \@defeqnswtrue % Has the user specified a hard-wired equation number? \newif\if@hardeqn \@hardeqnfalse % This is John Hobby's (hobby@research.att.com) version to fix up the spacing. \def\eqnarray{\stepcounter{equation}\let\@currentlabel=\theequation \if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi \tabskip\@centering\let\\=\@eqncr $$\halign to \displaywidth\bgroup\hfil\global\@eqcnt\z@ $\displaystyle\tabskip\z@{##}$&\global\@eqcnt\@ne \hfil$\displaystyle{{}##{}}$\hfil &\global\@eqcnt\tw@ $\displaystyle{##}$\hfil \tabskip\@centering&\llap{##}\tabskip\z@\cr} \@namedef{eqnarray*}{\@defeqnswfalse\global\@eqnswfalse\eqnarray} \@namedef{endeqnarray*}{\endeqnarray} \def\yesnumber{\@ifnextchar[{\@yesnumber}{\global\@eqnswtrue}} % ] % Set up to use an hardwired equation number in an eqnarray \def\@yesnumber[#1]{\global\@eqnswtrue \global\@hardeqntrue\let\ref=\expandableref\xdef\@hardeqndef{#1}} % This is taken out of \@@eqncr because TeX \if nesting was misdesigned - ews \def\make@eqnnum{\if@hardeqn{\def\theequation{ \@hardeqndef}\@eqnnum}\global\@har deqnfalse\else \@eqnnum\stepcounter{equation}\fi} \def\@@eqncr{\let\@tempa\relax \global\advance\@eqcnt by \@ne \ifcase\@eqcnt \def\@tempa{& & & &}\or \def\@tempa{& & &}\or \def\@tempa{& &}\or \def\@tempa{&}\else\fi \@tempa \if@eqnsw\make@eqnnum\fi \if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi \global\@eqcnt\z@\cr} % Several formulas like \eqalign (to go inside \begin{equation} % \end{equation} or $$ $$. \def\@eqnacr{{\ifnum0=`}\fi\@ifstar{\@yeqnacr}{\@yeqnacr}} \def\@yeqnacr{\@ifnextchar [{\@xeqnacr}{\@xeqnacr[\z@]}} \def\@xeqnacr[#1]{\ifnum0=`{\fi}\cr \noalign{\vskip\jot\vskip #1\relax}} \def\eqalign{\null\,\vcenter\bgroup\openup1\jot \m@th \let\\=\@eqnacr \ialign\bgroup\strut \hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil\crcr} \def\endeqalign{\crcr\egroup\egroup\,} % \cases \def\cases{\left\{\,\vcenter\bgroup\normalbaselines\m@th \let\\=\@eqnacr \ialign\bgroup$##\hfil$&\quad##\hfil\crcr} \def\endcases{\crcr\egroup\egroup\right.} % Several formulas like \eqalignno (doesn't need the $$ $$) \def\eqalignno{\stepcounter{equation}\let\@currentlabel=\theequation \if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi \let\\=\@eqncr $$\displ@@ \tabskip\@centering \halign to \displaywidth\bgroup \global\@eqcnt\@ne\hfil $\@lign\displaystyle{##}$\tabskip\z@skip&\global\@eqcnt\tw@ $\@lign\displaystyle{{}##}$\hfil\tabskip\@centering& \llap{\@lign##}\tabskip\z@skip\crcr} \def\endeqalignno{\@@eqncr\egroup \global\advance\c@equation\m@ne$$\global\@ignoretrue} \def\displ@@{\global\dt@ptrue\openup\jot\m@th % Remove \penalty from \displ@y \everycr{\noalign{\ifdt@p \global\dt@pfalse \vskip-\lineskiplimit \vskip\normallineskiplimit \fi}}} % Here's the eqalignno* environment: \@namedef{eqalignno*}{\@defeqnswfalse\eqalignno} \@namedef{endeqalignno*}{\endeqalignno} % Two formulas like \eqalignno (doesn't need the $$ $$) \def\eqaligntwo{\stepcounter{equation}\let\@currentlabel=\theequation \if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi \let\\=\@eqncr $$\displ@@ \tabskip\@centering \halign to \displaywidth\bgroup \global\@eqcnt\m@ne\hfil $\@lign\displaystyle{##}$\tabskip\z@skip&\global\@eqcnt\z@ $\@lign\displaystyle{{}##}$\hfil\qquad&\global\@eqcnt\@ne \hfil$\@lign\displaystyle{##}$&\global\@eqcnt\tw@ $\@lign\displaystyle{{}##}$\hfil\tabskip\@centering& \llap{\@lign##}\tabskip\z@skip\crcr} \def\endeqaligntwo{\@@eqncr\egroup \global\advance\c@equation\m@ne$$\global\@ignoretrue} \@namedef{eqaligntwo*}{\@defeqnswfalse\eqaligntwo} \@namedef{endeqaligntwo*}{\endeqaligntwo} % subequations %%% File: subeqn.sty %%% The subequations environment %%% % % Within the subequations environment, the only change is that % equations are labeled differently. The number stays the same, % and lower case letters are appended. For example, if after doing % three equations, numbered 1, 2, and 3, you start a subequations % environmment and do three more equations, they will be numbered % 4a, 4b, and 4c. 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(Makes a difference when % text Roman is not the same as textfont0 \def\@eqnnum{\mbox{\rm (\theequation)}} % modify \begin{equation} \end{equation} so it will take an optional arg. % \def\equation{$$ % $$ BRACE MATCHING HACK \@ifnextchar[% ] BRACE MATCHING HACK {\let\protect=\noexpand\@equation}% {\refstepcounter{equation}}} \def\expandableref#1{\@ifundefined{r@#1}{??}% {\expandafter\expandafter\expandafter\@car\csname r@#1\endcsname\@nil}} \let\@saveref=\ref \def\@equation[#1]{% \let\ref=\expandableref \edef\theequation{#1}% %\show\theequation \let\@currentlabel=\theequation \let\ref=\@saveref} \endinput \end{filecontents} \documentstyle[lutypaper,mydefs,12pt]{article} % ---------------------------------------------------------------------------- % Private definitions % ---------------------------------------------------------------------------- % Young tableaux \newcommand{\fund}{\drawsquare{6.5}{0.4}}% fundamental \newcommand{\symm}{\drawsquare{6.5}{0.4}\hskip-0.4pt% \drawsquare{6.5}{0.4}}% symmetric second rank tensor \newcommand{\asymm}{\raisebox{-3pt}{\drawsquare{6.5}{0.4}\hskip-6.9pt% \raisebox{6.5pt}{\drawsquare{6.5}{0.4}}}}% antisymmetric second rank \newcommand{\asymmfour}{\raisebox{-10pt}{\drawsquare{6.5}{0.4}}\hskip-6.9pt% \raisebox{-3.5pt}{\drawsquare{6.5}{0.4}}\hskip-6.9pt% \raisebox{3pt}{\drawsquare{6.5}{0.4}}\hskip-6.9pt% \raisebox{9.5pt}{\drawsquare{6.5}{0.4}}}% antisymmetric fourth rank \newcommand{\bb}[1]{\hbox{\bf #1}} \newcommand{\vu}{\avg{\bar{U}}} \begin{document} % ---------------------------------------------------------------------------- % Title page % ---------------------------------------------------------------------------- \begin{titlepage} \preprint{SLAC-PUB-7714\\ UMDHEP 98-68\\ UCB-PTH-97/62\\ LBNL-41148} \title{Composite Quarks and Leptons from\\\medskip Dynamical Supersymmetry Breaking\\\medskip without Messengers} \author{Nima Arkani-Hamed} \address{Stanford Linear Accelerator Center, Stanford University\\ Stanford, California 94309, USA\\ {\tt nima@slac.stanford.edu}} \author{Markus A. Luty\footnote{Sloan Fellow}} \address{Department of Physics, University of Maryland\\ College Park, Maryland 20742, USA\\ {\tt mluty@physics.umd.edu}} \author{John Terning} \address{Department of Physics, University of California\\ Berkeley, California 94720, USA\\ {\tt terning@alvin.lbl.gov}} \begin{abstract} We present new theories of dynamical \susy breaking in which the strong interactions that break \susy also give rise to composite quarks and leptons with naturally small Yukawa couplings. In these models, \susy breaking is communicated directly to the composite fields without ``messenger'' interactions. The compositeness scale can be anywhere between $10\TeV$ and the Planck scale. These models can naturally solve the \susc flavor problem, and generically predict sfermion mass unification independent from gauge unification. % % with the following features: % quarks and leptons from the first two generations are composite, % with Yukawa couplings arising from higher-dimension operators at % the \susy breaking scale; % \susy breaking dynamics near the GUT scale; % gaugino masses smaller than scalar masses; % unification of composite scalar masses without grand unification. % We also construct economical models of the ``more minimal'' type with % two composite generations. \end{abstract} \date{January 12, 1998} \end{titlepage} % ---------------------------------------------------------------------------- \section{Introduction} % ---------------------------------------------------------------------------- \Susy is arguably the most attractive framework for physics beyond the standard model, but a truly satisfactory and attractive model for \susy breaking has yet to emerge. One reason for dissatisfaction with present models is their ``modular'' structure: \susy is assumed to be broken in some new sector, and the information that \susy is broken is communicated to the observable fields via messenger interactions, which may be either (super)gravity \cite{SUGRA} or standard-model gauge interactions \cite{OldGaMed,NewGaMed}.% % \footnote{There has recently been important progress in simplifying models of gauge-mediated \susy breaking \cite{NewerGaMed}.} % While there is in principle nothing wrong with such modular schemes, it is interesting to ask whether there exist simpler models in which \susy is broken directly in the observable sector. An important obstacle in constructing such a model was pointed out by Dimopoulos and Georgi \cite{DimGeorgi}. They showed that if one assumes $(i)$ the gauge group is that of the standard model; $(ii)$ no higher-dimension operators in the \Kahler potential of the effective Lagrangian; and $(iii)$ tree approximation, then there is always a colored scalar lighter than the down quark. Any realistic model of \susy breaking must contain important effects that do not satisfy one of these assumptions. For example, gravity-mediated models violate $(ii)$, and gauge-mediated models violate $(iii)$. The effects that violate $(ii)$ and $(iii)$ are generally smaller than tree-level renormalizable effects, but the ``modular'' structure of these models guarantees that they are the \emph{leading} effects that communicate \susy breaking to the observable sector. An interesting way to evade the ``no go'' theorem of Dimopoulos and Georgi without introducing modular structure is to make the observable fields composite, in the sense that they couple to new strong dynamics at a scale $\La$ above the weak scale.