%Paper: 
%From: MARTIN@neuhep.hex.neu.edu
%Date: Thu, 15 Jul 1993 01:09 -0400 (EDT)


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%      Various internal macros
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\def\beginparmode{\endmode
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{\obeylines\gdef\
{}}
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  {\vskip 3pt plus 0.3fill \beginparmode
   \singlespace \narrower \noindent ABSTRACT: }
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  {\endpage                     %       This subsumes \body
   \body}
\def\body                       % Begin text body;  can be used to end
  {\beginparmode}               % \title, \author, \affil, \abstract,
                                % \reference, or \figurecaption modes
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   \raggedcenter \uppercase{#1}\par}
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%%\def\subhead#1{                 % Subhead;  NOTE enclose the text in {}
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%%  {\raggedcenter #1 \par}
%%   \nobreak\vskip 0.25truein\nobreak}
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  {\raggedcenter #1 \par}
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\def\references                 % Begin references -- basic format is Phys Rev
  {\subhead{References}         % I.e., volume, page, year(space after commas).
   \beginparmode
   \frenchspacing \parindent=0pt \leftskip=1truecm
   \parskip=8pt plus 3pt \everypar{\hangindent=\parindent}}
\gdef\refis#1{\indent\hbox to 0pt{\hss#1.~}}    % Ref list numbers.
\gdef\journal#1, #2, #3, 1#4#5#6{               % Journal reference. Comma sets
    {\sl #1~}{\bf #2}, #3, (1#4#5#6)}           % off: name, vol, page, year
\def\refstylenp{                % Nucl Phys(or Phys Lett) ref style: V, Y, P
  \gdef\refto##1{ [##1]}                                % Reference in text []
  \gdef\refis##1{\indent\hbox to 0pt{\hss##1)~}}        % Ref list numbers)
  \gdef\journal##1, ##2, ##3, ##4 {                     % Journal reference
     {\sl ##1~}{\bf ##2~}(##3) ##4 }}
\def\refstyleprnp{              % Input like pr, output like np!!
  \gdef\refto##1{ [##1]}                                % Reference in text []
  \gdef\refis##1{\indent\hbox to 0pt{\hss##1.~}}        % Ref list numbers)
  \gdef\journal##1, ##2, ##3, 1##4##5##6{               % Journal reference
    {\sl ##1~}{\bf ##2~}(1##4##5##6) ##3}}
\def\pr{\journal Phys. Rev., }
\def\prd{\journal Phys. Rev. D, }
\def\prl{\journal Phys. Rev. Lett., }
\def\prpts{\journal Phys. Rep., }
\def\np{\journal Nucl. Phys., }
\def\pl{\journal Phys. Lett., }
\def\plb{\journal Phys. Lett. B, }
\def\endreferences{\body}
\def\endpage                    %  Eject a page
  {\vfill\eject}
\def\endpaper                   %  Ways to say goodbye
  {\endmode\vfill\supereject}
\def\endit
  {\endpaper\end}
%%      Various user definitions
\def\ref#1{Ref. #1}                     %       for inline references
\def\Ref#1{Ref. #1}                     %       ditto
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%%		EQNORDER.TEX			11/05/85	Doug E.
\catcode`@=11
\newcount\tagnumber\tagnumber=0

\immediate\newwrite\eqnfile
\newif\if@qnfile\@qnfilefalse
\def\write@qn#1{}
\def\writenew@qn#1{}
\def\w@rnwrite#1{\write@qn{#1}\message{#1}}
\def\@rrwrite#1{\write@qn{#1}\errmessage{#1}}