\footnote{Recently, composite models including supersymmetry breaking (but with a ``modular" structure) have been constructed \cite{ModComp}.} If the strong dynamics also breaks \susy, assumption $(iii)$ will be violated (and the low-energy theory below the scale $\La$ will violate $(ii)$). We therefore look for a theory with a single sector that breaks \susy dynamically and generates composite fermions. More specifically, we have the following scenario in mind. Consider a model that breaks \susy by strong interactions at the scale $\La$, and suppose that the model has an unbroken global symmetry group $G$. If there are $G^3$ anomalies, the theory will have massless composite fermions in the low-energy spectrum to match the anomalies \cite{tHooft}. It is easy to find such models where the standard-model gauge group $G_{\rm SM}$ can be embedded in $G$, either because there are no $G_{\rm SM}^3$ anomalies, or because these anomalies are canceled by ``elementary'' states. In this case, some of the composite fermions will be charged under $G_{\rm SM}$ and may be identified with quarks and leptons. If there is no unbroken $\U1_R$ symmetry, standard model gaugino masses will be generated, suppressed compared to the mass of the composite scalars by a perturbative loop factor, and one must worry about gaugino masses being too small.% % \footnote{This killed the models of \Ref{comptry}, which were motivated by very similar considerations as those described above.} % One possibility is that the composite scalars are heavy enough that the gauginos are sufficiently heavy despite the loop suppression factor. This leads naturally to models with a low compositeness scale and a superpartner spectrum similar to that of the ``more minimal'' models \cite{CoKaNe}. Another alternative is to assume that the loop factor is compensated by a large multiplicity factor. In fact, in order to generate complete composite generations the global symmetry must be quite large, so a large multiplicity factor is hard to avoid. The large number of states also means that the standard-model gauge group is far from being asymptotically free, but the models can still accommodate perturbative gauge coupling unification if the scale $\La$ of non-perturbative composite dynamics is near the unification scale. A large value for $\La$ also helps avoid negative mass-squared terms for standard-model scalars, as we will explain in the text. The composite nature of some of the standard-model fermions can also help in understanding the small Yukawa couplings for the first two generations. If there are no Yukawa couplings generated by the strong dynamics, all Yukawa couplings must arise from flavor-dependent higher-dimension operators in the fundamental theory suppressed by powers of a scale $M > \La$. In the low-energy theory, these will become Yukawa couplings suppressed by powers of $\La / M$. This class of models makes two interesting generic predictions for the spectrum of superpartner masses. First, the gaugino masses will be lighter than the composite scalars. Second, the composite scalar masses are generated by the strong dynamics, and are therefore invariant under the global symmetry $G$ at the scale $\La$. This means that some or all of the soft masses for the composite fields unify at the scale $\La$. If supersymmetry is discovered, this prediction can be tested if the scalar masses are accurately measured. % ---------------------------------------------------------------------------- \section{New theories of dynamical SUSY breaking} % ---------------------------------------------------------------------------- In this Section, we describe \susc gauge theories that have local minima with dynamical \susy breaking and composite fermions. These models are similar in some ways to the models considered in \Ref{LutyTerning}, but have some features that are more favorable to the kind of model-building we are interested in. The models have gauge and flavor symmetry group \beq \SU{4} \times \SU{N} \times \left[ \SU{N} \times \U1 \times \U1_R \right] \eeq where the group in brackets is a global symmetry group. The matter content is \beq\bal Q &\sim (\fund, \fund) \times (\hbox{\bf 1}; 1, -\sfrac{N}{4} - 1)~, \\ L &\sim (\bar{\fund}, \hbox{\bf 1}) \times (\bar{\fund}; -1, \sfrac{N}{4} + 3 - \sfrac{8}{N})~, \\ \bar{U} &\sim (\hbox{\bf 1}, \bar{\fund}) \times (\fund; 0, \sfrac{8}{N})~, \\ A &\sim (\hbox{\bf 1}, \asymm\,) \times (\hbox{\bf 1}; \sfrac{4}{N-2}, 1)~. \eal\eeq The theory has a tree-level superpotential \beq W = \la L Q \bar{U}. \eeq Here $\la$ is a matrix that can be viewed as an adjoint spurion for the global $[\SU{N}]$ symmetry. Note that there are 4 $\fund$'s and $N$ $\bar{\fund}$'s under the global $[\SU{N}]$, so the theory has a nonzero $\left[ \SU{N} \right]^3$ anomaly for $N \ne 4$. The analysis of this model is somewhat different depending on whether $N$ is even or odd, so we consider both possibilities in turn. % ---------------------------------------------------------------------------- \subsection{Odd $N$ Models} We first consider the case where $N=2n+1$ is odd. %In this case, we write %\beq %N = 2n + 1 %\eeq %to simplify the expressions. If we include the effects of the tree-level superpotential, the theory has a classical moduli space that can be parameterized by the gauge invariants (we indicate the $\left[SU(N)\right]$ quantum numbers) \beq\bal L^4 &\sim \bar{\asymmfour} ~, %\bar{\{ 4 \}}, \quad\hbox{(for\ $N \ge 5$)} \\ A \bar{U}^2 &\sim \asymm ~, %\{ 2 \}, \\ \bar{U}^{N} &\sim \hbox{\bf 1} ~. \eal\eeq with the constraints \beq L^4 \cdot \bar{U}^{N} = 0, \quad L^4 \cdot A\bar{U}^2 =0. \eeq For $N \ge 5$, the classical moduli space has two branches: $\avg{\bar{U}^{N}} \ne 0$ with $\avg{L^4} = 0$ (``baryon branch''), and $\avg{L^4} \ne 0$ with $\avg{\bar{U}^{N}} = 0$ (``lepton branch''). For $N = 3$, only the baryon branch exists. We first analyze the baryon branch. In terms of the elementary fields, the \vevs can be written% % % \footnote{The analysis is simplified if one notes that there are no % holomorphic gauge invariants involving $Q$, and therefore one can % solve the $D$-flat constraints assuming $\avg{Q} = 0$.} % (up to gauge and flavor transformations) \beq \avg{Q} = 0, \quad \avg{L} = 0 \eeq \beq \avg{A} = \sqrt{2} \pmatrix{ a_1 \ep_2 & & & 0 \cr & \ddots & & \vdots \cr & & a_n \ep_2 & 0 \cr 0 & \cdots & 0 & 0 \cr}, \quad \avg{\bar{U}} = \pmatrix{ b_1 \hbox{\bf 1}_2 & & & 0 \cr & \ddots & & \vdots \cr & & b_n \hbox{\bf 1}_2 & 0 \cr 0 & \cdots & 0 & b \cr}, \eeq where $\hbox{\bf 1}_2$ is the $2 \times 2$ identity matrix, \beq \ep_2 \equiv \pmatrix{0 & 1 \cr -1 & 0 \cr}, \eeq and the \vevs satisfy \beq |b_j|^2 = |a_j|^2 + |b|^2, \quad j = 1, \ldots, n. \eeq We begin by analyzing the theory in the regime where $b, a_1, \ldots, a_n$ are all nonzero and large, so that a classical description is valid. In that case, the gauge group $\SU{N}$ is completely broken, while the $\SU{4}$ gauge group remains unbroken. The fields $Q$ and $L$ get masses $\sim \la \avg{\bar{U}}$. Integrating out these massive fields gives an effective theory consisting of $\SU{4}$ super-Yang--Mills theory and some singlets. $\SU{4}$ gaugino condensation gives rise to a dynamical superpotential% % \footnote{In this Section, we do not include factors of $4\pi$ and $N$ in our estimates for simplicity. These are included in the numerical estimates we give in the next Section.} % \beq\eql{magicW} W_{\rm dyn} \propto \det(\la \bar{U})^{1 / 4}. \eeq (Alternatively, the anomaly-free $U(1) \times U(1)_R$ symmetries can be used to show that this is the most general superpotential allowed.) For large values of $\avg{\bar{U}}$, the \Kahler potential is nearly canonical in $\bar{U}$, and so the potential slopes toward $\bar{U} = 0$ for $N \ge 5$. (For $N = 3$, the theory has a runaway \susc vacuum.) We are therefore led to analyze the theory near the origin of the moduli space.% % \footnote{If $b = 0$, $a_1, \ldots, a_n$ large, the analysis is different. In that case, the $\SU{4}$ gauge group has one light flavor that would run away if there were no further interactions. However, the would-be runaway direction is not $D$-flat, so there are no \susc minima with $b = 0$.} We analyze the dynamics near the origin of the moduli space assuming that $\SU{4}$ is weak at the scale $\La_{N}$ where the $\SU{N}$ becomes strong. (Note that this includes large values of $N$ where $\SU{4}$ is not asymptotically free.) The $\SU{N}$ theory is s-confining, and the effective theory can be written in terms of the fields \cite{Pouliot}% % \footnote{For a general analysis of s-confining theories, see \Ref{sconfine}.