\def\taghead#1{\gdef\t@ghead{#1}\global\tagnumber=0}
\def\t@ghead{}

\expandafter\def\csname @qnnum-3\endcsname
  {{\t@ghead\advance\tagnumber by -3\relax\number\tagnumber}}
\expandafter\def\csname @qnnum-2\endcsname
  {{\t@ghead\advance\tagnumber by -2\relax\number\tagnumber}}
\expandafter\def\csname @qnnum-1\endcsname
  {{\t@ghead\advance\tagnumber by -1\relax\number\tagnumber}}
\expandafter\def\csname @qnnum0\endcsname
  {\t@ghead\number\tagnumber}
\expandafter\def\csname @qnnum+1\endcsname
  {{\t@ghead\advance\tagnumber by 1\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+2\endcsname
  {{\t@ghead\advance\tagnumber by 2\relax\number\tagnumber}}
\expandafter\def\csname @qnnum+3\endcsname
  {{\t@ghead\advance\tagnumber by 3\relax\number\tagnumber}}

\def\equationfile{%
  \@qnfiletrue\immediate\openout\eqnfile=\jobname.eqn%
  \def\write@qn##1{\if@qnfile\immediate\write\eqnfile{##1}\fi}
  \def\writenew@qn##1{\if@qnfile\immediate\write\eqnfile
    {\noexpand\tag{##1} = (\t@ghead\number\tagnumber)}\fi}
}

\def\callall#1{\xdef#1##1{#1{\noexpand\call{##1}}}}
\def\call#1{\each@rg\callr@nge{#1}}

\def\each@rg#1#2{{\let\thecsname=#1\expandafter\first@rg#2,\end,}}
\def\first@rg#1,{\thecsname{#1}\apply@rg}
\def\apply@rg#1,{\ifx\end#1\let\next=\relax%
\else,\thecsname{#1}\let\next=\apply@rg\fi\next}

\def\callr@nge#1{\calldor@nge#1-\end-}
\def\callr@ngeat#1\end-{#1}
\def\calldor@nge#1-#2-{\ifx\end#2\@qneatspace#1 %
  \else\calll@@p{#1}{#2}\callr@ngeat\fi}
\def\calll@@p#1#2{\ifnum#1>#2{\@rrwrite{Equation range #1-#2\space is bad.}
\errhelp{If you call a series of equations by the notation M-N, then M and
N must be integers, and N must be greater than or equal to M.}}\else%
 {\count0=#1\count1=#2\advance\count1 by1\relax\expandafter\@qncall\the\count0,
  \loop\advance\count0 by1\relax%
    \ifnum\count0<\count1,\expandafter\@qncall\the\count0,%
  \repeat}\fi}

\def\@qneatspace#1#2 {\@qncall#1#2,}
\def\@qncall#1,{\ifunc@lled{#1}{\def\next{#1}\ifx\next\empty\else
  \w@rnwrite{Equation number \noexpand\(>>#1<<) has not been defined yet.}
  >>#1<<\fi}\else\csname @qnnum#1\endcsname\fi}

\let\eqnono=\eqno
\def\eqno(#1){\tag#1}
\def\tag#1$${\eqnono(\displayt@g#1 )$$}

\def\aligntag#1\endaligntag
  $${\gdef\tag##1\\{&(##1 )\cr}\eqalignno{#1\\}$$
  \gdef\tag##1$${\eqnono(\displayt@g##1 )$$}}

\let\eqalignnono=\eqalignno

\def\eqalignno#1{\displ@y \tabskip\centering
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    #1\crcr}}

\def\displayt@gpar(#1){(\displayt@g#1 )}

\def\displayt@g#1 {\rm\ifunc@lled{#1}\global\advance\tagnumber by1
        {\def\next{#1}\ifx\next\empty\else\expandafter
        \xdef\csname @qnnum#1\endcsname{\t@ghead\number\tagnumber}\fi}%
  \writenew@qn{#1}\t@ghead\number\tagnumber\else
        {\edef\next{\t@ghead\number\tagnumber}%
        \expandafter\ifx\csname @qnnum#1\endcsname\next\else
        \w@rnwrite{Equation \noexpand\tag{#1} is a duplicate number.}\fi}%
  \csname @qnnum#1\endcsname\fi}

\def\ifunc@lled#1{\expandafter\ifx\csname @qnnum#1\endcsname\relax}

\let\@qnend=\end\gdef\end{\if@qnfile
\immediate\write16{Equation numbers written on []\jobname.EQN.}\fi\@qnend}

\catcode`@=12
%%%%%%%%%%%%%%%%%%%%             REFORDER.TEX              %%%%%%%%%%%%%%%%%%%%
\refstyleprnp
\refstyleprnp
\catcode`@=11
\newcount\r@fcount \r@fcount=0
\def\refreset{\global\r@fcount=0}
\newcount\r@fcurr
\immediate\newwrite\reffile
\newif\ifr@ffile\r@ffilefalse
\def\w@rnwrite#1{\ifr@ffile\immediate\write\reffile{#1}\fi\message{#1}}

\def\writer@f#1>>{}
\def\referencefile{%			  Stuff to write .REF file
  \r@ffiletrue\immediate\openout\reffile=\jobname.ref%
  \def\writer@f##1>>{\ifr@ffile\immediate\write\reffile%
    {\noexpand\refis{##1} = \csname r@fnum##1\endcsname = %
     \expandafter\expandafter\expandafter\strip@t\expandafter%
     \meaning\csname r@ftext\csname r@fnum##1\endcsname\endcsname}\fi}%
  \def\strip@t##1>>{}}
\let\referencelist=\referencefile

\def\citeall#1{\xdef#1##1{#1{\noexpand\cite{##1}}}}
\def\cite#1{\each@rg\citer@nge{#1}}	% Variable No. of args, separatedby ","

\def\each@rg#1#2{{\let\thecsname=#1\expandafter\first@rg#2,\end,}}
\def\first@rg#1,{\thecsname{#1}\apply@rg}	% each@ag is a general purpose
\def\apply@rg#1,{\ifx\end#1\let\next=\relax%	  variable no. of arg. macro.
\else,\thecsname{#1}\let\next=\apply@rg\fi\next}% args separated by commas

\def\citer@nge#1{\citedor@nge#1-\end-}	% Check for M-N range (M and N numbers)
\def\citer@ngeat#1\end-{#1}
\def\citedor@nge#1-#2-{\ifx\end#2\r@featspace#1 % Single argument
  \else\citel@@p{#1}{#2}\citer@ngeat\fi}	% M-N range of arguments
\def\citel@@p#1#2{\ifnum#1>#2{\errmessage{Reference range #1-#2\space is bad.}
    \errhelp{If you cite a series of references by the notation M-N, then Mand
    N must be integers, and N must be greater than or equal to M.}}\else%
 {\count0=#1\count1=#2\advance\count1 by1\relax\expandafter\r@fcite\the\count0,
  \loop\advance\count0 by1\relax%	  Loop from M to N
    \ifnum\count0<\count1,\expandafter\r@fcite\the\count0,%
  \repeat}\fi}

\def\r@featspace#1#2 {\r@fcite#1#2,}	% Eat spaces at beginning or end of arg
\def\r@fcite#1,{\ifuncit@d{#1}%		  Cite individual reference
    \newr@f{#1}%
    \expandafter\gdef\csname r@ftext\number\r@fcount\endcsname%
                     {\message{Reference #1 to be supplied.}%
                      \writer@f#1>>#1 to be supplied.\par}%
 \fi%
 \csname r@fnum#1\endcsname}
\def\ifuncit@d#1{\expandafter\ifx\csname r@fnum#1\endcsname\relax}%
\def\newr@f#1{\global\advance\r@fcount by1%
    \expandafter\xdef\csname r@fnum#1\endcsname{\number\r@fcount}}

\let\r@fis=\refis			% Save old \refis, redefine
\def\refis#1#2#3\par{\ifuncit@d{#1}%      Use two params #2 #3 to strip blank
   \newr@f{#1}%
   \w@rnwrite{Reference #1=\number\r@fcount\space is not cited up to now.}\fi%
  \expandafter\gdef\csname r@ftext\csname r@fnum#1\endcsname\endcsname%
  {\writer@f#1>>#2#3\par}}