} % \beq\bal (Q\bar{U}) &\sim \fund \times \fund ~, \\ (Q A^n) &\sim \fund \times \hbox{\bf 1} ~, \\ (Q^3 A^{n - 1}) &\sim \bar{\fund} \times \hbox{\bf 1} ~, \\ (A \bar{U}^2) &\sim \hbox{\bf 1} \times \asymm ~, %\{ 2 \}, \\ (\bar{U}^{N}) &\sim \hbox{\bf 1} \times \hbox{\bf 1} ~, \\ L &\sim \bar{\fund} \times \bar{\fund} ~, \eal\eeq where we have given the transformation properties under $\SU{4} \times \left[ \SU{N} \right]$. The parentheses indicate that these are elementary fields in the effective Lagrangian with the same quantum numbers as the composite operators inside the parentheses. The \Kahler potential is smooth in terms of the effective fields, \eg \beq K_{\rm eff} \sim \frac{1}{\La_{N}^{2N - 2}} \left| (\bar{U}^{N}) \right|^2 + \cdots \eeq The effective superpotential is given by the sum of the tree superpotential and a dynamical superpotential \cite{Pouliot} \beq\bal W_{\rm eff} \sim \frac{1}{\La_{N}^{2N - 1}} &\biggl[ (Q A^n) (Q \bar{U})^3 (A \bar{U}^2)^{n - 1} + (Q^3 A^{n - 1}) (Q\bar{U}) (A \bar{U}^2)^n \\ &\quad + (\bar{U}^{N}) (Q A^n) (Q^3 A^{n - 1}) \biggr] + \la L (Q\bar{U}). \eal\eeq The trilinear term has become a mass term for $L$ and $(Q\bar{U})$; integrating out these fields gives an $\SU{4}$ gauge theory with 1 flavor $(Q A^n)$, $(Q^3 A^{n - 1})$ and singlets $(A \bar{U}^2)$, $(\bar{U}^{N})$, with a trilinear effective superpotential \beq W_{\rm eff} \sim \frac{1}{\La_{N}^{2N - 1}} (\bar{U}^{N}) (Q A^n) (Q^3 A^{n - 1}). \eeq If this were a theory of fundamental fields, it would have a runaway vacuum with $(\bar{U}^{N}) \to \infty$. This can be described by a superpotential of the form \Eq{magicW}, but %(note the automatic equation numbering is not working...) in the regime we are now considering the \Kahler potential is smooth in terms of the field $(\bar{U}^{N})$. But if $\avg{\bar{U}}$ is large compared to $\La_{N}$, we can no longer treat $(\bar{U}^{N})$ as an elementary field; instead, we must use the analysis above, which shows that the potential slopes toward $\bar{U} = 0$ for $\avg{\bar{U}} \gg \La_{N}$. We see that there is no \susc vacuum for either large or small values of $\bar{U}$ on the baryon branch, so there must be at least a local \susy-breaking minimum for $\avg{\bar{U}} \sim \La_{N}$. This is the mechanism for \susy breaking found in the models of \Refs{LutyTerning,prevbreak}. Note that there is no unbroken $\U1_R$ symmetry, so that when we gauge a subgroup of the global $[\SU{N}]$ symmetry, gaugino masses can be generated. We see that the baryon branch of this model has two descriptions. There is a ``Higgs'' description in which the gauge group $\SU{N}$ is broken (valid for large $\avg{\bar{U}}$), and a ``confining'' description in which $\SU{N}$ confines (valid near $\avg{\bar{U}} = 0$). Neither of these descriptions is under control near the local minimum found above, but both pictures are expected to be a reliable guide to the qualitative features of the low-energy dynamics \cite{compliment}. We know that $b \sim \La_{N}$ (using the Higgs description), but we cannot determine whether $a_1, \ldots, a_n$ are nonzero. In this paper, we will make the dynamical assumption that \beq \avg{A} = 0. \eeq This corresponds to the largest possible unbroken global symmetry \beq\eql{gaugetoglobal} \SU{N} \times \left[ \SU{N} \right] \to \left[ \SU{N} \right]. \eeq This is reasonable, since points of maximal symmetry are generically stationary points of the energy, but it is an assumption nonetheless. (The assumption is equivalent to the statement that certain mass-squared terms in the effective theory are positive.) With this assumption, we see that the fermionic components of $A$ must remain massless in order to match the anomalies of the unbroken global $\left[ \SU{N} \times \U1 \right]$ symmetry. (In the Higgs description we are using, $A$ is charged under the global symmetry due to the symmetry breaking in \Eq{gaugetoglobal}.) If we use the confined description, we find that the fermionic components of the composite field $(A \bar{U}^2)$ are massless. In either description, we find that there are massless fermions transforming as a $\asymm$ %$\{ 2 \}$ under the unbroken $[\SU{N}]$ global symmetry. Later we will identify some of these fermions with composite quarks and leptons. Note that if the dynamical assumption above is false, we can use this model to construct models of direct gauge-mediated \susy breaking with composite messengers, along the lines suggested in \Ref{LutyTerning}. In this case, we add higher-dimension terms to the superpotential that give a \susc mass to the composite fields that stabilizes the vacuum at $\avg{A} = 0$, and gauge a subgroup of the $[\SU{N}]$ global symmetry of this model with the standard-model gauge group. The negative \susy-breaking mass-squared terms that result from the non-perturbative dynamics then induce {\it positive} mass-squared terms for the squarks and sleptons from gauge loops. We will not pursue this possibility further in this paper. In the remainder of this Subsection, we will show that for $N > 5$ there is a runaway \susc vacuum on the lepton branch of the classical moduli space. When we consider even $N$, we will find that the story is much the same: there are three branches of the classical moduli space, and there is a local \susy-breaking minimum on the ``baryon'' branch whose description is identical to the one found for $N$ odd, and there are runaway \susc vacua on the other two branches. In the remainder of the paper, we will build models assuming that the universe lives in the false vacuum on the baryon branch.% \footnote{The rate for tunneling from the false to one of the \susy vacua is shown to be negligibly small in Subsection 4.5.} % The rest of this Section is therefore not necessary to understand the main results of the paper. The reader interested primarily in model-building is strongly encouraged to skip to Section 3 at this point. % % (We will show that the tunneling rate into the true vacuum is negligible % in these models. % While it may make us uncomfortable to live in a false vacuum, it is % fair to say that this possibility is not precluded by our present % understanding of cosmology. We now analyze the lepton branch of the classical moduli space. In terms of the elementary fields, the \vevs can be written (up to gauge and flavor transformations) \beq \avg{Q} = 0, \quad \avg{L} = \pmatrix{ & 0 & \cdots & 0 \cr \ell \hbox{\bf 1}_4 & \vdots & & \vdots \cr & 0 & \cdots & 0 \cr}, \eeq \beq \!\!\!\!\!\!\!\!\! \avg{A} = \pmatrix{ \hbox{\bf 0}_5 & & & \cr & a_1 \ep_2 & & \cr & & \ddots & \cr & & & a_{n - 2} \ep_N \cr}\!, \ \ \avg{\bar{U}} = \sqrt{2} \pmatrix{ \hbox{\bf 0}_5 & & & \cr & a_1 \hbox{\bf 1}_2 & & \cr & & \ddots & \cr & & & a_{n - 2} \hbox{\bf 1}_2 \cr}\!. \eeq We begin by analyzing the theory in the regime where $\ell, a_1, \ldots, a_{n - 2}$ are all nonzero and large, so that a classical description is valid. In that case, the gauge group $\SU{4}$ is completely broken, and $\SU{N}$ is broken down to $\SU{5}$. $\avg{L} \ne 0$ gives mass to all of the $Q$'s and 4 flavors of $\bar{U}$'s, and most of the components of $A$ and $\bar{U}$ are eaten. The effective $\SU{5}$ gauge theory has matter content $\bar{\fund} \oplus \asymm$ %\{ 2 \}$ plus singlets, with no superpotential. If this were a theory of fundamental fields, this would break \susy \cite{fivebarten}, giving a vacuum energy proportional to $\La_{5,{\rm eff}}^4$, where $\La_{5,{\rm eff}}$ is determined by 1-loop matching to be \beq \La_{5,{\rm eff}} = \La_{N}^{(4N + 1)/13} \ell^{4/13} (a_1 \cdots a_{n-2})^{-(4N - 8) / (13(n - 2))}. \eeq For $N > 5$, the vacuum energy goes to zero as the $a$'s go to infinity, and there are runaway vacua on the lepton branch. For $N = 5$, the classical constraints force $\avg{A} = 0$, and there are no runaway directions; in this case \susy is broken on the lepton branch as well as the baryon branch. For $N > 5$, we could lift the runaway directions by adding higher-dimension terms to the superpotential (see \cite{ADStoo}). However, these will partially break the global symmetry, and can be shown to have lower energy than the local minima on the baryon branch. % ---------------------------------------------------------------------------- \subsection{Even $N$ Models} We now consider the model for even $N = 2n$. The analysis closely parallels that of the odd $N$ models, and the reader interested mainly in our models is encouraged to skip to Section 3. The classical moduli space can be parameterized by (again indicating the global $[\SU{N}]$ quantum numbers) \beq\bal L^4 &\sim \bar{\asymmfour} ~, \qquad\hbox{(for\ $N \ge 4$)} %\bar{\{ 4 \}}, \\ A \bar{U}^2 &\sim \asymm ~, %\{ 2 \} ~, \\ Q^4 A^{n - 2} &\sim \hbox{\bf 1} ~, \\ \bar{U}^{N} &\sim \hbox{\bf 1} ~, \\ A^n &\sim \hbox{\bf 1} ~, \eal\eeq with the constraints \beq\bal L^4 \cdot \bar{U}^{N} &= 0, \quad L^4 \cdot A\bar{U}^2 = 0, \quad L^4 \cdot Q^4 A^{n - 2} = 0, \quad \bar{U}^{N} \cdot Q^4 A^{n - 2} = 0. \eal\eeq This moduli space has three branches: $\avg{L^4} \ne 0$, $\avg{\bar{U}^{N}}, \avg{Q^4 A^{n - 2}} = 0$ (``lepton branch''), $\avg{\bar{U}^{N}} \ne 0$, $\avg{L^4}, \avg{Q^4 A^{n - 2}} = 0$ (``baryon branch''), and $\avg{Q^4 A^{n - 2}} \ne 0$, $\avg{L^4}, \avg{\bar{U}^{N}} = 0$ (``mixed branch''). We first analyze the baryon branch. On this branch, the \vevs can be written (up to gauge and flavor transformations) \beq \avg{L} = 0, \quad \avg{Q} = 0, \eeq \beq \avg{A} = \pmatrix{a_1 \ep_2 & & \cr & \ddots & \cr & & a_n \ep_2 \cr}, \quad \avg{\bar{U}} = \sqrt{2} \pmatrix{b_1 \hbox{\bf 1}_2 & & \cr & \ddots & \cr & & b_n \hbox{\bf 1} \cr}, \eeq where \beq |a_j|^2 - |b_j^2| = c, \quad j = 1, \ldots, n. \eeq We begin by analyzing the theory in the region of moduli space where $a_1, \ldots a_n$ are all nonzero and large, so that a classical description is valid. In that case, the gauge group $\SU{N}$ is completely broken, while the $\SU{4}$ gauge group remains unbroken. The fields $Q$ and $L$ get masses $\sim \la\avg{\bar{U}}$, and the low-energy theory is $\SU{4}$ super-Yang--Mills with singlets. Gaugino condensation in this theory gives rise to a dynamical superpotential \beq W_{\rm dyn} \propto \det(\la \bar{U})^{1 / 4}. \eeq % (Alternatively, the anomaly-free $U(1) \times U(1)_R$ symmetries can be used % to show that this is the most general superpotential allowed.) For large values of $\avg{\bar{U}}$, the \Kahler potential is nearly canonical in $\bar{U}$, and so the potential slopes toward $\bar{U} = 0$ for $N > 4$. For $N = 2$, there is a runaway \susc vacuum. For $N = 4$, the superpotential is linear in $\bar{U}$, and the location of the true vacuum depends on the form of the \Kahler potential. For large values of $\vu$, the \Kahler potential can be computed in perturbation theory, and one finds that 1-loop corrections involving the Yukawa coupling $\la$ tend to push the the vacuum away from the origin, while 1-loop corrections involving the gauge couplings have the opposite sign. These effects can give rise to a local minimum for large values of $\vu$ for a range of parameters. (This is the inverted hierarchy mechanism \cite{inverted}.) %Yukawa couplings give corrections to the \Kahler potential that tend to %push the field to the origin of moduli space, while gauge corrections %produce the opposite effect. %Since the $\SU{N=4}$ interactions are asymptotically free, the contribution %from the %Yukawa coupling will dominate for large %$\bar{U}$, so $\bar{U}$ is pushed towards the origin for this case as well. For any $N \ge 4$, we see that there is no \susc vacuum for large $\vu$, and we are led to analyze the theory near the origin of the moduli space. We now analyze the dynamics near the origin of the moduli space assuming that $\La_{N} \gg \La_4$. The $\SU{N}$ theory is s-confining, and the effective theory can be written in terms of the fields \cite{Pouliot} \beq\bal (Q\bar{U}) &\sim \fund \times \fund ~, \\ (A^n) &\sim \hbox{\bf 1} \times \hbox{\bf 1} ~, \\ (Q^2 A^{n - 1}) &\sim \asymm %\{ 2 \} \times \hbox{\bf 1} ~, \\ (Q^4 A^{n - 2}) &\sim \hbox{\bf 1} \times \hbox{\bf 1} ~, \\ (A \bar{U}^2) &\sim \hbox{\bf 1} \times \asymm ~, %\{ 2 \}, \\ (\bar{U}^{N}) &\sim \hbox{\bf 1} \times \hbox{\bf 1} ~, \\ L &\sim \bar{\fund} \times \bar{\fund} ~, \eal\eeq where we have given the transformation properties under $\SU{4} \times \left[ \SU{N} \right]$. The superpotential is given by the sum of the tree superpotential and a dynamical superpotential \cite{Pouliot}. The tree-level superpotential turns into a mass term for $L$ and $(Q\bar{U})$. Integrating out these states gives an effective theory with gauge group $\SU{4}$, a field $(Q^2 A^{n - 1}) \sim \asymm\,$, and singlets, with effective superpotential \beq\bal W_{\rm eff} \sim \frac{1}{\La_{N}^{2N - 1}} &\biggl[ (Q^4 A^{n - 2}) (A \bar{U}^2)^n + (\bar{U}^{N}) (A^n) (Q^4 A^{n - 2}) \\ &\quad + (\bar{U}^{N}) (Q^2 A^{n - 1})^2 \biggr]. \eal\eeq For $\avg{\bar{U}^{N}} \ne 0$, $(Q^2 A^{n - 1})$ is massive and $\SU{4}$ gaugino condensation pushes $(\bar{U}^{N})$ away from the origin. If this were a theory of fundamental fields, it would have a runaway vacuum with $(\bar{U}^{N}) \to \infty$, but this description breaks down for large values of $\avg{\bar{U}}$. We see that this theory has a local \susy-breaking minimum on the baryon branch through a mechanism identical to that in the odd $N$ case. As before, we make the dynamical assumption that \beq \avg{A} = 0, \eeq so that there is an unbroken $[\SU{N}]$ global symmetry, and the theory has massless composite fermions transforming as a $\asymm$ %$\{ 2 \}$ under $\SU{N}$. We now turn to the lepton branch. On this branch, the \vevs are \beq \avg{L} = \pmatrix{& 0 & \cdots & 0 \cr \ell \hbox{\bf 1}_4 & \vdots & & \vdots \cr & 0 & \cdots & 0 \cr}, \quad \avg{Q} = 0, \eeq \beq \!\!\!\!\!\!\!\! \avg{A} \! = \! \pmatrix{a \epsilon_2 \!\! & & & & \cr & a \epsilon_2 \!\! & & & \cr & & a_1 \epsilon_2 \!\! & & \cr & & & \ddots & \cr & & & & a_{n-2} \ep_2 \!\! }\!, \ \avg{\bar{U}} \! = \! \sqrt{2} \pmatrix{\hbox{\bf 0}_4 & & & \cr & b_1 \hbox{\bf 1}_2 & & \cr & & \ddots & \cr & & & b_{n - 2} \hbox{\bf 1}_2 \cr}\!. \eeq with \beq |a|^2 = |a_j|^2 - |b_j|^2, \quad j = 1, \ldots, n - 2. \eeq We analyze the theory in the region of moduli space where $\ell, a, a_1, \ldots, a_{n - 2}$ are all nonzero and large. In that case, the $\SU{4}$ gauge group is completely broken, and the $\SU{N}$ gauge group is broken down to $\Sp{4}$. After taking into account the effects of the superpotential and the eaten fields, there are no charged fields under the unbroken $\Sp{4}$. Gaugino condensation in $\Sp{4}$ then pushes $a, a_1, \ldots, a_{n-2}$ away from the origin \cite{ADStoo}, and so there is a runaway \susc vacuum in this branch. Finally, we analyze the mixed branch. On this branch, the \vevs are \beq \avg{L} = 0, \quad \avg{Q} = \sqrt{2} \pmatrix{& 0 & \cdots & 0 \cr q \hbox{\bf 1}_4 & \vdots & & \vdots \cr & 0 & \cdots & 0 \cr}\!, \eeq \beq \!\!\!\!\!\!\!\! \avg{A} \! = \! \pmatrix{a \epsilon_2 \!\! & & & & \cr & a \epsilon_2 \!\! & & & \cr & & a_1 \ep_2 \!\! & & \cr & & & \ddots & \cr & & & & a_{n - 2} \ep_2 \!\! \cr}\!, \ \avg{\bar{U}} \! = \! \sqrt{2} \pmatrix{\hbox{\bf 0}_4 & & & \cr & b_1 \hbox{\bf 1}_2 & & \cr & & \ddots & \cr & & & b_{n - 2} \hbox{\bf 1}_2 \cr}, \eeq where \beq |a|^2 + |q|^2 = |a_j|^2 - |b_j|^2, \quad j = 1, \ldots, n - 2. \eeq As on the lepton branch, the low-energy theory is a pure $Sp(4)$ gauge symmetry, and gaugino condensation pushes $a,a_j$ away from the origin, and so there are additional runaway vacua on this branch. % ---------------------------------------------------------------------------- \section{Numerical Estimates} % ---------------------------------------------------------------------------- We now consider the numerical estimates of various quantities of interest in these models. The models are non-calculable, but we can make estimates using dimensional analysis, and also keep track of factors of $4\pi$ and $N$, which are potentially large. Neither the ``Higgs'' nor the ``confining'' descriptions of these theories is weakly coupled in the local \susy-breaking vacuum we consider. We find it simplest to make estimates using the ``Higgs'' description that uses the elementary fields of the theory. We estimate the size of various effects by assuming that loop corrections are the same size as leading effects in perturbation theory. This is the philosophy of ``na\"\i ve dimensional analysis" \cite{fourpi,fourpiSUSY}. We use these considerations to argue that the strong dynamics preserves an approximate $[\SU{N}]$ flavor symmetry even if the Yukawa matrix $\la$ in the tree-level superpotential is completely arbitrary.% % \footnote{It is a consequence of our dynamical assumption that $\avg{A} = 0$ that the flavor symmetry is not spontaneously broken by the strong dynamics in the limit where $\la$ is proportional to the identity.} % This is important for naturally suppressing flavor-changing neutral currents in the models we construct below.% % \footnote{We thank M. Schmaltz for emphasizing this point.} % First of all, the dynamical superpotential \Eq{magicW} depends only on $\det(\la)$, and so has no flavor dependence. This means that all flavor dependence appears in the effective \Kahler potential. In the Higgs description, the $\la$ dependence in the \Kahler potential comes from diagrams with $\la$ vertices, and through the Dirac mass matrix of $Q$ and $L$, which is proportional to $\la$. Diagrams with $\la$ vertices are suppressed by $\la^2 / (16\pi^2)$, so these give only small flavor violation. Internal $Q$ and $L$ loops without $\la$ vertices do not contribute to flavor violation because they always involve traces of the mass matrix. (We are not interested in diagrams with external $Q$ and $L$ lines because the only light matter states correspond to $\tr\bar{U}$ and $A$; see below.) This shows that the flavor symmetry is preserved up to corrections of order $\la^2 / (16\pi^2) \lsim 10^{-2}$. We now estimate $\avg{\bar{U}}$. Na\"\i ve dimensional analysis tells us that $\avg{\bar{U}}$ must be close to the value for which perturbation theory breaks down. The $\SU{N}$ gauge dynamics becomes strong at the scale $\La_N$, where \beq g_N(\mu \sim \La_N) \sim \frac{4\pi}{\sqrt{N}}. \eeq The perturbative description breaks down when the massive gauge bosons (and the states that get a mass due to the $\SU{N}$ $D$-term potential) have masses of order $\La_N$, which gives \beq \avg{\bar{U}} \sim \La_N \frac{\sqrt{N}}{4\pi} \hbox{\bf 1}_N. \eeq The $F$ component of $\bar{U}$ is estimated to be% % \footnote{For a discussion of the factors of $4\pi$ in $W_{\rm eff}$, see \Ref{fourpiSUSY}.