\def\ignoreuncited{%   redefine \refis if ignoring uncited references
   \def\refis##1##2##3\par{\ifuncit@d{##1}%
     \else\expandafter\gdef\csname r@ftext\csname r@fnum##1\endcsname\endcsname
     {\writer@f##1>>##2##3\par}\fi}}

\def\r@ferr{\endreferences\errmessage{I was expecting to see
\noexpand\endreferences before now;  I have inserted it here.}}
\let\r@ferences=\references
\def\references{\r@ferences\def\endmode{\r@ferr\par\endgroup}}

\let\endr@ferences=\endreferences
\def\endreferences{\r@fcurr=0%		  Save old \endreferences, redefine
  {\loop\ifnum\r@fcurr<\r@fcount%	  Loop over refnum and produce text
    \advance\r@fcurr by 1\relax\expandafter\r@fis\expandafter{\number\r@fcurr}
    \csname r@ftext\number\r@fcurr\endcsname%
  \repeat}\gdef\r@ferr{}\global\r@fcount=0\endr@ferences}

\let\r@fend=\endpaper\gdef\endpaper{\ifr@ffile
\immediate\write16{Cross References written on []\jobname.REF.}\fi\r@fend}

\catcode`@=12

\def\reftorange#1#2#3{$^{\cite{#1}-\setbox0=\hbox{\cite{#2}}\cite{#3}}$}

\citeall\refto		% These macros will generate citations
\citeall\ref		%
\citeall\Ref		%

\referencefile
\def\uY{{U(1)_{\ssc Y}}}
\def\uYp{{U(1)_{\ssc Y^{\prime}}}}
\def\uYpp{{U(1)_{\ssc Y^{\prime\prime}}}}
\def\uEM{{U(1)_{\ssc EM}}}
\def\suL{{SU(2)_{\ssc L}}}
\def\suR{{SU(2)_{\ssc R}}}
\def\suc{{SU(3)_{\ss C}}}
\def\sucp{{SU(3)_{\ss {C^\prime}}}}
\def\sucpp{{SU(3)_{\ss {C^{\prime\prime}}}}}
\def\sucppp{{SU(3)_{\ss {C^{\prime\prime\prime}}}}}
\def\suPS{{SU(4)_{\ssc PS}}}
\def\suPSp{{SU(4)_{\ssc PS^\prime}}}
\def\tc{{SU(4)_{\ss TC}}}
\def\uBL{{U(1)_{\ssc B-L}}}
\def\uX{{U_1^{\ssc X}}}
\def\uA{{U(1)_{\ssc A}}}
\def\uR{{U(1)_{\ssc R}}}
\def\H{{\bar H}}
\def\VEV#1{\left\langle #1\right\rangle}
\def\vev#1{\langle #1\rangle}
\def\ssc{\scriptscriptstyle}
\def\ss{\scriptscriptstyle}
\def\qhat{\int d{\hat q}}
\def\khat{\int d{\hat k}}
%%\def\half{{\textstyle{ 1\over 2}}}
\def\half{1/2}
%%\def\frac#1/#2{{\textstyle{#1 \over #2}}}
\def\frac#1/#2{#1 / #2}
\def\nugrant{This work was supported in part by the National Science
Foundation Grants PHY-90-01439 and PHY-917809.}
\def\nub{Department of Physics\\Northeastern University\\Boston MA 02115}
\def\uof{Department of Physics\\University of Florida\\Gainesville FL 32611}
\def\neu{Physics Department, Northeastern University, Boston MA 02115 USA}
\def\oneandonefourthspace{\baselineskip=\normalbaselineskip
  \multiply\baselineskip by 5 \divide\baselineskip by 4}
\def\oneandthreefifthsspace{\baselineskip=\normalbaselineskip
  \multiply\baselineskip by 8 \divide\baselineskip by 5}
\def\twoandonefifthsspace{\baselineskip=\normalbaselineskip
  \multiply\baselineskip by 11 \divide\baselineskip by 5}
\def\twoandoneeigthsspace{\baselineskip=\normalbaselineskip
  \multiply\baselineskip by 17 \divide\baselineskip by 8}
\font\titlefont=cmr10 scaled\magstep3
\def\bigtitle                      %  Title on title page
  {\null\vskip 3pt plus 0.2fill
   \beginlinemode \doublespace \raggedcenter \titlefont}