} % \beq \avg{F_{\bar{U}}} \sim \left\langle \frac{\partial W_{\rm eff}}{\partial \bar{U}} \right\rangle \sim \frac{1}{4\pi} \frac{\sqrt{N}}{4} \left( \det(\sqrt{N}\la) \right)^{1/4} \La_4^{3 - N/4} \La_N^{N/4 - 1} \hbox{\bf 1}_N. \eeq This shows that as long as $\SU{4}$ is weak at the scale where $\SU{N}$ becomes strong, we have $\avg{F_{\bar{U}}} \ll \avg{\bar{U}}^2$. If $N < 12$, $\SU{4}$ is asymptotically free and the condition for $\SU{4}$ to be weak at the scale $\La_N$ is $\La_4 \ll \La_N$. For $N \ge 12$, $\La_4$ is the ultraviolet Landau pole of $\SU{4}$, and so the condition that $\SU{4}$ is weak at $\La_N$ is $\La_4 \gg \La_N$. As a consequence of our dynamical assumption, both $\avg{\bar{U}}$ and $\avg{F_{\bar{U}}}$ are proportional to the $N \times N$ unit matrix, so that the $\SU{N} \times \left[ \SU{N} \right]$ symmetry is broken down to a global $[\SU{N}]$. The superpotential gives a \susc mass to the fields $Q$ and $L$ of order \beq m_{Q,L} \sim \la \avg{\bar{U}} \sim \frac{\sqrt{N} \la}{4\pi} \La_N. \eeq (This mass does not become large compared to $\La_N$ for large $N$ because the Yukawa coupling must be $\la \sim 1/\sqrt{N}$ in order to have a good large-$N$ limit.) There are also \susy breaking $B$-type mass terms of order $\avg{F_{\bar{U}}}$. Below the scale $m_{Q,L}$, the only light fields are the $\SU{4}$ gauge bosons, $\tr\bar{U}$ and $A$. (In the confined description, these fields correspond to $(\bar{U}^N)$ and $(A\bar{U}^2)$, respectively.) The field $\tr\bar{U}$ is a singlet, and $A$ transforms as a $\asymm$ under the unbroken $[\SU{N}]$ global symmetry. The scalar and fermion components of $\tr\bar{U}$ get masses of order \beq m_{\tr\bar{U}} \sim \left\langle \frac{\partial^2 W_{\rm eff}}{\partial\bar{U}^2} \right\rangle \sim \frac{\avg{F_{\bar{U}}}}{\avg{\bar{U}}} \equiv M_{\rm comp}. \eeq We will see that the scale $M_{\rm comp}$ sets the scale for all \susy breaking masses in this model. The scalar components of the field $A$ receives (strong \SU{N} gauge-mediated) loop contributions both from the \susy breaking in the $Q,L$ spectrum and from the induced \susy breaking in the fields at the scale $\La_N$. These contributions can be most easily estimated using the method of Giudice and Rattazzi \cite{GiudRatt}.% % \footnote{This method can be extended to all orders in perturbation theory \cite{SonOfGiudRatt}.} % In this method, one computes the wavefunction renormalization factor $Z_A$ as a function of the threshold $m$ where heavy states are integrated out, and then makes the replacement $m \to 4\pi \sqrt{\bar{U}^\dagger \bar{U}} / \sqrt{N}$ to find the dependence on $\avg{\bar{U}}$ and $\avg{F_{\bar{U}}}$ to leading order in $\avg{F_{\bar{U}}}/\avg{\bar{U}}$. The $A$ scalar mass is then obtained from the $\theta^2 \bar{\theta}^2$ component of $\ln Z_{A}$. The quantity $\ln Z_{A}$ satisfies a renormalization group equation \beq \mu \frac{d}{d\mu} \ln Z_{A} = f\left(\frac{N g_N^2}{16\pi^2}\right), \eeq where $f$ is a function with no large parameters. (Note that there are $N$ ``flavors'' of $\bar{U}$, so loops of $\bar{U}$ fields are not suppressed for large $N$.) Since $N g_N^2 / (16\pi^2) \sim 1$, we obtain simply \beq m^2_{\phi_A} \sim \left( \frac{\avg{F_{\bar{U}}}}{\avg{\bar{U}}} \right)^2 = M_{\rm comp}^2. \eeq If we identify the composite fermions with quarks and leptons, this gives the mass of the corresponding scalar superpartners. We now assume that the standard-model gauge group is embedded into the $[\SU{N}]$ global symmetry and estimate the standard-model gaugino and elementary scalar masses. We can compute these using the method of \Ref{GiudRatt}, or by simply estimating the corresponding perturbative diagrams. There are of order $N$ messengers, so we obtain \beq m_{\la_{\rm SM}} \sim N \frac{g^2_{\rm SM}}{16\pi^2} M_{\rm comp} \eeq for the standard-model gaugino masses. In addition, the scalars will receive a gauge-mediated contribution \beq \de m^2_{\phi,{\rm gauge\,med}} \sim N \left( \frac{g^2_{\rm SM}}{16\pi^2} \right)^2 M_{\rm comp}^2. \eeq For the composite fields, this is a small correction; for the elementary fields, this is the dominant contribution to the scalar mass. (We will see below that there is also a flavor-dependent contribution to the scalar masses that can be comparable.) In the models we consider, there is a scale of new physics $M$ that is not far above the scale $\La_N$. In the effective theory at the scale $\La_N$, there will therefore be higher-dimension operators suppressed by powers of $1/M$. For example, the following terms in the Lagrangian are compatible with all symmetries: \beq\eql{softmassbreak} \de\scr{L} \sim \myint d^4\th \left[ \frac{c_1}{M^2} \tr( \bar{U}^\dagger \bar{U} ) A^\dagger A + \frac{c_2}{M^2} \tr( \bar{U}^\dagger \bar{U} ) \Phi^\dagger \Phi \right], \eeq where $\Phi$ is an elementary quark or lepton field. In the ``Higgs'' picture we are using, we can estimate the terms in the effective Lagrangian for the composite fields by simply replacing $\bar{U}$ by its \vev.% % \footnote{When expressed in terms of the scale $\La_N$, this gives results with $4\pi$ dependence in agreement with a ``confined'' description \cite{fourpiSUSY}.} % We therefore obtain an additional contribution to the elementary and composite scalar masses of order \beq\eql{seec} \de m_{\phi,{\rm new\,phys}}^2 \sim \frac{c_{1,2} N \avg{F_{\bar{U}}}^2}{M^2} \sim \frac{c_{1,2} N \avg{\bar{U}}^2}{M^2} M_{\rm comp}^2. \eeq On general grounds, we might expect $c_{1,2} \sim 1$; alternatively, if the model has a good large-$N$ limit with $M$ held fixed, we expect $c_{1,2} \sim 1/N$. There are additional higher-dimension operators in the models we construct. We can easily estimate their effects on the composite fields in the Higgs description by simply replacing $\bar{U}$ by appropriate scalar or $F$-component \vevs. % ---------------------------------------------------------------------------- \section{Composite Quarks and Leptons} % ---------------------------------------------------------------------------- We now build models of composite quarks and leptons using the models analyzed above as building blocks. Because the Yukawa couplings arise from high-dimension operators, they are naturally small compared to unity. This means that the top quark cannot be composite in the models we construct.% % \footnote{It would be interesting to find \susy-breaking models where the top-quark Yukawa coupling arises as a term in a dynamical superpotential. In that case, the top-quark Yukawa coupling is of order $4\pi$ at the compositeness scale, and runs down to a quasi-fixed point value at the weak scale \cite{fixed}. The top-quark Yukawa coupling arises in this way in the models of Nelson and Strassler \cite{NelsonStrass}, but the composite dynamics does not break \susy in these models.} % In models of this type, the masses of the gauginos and elementary scalars are suppressed by a loop factor compared to the composite scalar masses: \beq \frac{m_{\la_{\rm SM}}}{M_{\rm comp}} \sim \frac{N g_{\rm SM}^2}{16\pi^2}. \eeq If $N$ is not large, then this can be realistic only if the composite scalars are very heavy. As we will explain below, this leads naturally to models with a low compositeness scale. On the other hand, we can consider models where the loop suppression is overcome by the large multiplicity factor $N$, allowing models with a high compositeness scale.% % \footnote{We do not consider the possibility that the gauginos may be ultra-light \cite{Glennys}.} % ---------------------------------------------------------------------------- \subsection{Embedding the Standard Model} Before turning to the models, we discuss some aspects of embedding the standard model gauge group into the global $[\SU{N}]$ symmetry. Because we want to preserve perturbative unification, we will consider only embeddings where the preons fall into complete $\SU{5}_{\rm SM}$ multiplets, even if only the standard-model subgroup is gauged.% % \footnote{The interesting possibility that the preons fall into complete representations of the ``trinification'' group $\SU{3}^3 / Z_3$ will not be explored in this paper.} Because our models generate composite states transforming as a $\asymm$ of a global $[\SU{N}]$ symmetry, it is tempting to generate a $\bb{10}$ of $\SU{5}_{\rm SM}$ from the antisymmetric product $(\bb{5} \otimes \bb{5})_{\rm asymm}$. However, it is easy to see that there is no way of assigning baryon number to the preons to obtain the correct baryon numbers for the states of the the composite $\bb{10}$.% % \footnote{It is possible to obtain a ${\bf 10}$ from the antisymmetric product of two \emph{different} ${\bf 5}$'s, but this leads to rather uneconomical models.} % Since baryon number is not a good quantum number of the strong dynamics, we expect baryon-number violating operators suppressed by powers of $\La_N$ in the low-energy theory, so this kind of embedding cannot be used in models where the compositeness scale is below the grand-unified theory (GUT) scale. It is not hard to construct baryon-number conserving as well as baryon-number violating embeddings, and we will consider both types below. The first embedding we consider is based on the model with $N = 11$. $\SU{5}_{\rm SM}$ is embedded into $[\SU{11}]$ so that the $\fund = {\bf 11}$ representation decomposes as \beq \fund \to \bb{5} \oplus \mybar{\bb{5}} \oplus \bb{1}. \eeq The composite states then decompose under $\SU{5}_{\rm SM}$ as \beq \asymm \to \bb{10} \oplus \mybar{\bb{5}} \oplus \bb{1} \oplus \left[ \bb{24} \oplus \mybar{\bb{10}} \oplus \bb{5} \right]. \eeq The composite states include a complete generation (including a right-handed neutrino), together with the exotic states in square brackets. Baryon number is violated at the scale $\La_N$. We can remove the unwanted exotic states by adding an additional elementary generation $\bb{10} \oplus \mybar{\bb{5}}$ to the theory and including higher-dimension operators of the form \beq \de\scr{L}_{\rm eff} \sim \myint d^2\th \left[ \frac{1}{M} (A \bar{U}^2)_{\mybar{\bf 5}} X_{\vphantom{\mybar{\bf 5}}\bf 5} + \frac{1}{M} (A \bar{U}^2)_{\vphantom{\mybar{\bf 5}}\bf 10} X_{\mybar{\bf 10}} + \frac{1}{M^3} (A \bar{U}^2)_{\bf 24}^2 \right] + \hc, \eeq which gives rise to masses \beq m_{\mybar{\bf 5}, {\bf 10}} \sim \frac{\vu^2}{M}, \quad m_{\bf 24} \sim \frac{\vu^4}{M^3}. \eeq One can obtain a model with two composite generations by considering a model with gauge group $\left[ \SU{4} \times \SU{11} \right]^2/Z_2$. This may not be unnatural, since whatever explains the replication of families may also give rise to a replicated group structure. A simple way to conserve baryon number is to have only composite $\mybar{\bb{5}}$'s. The simplest such model is based on $N = 5 + k = 7$ with $\SU{5}_{\rm SM}$ acting on the preons as \beq \fund \to \mybar{\bb{5}} \oplus (k \times \bb{1}). \eeq The composite states decompose as \beq \asymm \to (k \times \mybar{\bb{5}}) \oplus \left[ \mybar{\bb{10}} \oplus \left( \sfrac{k(k - 1)}{2} \times \bb{1} \right) \right]. \eeq This gives rise to $k = 2$ composite $\mybar{\bb{5}}$'s and some unwanted states that can be eliminated by adding higher-dimension operators similar to those described above. %Note that there is a $\SU{k}$ global symmetry acting on the preons %that turns into an $\SU{k}$ flavor symmetry acting on the composite %states in the low-energy theory. Finally, we consider a more elegant embedding that naturally replicates generations and conserves baryon number. We consider the theory with $N = 15 + k = 18$, with $\SU{5}_{\rm SM}$ acting on the preons as \beq \fund \to \bb{10} \oplus \mybar{\bb{5}} \oplus (k \times \bb{1}). \eeq Then the composite states decompose as \beq \asymm \to (k \times \bb{10}) \oplus (k \times \mybar{\bb{5}}) \oplus \left[ {\bf 45} \oplus \mybar{{\bf 45}} \oplus \mybar{{\bf 10}} \oplus \bb{5} \oplus \left( \sfrac{k(k - 1)}{2} \times {\bf 1} \right) \right]. \eeq If we now write down the most general superpotential involving the composite states, we will generate Dirac masses marrying $\bb{45}$ and $\mybar{\bb{45}}$, as well as marrying one of the composite generations with the antigeneration, leaving us with $(k -1)$ complete composite generations. We now address the question of Yukawa couplings. Yukawa couplings involving the composite fermions must arise from higher-dimension operators in the fundamental theory. We therefore assume that the new physics at the scale $M$ induces terms in Lagrangian such as \beq\eql{YukOp} \de\scr{L} \sim \myint d^2\th \left[ \frac{b_1}{M^4} (A \bar{U}^2)_{\mybar{\bf 5}} (A \bar{U}^2)_{\vphantom{\mybar{\bf 5}}\bf 10} H + \frac{b_2}{M^2} (A \bar{U}^2)_{\mybar{\bf 5}} \Phi_{\vphantom{\mybar{\bf 5}}\bf 10} H + \cdots \right] + \hc \eeq where $H$ is a fundamental Higgs field and $\Phi$ is a fundamental matter field. This gives Yukawa couplings to the composite fields \beq \de\scr{L}_{\rm eff} \sim \myint d^2\th \left[ \frac{b_1 \avg{\bar{U}}^4}{M^4} A_{\mybar{\bf 5}} A_{\vphantom{\mybar{\bf 5}}\bf 10} H + \frac{b_2 \avg{\bar{U}}^2}{M^2} A_{\mybar{\bf 5}} \Phi_{\vphantom{\mybar{\bf 5}}\bf 10} H + \cdots \right] +\hc \eeq Note that if the second generation quarks are to be composite, we require $\avg{\bar{U}} / M \sim \sfrac{1}{3}$, so the scale of new physics is not far above the scale of strong dynamics. This problem appears particularly worrisome if we note that the scale $\avg{\bar{U}}$ is smaller than $\La_N$ (the scale of strong dynamics) for moderate $N$. However, the example of the charm quark in QCD suggests that it is not absurd to integrate out particles with masses near the scale $\La_N$. (The charm quark mass and the scale $\La$ in QCD are both near 1~GeV.) Note that, in all of the above models, an approximate flavor symmetry of the strong dynamics (the $Z_2$ in the $N = 11$ theory, $\SU{k}$ in the $N = 5 + k$ and $N = 15 + k$ theories) guarantees equal soft masses for all the composite states. While this is somewhat artificial in the $N = 11$ case, it is quite natural in the $N = 5 + k$ and $15 + k$ cases. In particular, in the $N = 15 + k$ case, {\it all} soft masses for the first two generation scalars are degenerate at leading order. Of course, the flavor physics responsible for generating the correct pattern of Yukawa couplings must distinguish between the first two generations and will necessarily break the flavor symmetry of the strong dynamics.% % \footnote{Flavor violation in the the $\la$ matrix does not break the chiral symmetries acting on the composite quarks and leptons, and therefore does not give rise to Yukawa couplings.} % The corrections to the soft masses induced by this flavor physics are model-dependent, but are at least suppressed by the same small parameters that control the small Yukawa couplings for the light generations. We will see that this suppression is already sufficient for marginal consistency with flavor-changing neutral current (FCNC) constraints, so the \susc flavor problem is very mild in these models. % ---------------------------------------------------------------------------- \subsection{Low-scale Composite Models} If the multiplicity factor $N$ is not large, then the composite scalars must have masses of order $10\TeV$ or more in order to have gaugino and elementary squark and slepton masses of order $100\GeV$. In this case, there are negative 2-loop contributions to the elementary scalar mass-squared from the composite scalar masses \cite{NimaHitoshi}. These contributions are dangerous because they are enhanced by $\ln(\La_N / M_{\rm comp})$ compared to the usual gauge-mediated contributions. To avoid these, we must require that $\La_N$ is not far above $M_{\rm comp} \sim 10$--$100 \TeV$. Independently of these considerations, we are interested in the possibility of a low compositeness scale because it holds out the possibility of rich phenomenology. One possibility is to use the $N = 5 + k = 7$ model, which gives rise to the composite states $\mybar{\bb{10}} \oplus (2 \times \mybar{\bb{5}}) \oplus \bb{1}$. We identify the two composite $\mybar{\bb{5}}$ fermions with quarks and leptons, and eliminate the unwanted composite fermions by combining them with elementary fields transforming as $\bb{10} \oplus \bb{1}$. In order to obtain sufficiently heavy masses for the elementary squarks and sleptons, we take the mass of the composite scalars to be in the $10$--$100\TeV$ range. For purposes of running the standard-model gauge couplings, this model adds an equivalent of $6 \times (\bb{5} \oplus \mybar{\bb{5}})$ to the theory above the scale $\La_7$ of the strong $\SU{7}$ dynamics, and so it is marginally compatible with unification if $\La_7 \gsim 200\TeV$. If we assume that the Yukawa couplings are generated by new physics at a scale $M$ from operators of the form \Eq{YukOp}, we find that in order to generate Yukawa couplings of order $10^{-3}$ (for the composite $s$ and $\mu$), we require $\vu / M \sim 3 \times 10^{-2}$. This gives an explanation of the smallness of the down-type Yukawa couplings of the first two generations, but it does not explain why the up-type Yukawa couplings are also small. We now discuss FCNC's in this model. Note that there is a global $\SU{2}$ acting on the $SU(5)_{\rm SM}$ singlet preons in this model, which becomes a $\SU{2}$ flavor symmetry acting on the composite $\bb{5}$'s in the low-energy theory. We can therefore envision that the flavor breaking in the preon theory has a GIM mechanism acting on the first two generations that would align the flavor structure in the scalar and fermion sectors \cite{U(2),NS,techniGIM}. In the absence of such a mechanism, this model has FCNC's. Because the up-squarks are elementary, their mass arises dominantly from gauge-mediation, and this is not large enough to naturally suppress FCNC's. For example, \beq \frac{\de m_{\tilde{u} \tilde{c}}^2}{m_{\tilde{u}}^2} \sim \left( \frac{\vu}{M} \right)^2 \left( \frac{g_3^2}{16\pi^2} \right)^{-2} \sim 1, \eeq where we use $\vu^2 / M^2 \sim y_{uc} \sim \sqrt{y_u y_c}$. This is incompatible with the bound from $D$--$\bar{D}$ mixing, which requires $\de m_{\tilde{u} \tilde{c}}^2 / m_{\tilde{u}}^2 \lsim 10^{-2}$. There are also problems with $K$--$\bar{K}$ mixing. We next consider a model based on the $N = 15 + k = 18$ embedding described above. With such a large value for $N$, it may not be necessary to have a low value for $\vu$ to avoid negative third generation scalar masses, but we can consider the possibility of a low compositeness