\preprintno{NUB-3063-93TH}
\oneandonefourthspace
\bigtitle{Automatic Gauged R-Parity} \bigskip
\author Stephen P. Martin
\affil\nub\body
\abstract
In the minimal supersymmetric standard model, the existence of R-parity is not
required for the internal consistency of the theory and might therefore be
regarded as {\it ad hoc}. I catalog some simple conditions which are sufficient
to guarantee that R-parity survives as an unbroken gauged discrete subgroup of
the continuous gauge symmetry in certain extensions of the minimal
supersymmetric standard model. If these criteria are met, R-parity is
automatic. [Based on a talk given at the International Workshop on
Supersymmetry and Unification of Fundamental Interactions (SUSY93) at
Northeastern University, Boston, March 29-April 1 1993.]

\body

\endtitlepage
\oneandonefourthspace

Low energy $N=1$ supersymmetry has been proposed as a cure for the fine-tuning
problem associated with the Higgs scalar boson[\cite{reviews}]. However,
in the minimal supersymmetric extension of the standard model, proton
decay might be expected to proceed at an unacceptable rate due to the virtual
exchange of the superpartners of the standard model
states[\cite{rp}].
To see this, we can write all of the renormalizable and gauge-invariant
terms which might occur in the superpotential:
$$
\eqalign{
& W = W_0 + W_1 + W_2
\cr
& W_0 = \mu  H \H + y_u Q \H u + y_d Q H d + y_e LH e
\cr
& W_1 = \lambda_1 udd
\cr
& W_2 = \mu^\prime L \H +
\lambda_2 QLd +
\lambda_3 LLe
\>\>\> .
\cr
}
$$
[Here $Q$ and $L$ are chiral superfields for the $\suL$-doublet quarks and
leptons; $u$, $d$, $e$ are chiral superfields for the $\suL$-singlet quarks
and leptons, and $H$, $\H$ are the two $\suL$-doublet Higgs chiral superfields.
Family and gauge indices are suppressed. It is possible to eliminate
$\mu^\prime$ by a suitable rotation among the superfields $H$ and $L$;
but we choose not to do this because in most extensions of
the minimal supersymmetric standard model $H$ and $L$ will not
have the same quantum numbers.]
The terms in $W_0$ are just the supersymmetric versions of the usual standard
model Yukawa couplings and Higgs mass, and they conserve baryon number ($B$)
and lepton number ($L$). However, $W_1$
violates $B$ by one unit and $W_2$
violates $L$ by one unit. To prevent the proton from decaying in short order,
either $(\lambda_1)$ or $(\mu^\prime, \lambda_2, \lambda_3)$
must be very small. (For precise constraints, see refs. [1-\cite{FIQ}]
and others listed in [\cite{constraints}].)

The simplest and most popular
way to save the proton and avoid other phenomenological
disasters is to just banish all
of the terms occuring in $W_1$ and $W_2$
by means of a discrete $Z_2$ symmetry
known as R-parity[\cite{rp}]. All of the standard model
states are taken to be even under R-parity and their superpartners are taken
to be odd. All interactions are required to have even R-parity. This means
that particles with odd R-parity are always produced in pairs, and that the
lightest particle with odd R-parity must be stable. At the level of the
chiral superfields, this may be implemented by assigning $R_p = -1$ to
$Q,L,u,d,e$ and $R_p = +1$ to $H, \H$.
(This $R_p$ is trivially related to R-parity by a factor
of $-1$ for fermions and is usually called matter parity.)
Then the terms in $W_1$ and $W_2$
are forbidden because they are $R_p$-odd,
while the terms in $W_0$ are $R_p$-even and allowed.
The $R_p$ symmetry also forbids some $B$ and $L$-violating operators of
dimension five and higher.