scale nonetheless. This model produces 2 complete composite generations of quarks and leptons, but contains a large number of fields charged under the standard model gauge group above the scale $\La_{18}$ of the strong $\SU{18}$ dynamics. The standard-model gauge couplings have a Landau pole at a few times $\La_{18}$ in this model, so it is certainly not compatible with perturbative unification. Since the Landau pole is so close to $\La_{18}$ is not clear that this model makes sense as an effective theory at the scale $\La_{18}$. However, the strong dynamics at the Landau scale may have an interpretation in terms of a dual theory \cite{dual}, and we expect such a theory to behave qualitatively the same as what we find here. We can also hope that models with a more favorable group-theory structure will be found. The problems with perturbative unification lead us naturally to consider high-scale models with large values of $\La_N$. The high-scale and low-scale models with two composite generations have a similar phenomenology, and we will discuss this after we have introduced the high-scale models. % ---------------------------------------------------------------------------- \subsection{High-scale Composite Models} We now discuss the possibility that the compositeness scale $\Lambda_N$ is near or above $M_{\rm GUT} \sim 10^{16}\GeV$, allowing perturbative unification even if $N$ is large. If the scale $\La_N$ is large, we must address the dangerous negative contributions to the third generation scalar masses coming from the scalars of the first two generations \cite{NimaHitoshi}. These arise from the renormalization group equations \beq \mu \frac{d m_3^2}{d\mu} = \frac{8 g^2}{16\pi^2} C_2 \left[ \frac{3 g^2}{16\pi^2} m_{1,2}^2 - m_\la^2 \right], \eeq where we have assumed that one gauge group dominates and specialized to the case of two composite generations. (Here, $m_3$ is the third-generation scalar mass, $m_{1,2}$ are the scalar masses of the first two generations, and $m_\la$ is gaugino mass. $C_2$ is the quadratic Casimir, with the $\U1_Y$ generator in $\SU{5}$ normalization.) We see that the contribution to the gaugino mass dominates provided that \beq m_\la \gsim \frac{m_{1,2}}{10}, \eeq which agrees with the detailed analysis of \Ref{NimaHitoshi}. This condition is plausibly satisfied in our models if $N \gsim 10$. Since dimension-6 $B$-violating operators suppressed by such high scales are safe, we consider both the $B$-violating ``squared" $N = 11$ as well as the $B$-conserving $N = 18$ theories. Both of these theories give rise to two complete composite generations. It is believed that new physics at the Planck scale will give rise to higher-dimension operators suppressed by the reduced Planck scale $M_* \sim 10^{18}\GeV$. It is therefore natural to consider the possibility that it is these effects that give rise to the higher-dimension operators that are required to make the theory realistic, and identify $M = M_*$. In this case, $\La_N$ will be \emph{above} $M_{\rm GUT}$, and even those extra charged states that become massive due to higher dimension operators are massive enough (within one or two decades of $M_{GUT}$) in order to leave perturbative gauge coupling unification (marginally) intact. Finally, even though $N = 11$ or $N = 18$ is plausibly large enough to overcome the problem of negative third-generation scalar masses even in high-scale theories, we note that the new physics at the scale $M$ gives rise to third-generation scalar masses of order $\de m_3 \sim \sqrt{cN} (\vu/M) M_{\rm comp}$, which can be in the range $100\GeV$--$1\TeV$. If this contribution is positive, it may improve the problem with the negative log-enhanced contributions to the third-generation scalar masses. The contributions of new physics at the scale $M_*$ can give the gravitino a mass of order $100\GeV$ in high-scale models, so the gravitino need not be lightest \susc particle (LSP) in these models. The LSP is most likely a neutralino with a mass in the $100\GeV$ range, which is a traditional favorite candidate for cold dark matter. % ---------------------------------------------------------------------------- \subsection{Implications for Flavor Physics} We now turn to the phenomenological implications of the models with two composite generations, concentrating mainly on flavor physics.% % \footnote{Theories of flavor exploiting compositeness (but not addressing \susy breaking) have been constructed in \cite{KaplanSchmaltz}.} % If we assume that new physics at the scale $M$ is responsible for the Yukawa couplings, then the Yukawa couplings will arise from operators of the form \Eq{YukOp}.% % \footnote{Models with dynamical \susy breaking and composite states with large global symmetries were also found in \Ref{LutyTerning}. However, the composite states were high-dimension baryons, and the Yukawa couplings for the composite generations are suppressed by the ratio of the compositeness scale to the higher scale $M$ raised to the $30^{\rm th}$ power. Therefore, these models cannot naturally produce large enough Yukawa couplings even for the light generations.} % This gives rise to Yukawa matrix with the skeletal form \beq y \sim \pmatrix{\ep^2 & \ep^2 & \ep \cr \ep^2 & \ep^2 & \ep \cr \ep & \ep & 1 \cr}, \quad \ep \sim \left( \frac{\vu}{M} \right)^2. \eeq It is clear that additional structure is needed to construct fully realistic Yukawa matrices. However, this is certainly a good starting point for constructing a theory of flavor, and the automatic $\epsilon$ suppressions due to the composite nature of the first two generations leave a milder hierarchy in the coefficients of the higher-dimension operators that needs to be explained. For $\ep$ in the range $10^{-2}$ to $10^{-1}$, realistic fermion masses can be obtained with simple textures and hierarchies of order $10$ in the effective coupling constants. Let us turn to the issue of FCNC's due to non-degeneracy of the scalar masses of the first two generations. We emphasize again that, due to approximate flavor symmetries of the strong dynamics, the leading contribution to the soft masses is equal for the first two generations, and the issue is whether sufficient degeneracy is maintained to avoid FCNC constraints after the effects of the higher dimension operators are included.% % \footnote{Flavor symmetries have been used to constrain both the form of the Yukawa matrices and the scalar mass matrices, thereby addressing both the \susc and usual flavor problems \cite{U(2),NS}. In our case, however, the approximate flavor symmetry guaranteeing scalar degeneracy need not be respected by the higher dimension operators generating the Yukawa couplings.} % The size of the corrections depends on the flavor physics at the scale $M$. For example, we have already pointed out that it is possible that the flavor physics has a GIM mechanism that suppresses FCNC's. We now analyze the possibility that there is no alignment mechanism at the scale $M$, so the off-diagonal scalar masses are suppressed only by the powers of $\La / M$ that suppress the corresponding Yukawa couplings. The mixing contributions to the soft mass matrices come from operators such as \beq \de\scr{L} \sim \myint d^4\th \left[ \frac{c}{M^2} (A \bar{U})^\dagger (A \bar{U}) \right], \eeq which give \beq\eql{compsupp} \frac{\de m_{jk}^2}{M_{\rm comp}^2} \sim c \left( \frac{\vu}{M} \right)^2 \sim c \sqrt{y_{jk}}. \eeq (Note that the operator of \Eq{softmassbreak} is enhanced by a factor of $N$, but is flavor-diagonal.) The most stringent FCNC bounds come from the $K$--$\bar{K}$ system, and can be summarized as \beq \mbox{Re}\left(\frac{\de m_{\tilde{d}\tilde{s}}^2}{M_{\rm comp}^2} \right) \lsim 10^{-1} \frac{M_{\rm comp}}{10\TeV}, \quad \mbox{Im} \left(\frac{\de m_{\tilde{d}\tilde{s}}^2}{M_{\rm comp}^2} \right) \lsim 10^{-2} \frac{M_{\rm comp}}{10\TeV}. \eeq The constraint from Re($\de m^2_{\tilde{d}\tilde{s}}$) gives (using $y_{ds} \sim \sqrt{y_d y_s}$) \beq\eql{cbound} c \lsim 5\, \frac{M_{\rm comp}}{10\TeV}, \eeq which is plausibly satisfied for $M_{\rm comp}$ as low as $1\TeV$ given the uncertainties. In order to also evade the bounds from $\Im(\de m^2_{\tilde{d}\tilde{s}})$, we must assume that the \CP-violating phase in this quantity is somewhat small, of order $\frac{1}{10}$. Alternately, $M_{\rm comp} \sim 10\TeV$ is completely safe from all constraints. Even if the induced non-degeneracies between the first two generation sfermions are small enough to avoid present FCNC constraints, there is still a rich spectrum of flavor changing signals due to the non-degeneracy between the first two and third generation sfermions. If the sfermion mixing angles are CKM-like, flavor-violating signals are expected at experimentally interesting levels in a wide variety of processes such as $\mu \to e \gamma$, $\mu \to 3 e$, $B$--$\bar{B}$ mixing, and electron/neutron electric dipole moments \cite{BarbHall}. Finally, we note that the new physics at the scale $M$ may provide a solution to the ``$\mu$ problem.'' If the low-energy theory contains the terms \beq\eql{muterm} \de\scr{L} = \myint d^4\th\, \frac{c'}{M} \tr(\bar{U}^\dagger \bar{U}) ( H \bar{H} + \hc), \eeq where $H$, $\bar{H}$ are the standard-model Higgs fields, then the low-energy theory contains $\mu$ and $B\mu$ terms of order \beq \mu \sim c' N \frac{\avg{F_{\bar{U}}} \vu}{M^2} \sim c' N \sqrt{y} M_{\rm comp}, \quad B\mu \sim c' N \frac{\avg{F_{\bar{U}}}^2}{M^2} \sim c' N \sqrt{y} M_{\rm comp}^2, \eeq where $y \sim (\vu / M)^4$ is the magnitude of a Yukawa coupling generated at the scale $M$. If we want $\mu^2 \sim B\mu$, then we need $c' N \sim 1/\sqrt{y}$, which is plausible for large $N$. (The parameter $c'$ is of order 1 or $1/N$, as in the discussion below \Eq{seec}.) In this case, both $\mu$ and $B\mu$ are naturally near $M_{\rm comp}$, which is somewhat large for electroweak symmetry breaking even if $M_{\rm comp} \sim 1\TeV$. However, given the large uncertainties and model-dependence in these estimates, this mechanism may work in a detailed model. % ---------------------------------------------------------------------------- \subsection{Decay of the False Vacuum} All of the models above require that the universe live in a false vacuum on the ``baryon'' branch, and so we must consider the possibility of the decay of the vacuum. All of the \susc vacua occur at infinite field values on other branches of the moduli space. Therefore, the energy difference between the false vacuum and the true vacuum is small compared to the distance in field space to the classical escape point. We can therefore give a conservative bound on the tunneling rate by approximating the potential as completely flat. In that case, the Euclidean tunneling action is \cite{flatbounce} \beq S_{\rm tunnel} \simeq 2\pi^2 \frac{(\De\phi)^4}{V}, \eeq where $\De\phi$ is the distance in field space to the classical escape point, and $V \sim \avg{F_{\bar{U}}}^2$ is the value of the energy density in the false vacuum. Since $(\De\phi)^2 \gg \avg{F_{\bar{U}}}$ in our models, this always gives a negligible tunneling rate. % ---------------------------------------------------------------------------- \section{Discussion and Conclusions} % ---------------------------------------------------------------------------- We have presented new models of dynamical \susy breaking in which the same strong dynamics breaks \susy and gives rise to massless composite fermions that we identify with quarks and leptons of the first two generations. Since the corresponding composite squarks and sleptons arise directly from the \susy breaking sector, they receive \susy-breaking soft masses directly, without ``mediation'' via gravitational or SM gauge interactions. In this sense, these models provide an alternative to the ``modular'' structure of realistic models of \susy breaking, where \susy is broken in a separate sector of the model and communicated by messenger interactions to the observed particles. It is also pleasing that the models we construct are quite simple. As an illustration, we write the complete $N = 18$ model below. The gauge group is \beq \SU{4} \times \SU{18} \times \left[ \SU{18} \right] \eeq where $\SU{5}_{\rm SM}$ (the usual embedding of the standard-model group) is embedded into $[\SU{18}]$ so that ${\bf 18} \to {\bf \bar{5}} + {\bf 10} + (3 \times {\bf 1})$. The field content is \beq\bal Q &\sim (\fund, \fund, {\bf 1})~, \\ L &\sim (\bar{\fund}, {\bf 1}, \fund)~, \\ \bar{U} &\sim ({\bf 1}, \bar{\fund}, \bar{\fund})~, \\ A &\sim ({\bf 1}, \asymm, {\bf 1})~, \eal\eeq together with a single (third) generation $\Phi_{\mybar{\bf 5}}$, $\Phi_{\bf 10}$ and Higgs fields $H$, $\bar{H}$. The model has a superpotential of the form \beq\bal W &\sim L Q \bar{U} + H \Phi \Phi + \frac{1}{M^2} (A\bar{U}^2) H \Phi \\ &\quad + \frac{1}{M^3} (A\bar{U}^2) (A\bar{U}^2) + \frac{1}{M^4} (A\bar{U}^2) (A\bar{U}^2) H \eal\eeq where we have omitted indices for simplicity. The higher-dimension operators generate Yukawa couplings involving the composite states and eliminate unwanted composite fermions from the low-energy spectrum. This model generates two composite generations of quarks and leptons with small Yukawa couplings, breaks \susy, communicates \susy breaking directly to the composite squarks and sleptons, and gives sufficiently large gaugino masses through gauge loops. It is striking that a simple model such as this can be completely realistic, with the compositeness scale ranging anywhere from $10\TeV$ to the Planck scale. The leading contribution to the scalar masses is naturally flavor-diagonal due to an approximate symmetry of the strong dynamics that is present even if $\la$ has arbitrary flavor structure; this symmetry is violated only by ``perturbative'' corrections of order $\la^2 / (16\pi^2) \sim 10^{-4}$. These global symmetries also lead to the striking prediction that (depending on the model) some or all of the scalar masses of the first two generations unify at the compositeness scale, which need not be close to the GUT scale. (Models with flavor symmetries can also predict scalar unification at some level, but they cannot naturally explain unification between scalars with different gauge quantum numbers below the GUT scale.% % \footnote{Even if the scalar unification scale is close to the GUT scale, the model above predicts that \emph{all} squarks and sleptons from the first two generations unify. Even the degeneracy of squark and slepton masses within a generation is not easy to understand in $\SO{10}$, since it is broken by $D$ terms corresponding to broken generators.}) % We emphasize that these features are present in our model without the need to impose any flavor symmetry on the underlying theory. The Yukawa couplings are generated by new physics at a scale above the compositeness scale, naturally explaining why the fermion masses of the first two generations are small, while the corresponding scalar masses are large. % Off-diagonal scalar masses induced by the flavor physics may be suppressed % by a GIM mechanism. In the absence of any flavor alignment mechanism, the off-diagonal terms are just compatible with existing constraints on \CP-conserving FCNC's if the scalar masses are the $1\TeV$ range. (Consistency with $\ep_K$ requires scalar masses of order $10\TeV$.) In either case, one expects FCNC's that may be observed with increased experimental sensitivity. The models require a dynamical assumption regarding the sign of a dynamically-generated mass term. (If the sign is opposite to what is assumed here, one can use the dynamics to build a composite messenger model of direct gauge-mediated \susy breaking along the lines of \Ref{LutyTerning}.) We close with some speculations on how to build more attractive models based on the ideas presented here. The models discussed in this paper have a large number of states charged under the standard-model gauge group above the compositeness scale, resulting in a Landau pole close to the compositeness scale. Also, the scale of flavor physics must be very close to the compositeness scale in order to generate sufficiently large Yukawa couplings. Both of these potential difficulties may be alleviated if one could find models where the composite states correspond to dimension-2 ``meson'' operators of the form $P_1 P_2$, where $P_{1,2}$ are strongly-coupled preons. In that case, Yukawa couplings involving the composite states arise from terms in the Lagrangian of the form \beq \de\scr{L} \sim \myint d^2\th \left[ \frac{1}{M^2} (P_1 P_2)^2 H + \frac{1}{M} (P_1 P_2) \Phi H \right] + \hc, \eeq where $H$ is an elementary Higgs field and $\Phi$ is an elementary third-generation quark or lepton field. This gives rise to Yukawa couplings for the composite fields of order \beq y_q \sim \frac{\avg{P}^2}{M^2} \eeq compared to $\avg{P}^4 / M^4$ in our models. This would allow the flavor scale to be larger compared to the compositeness scale. Off-diagonal scalar mass terms for the composite fields arise from \beq \de\scr{L} \sim \myint d^4\th\, \frac{1}{M^2} (P_1^\dagger P^{\vphantom{\dagger}}_1) (P_2^\dagger P^{\vphantom{\dagger}}_2), \eeq giving rise to off-diagonal scalar masses for the composite states \beq \frac{\de m^2_{jk}}{M_{\rm comp}^2} \sim \frac{\avg{F_{P}}^2}{M^2} \sim \frac{\avg{P}^2}{M^2} M_{\rm comp}^2 \sim y_{jk} M_{\rm comp}^2, \eeq where $y_{jk}$ is the corresponding off-diagonal Yukawa coupling. This is an extra suppression by $\sqrt{y_{jk}}$ compared to our models, which makes FCNC's completely safe. Finally, one might hope that the group-theory structure of such models allows more economical models with a higher Landau pole for the standard-model interactions. It is also interesting to see if models of this type can give rise to realistic theories of flavor. We believe that these are promising directions, and work along these lines is in progress \cite{wip}. % ---------------------------------------------------------------------------- \section{Acknowledgments} % ---------------------------------------------------------------------------- N.A-H. thanks L. Hall and H. Murayama for useful discussions. M.A.L. thanks the theory group at LBNL for hospitality during the initial stages of this work. J.T. thanks M. Schmaltz for useful discussions. N.A-H. is supported by the Department of Energy under contract DE-AC03-76SF00515. M.A.L. is supported by a fellowship from the Alfred P. Sloan Foundation. 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