At the level of the minimal supersymmetric standard model, the assumption
of R-parity
appears {\it ad hoc}, in the sense that nothing goes wrong with the
internal consistency of the theory if $R_p$ is not imposed.
In contrast, $R_p$ is actually
automatic in certain extensions of the minimal supersymmetric standard
model  which have gauged $B-L$ (e.g. some supersymmetric
grand unified theories), and moreover can survive the spontaneous breakdown of
the continuous gauge invariance to the standard model gauge group. This will
occur if certain surprisingly mild conditions are met
by the order parameters of the theory. This seems to have
been underemphasized in the literature.
I will catalog some of the simple criteria which are
sufficient to guarantee that $R_p$ is an unbroken discrete gauge symmetry
for various choices of the gauge group, by classifying the possible gauge
transformation properties of the order parameters of the
theory as ``safe" or ``unsafe" for $R_p$. For our purposes, it is most
convenient to note that for each chiral superfield,
$$
R_p = (-1)^{3(B-L)}
\>\>\> .
\eqno(rp)
$$
This strongly suggests that we obtain gauged $R_p$ as the discrete
remnant of a gauged $\uBL$. In fact, with the $\uBL$ assignments
$Q\sim\frac 1/3$;
$L\sim -1$; $u,d \sim -\frac 1/3$; $e\sim 1$; and $H,\H \sim 0$, it is clear
that unbroken $\uBL$ forbids each of the terms in $W_1$ and $W_2$.
To guarantee that $R_p$ remains unbroken even after
$\uBL$ is broken, it is necessary and sufficient to require that all
Higgs vacuum expectation values (or other order parameters)
carry $3(B-L)$ charges which are even integers[\cite{FIQ}].
Following the general arguments of Krauss and Wilczek[\cite{KW}], $\uBL$
then breaks down to a gauged  $Z_2$ subgroup which, in view of Eq.~\(rp),
is nothing other than $R_p$. Unlike a global symmetry, such a gauged
discrete symmetry cannot be violated by Planck-scale effects[\cite{KW}].

A natural setting for gauged $\uBL$ is in the Pati-Salam unification
of color and lepton number: $\suPS \supset \suc \times \uBL$.
Under the gauge group $ \suPS \times \suL \times \uR$, the standard model quark
and lepton superfields transform as  $Q,L \sim ({\bf 4}, {\bf 2}, 0)$ and
$d,e \sim ({\bf \overline 4}, {\bf 1}, \half)$ and
$u,\nu \sim ({\bf \overline 4}, {\bf 1}, -\half)$. [Here $\nu$ is the
superfield for a neutrino which transforms as a singlet under the
standard model gauge group.] With unbroken $\suPS$, the couplings $\lambda_1$,
$\lambda_2$, and $\lambda_3$ clearly vanish by gauge
invariance, since the $\suPS$ direct products
${\bf \overline 4} \times {\bf \overline 4} \times {\bf \overline 4}$
and ${\bf  4} \times {\bf  4} \times {\bf \overline 4}$ contain
no singlets. (All group theory conventions and facts used here may be
found in [\cite{Slansky}].)
Also, gauge invariance of the allowed Yukawa
couplings in $W_0$ requires that $H$ transforms as a linear combination of
$( {\bf 1}, {\bf 2}, -\half)$ and the color singlet part of
$( {\bf 15}, {\bf 2}, -\half)$, and that $\H$ transforms
as a linear combination of $( {\bf 1}, {\bf 2}, \half)$ and
the color singlet part of
$( {\bf 15}, {\bf 2}, \half)$. It then follows that $\mu^\prime$ vanishes
as well. So unbroken $\suPS$ prohibits the same terms in
$W_1$ and $W_2$ that $R_p$ does. This is hardly a surprise, since
$R_p$ is a discrete subgroup of $\uBL$ which is contained in $\suPS$.

Of course, $\suPS$ and $\uBL$ must be broken if we are to obtain the standard
model gauge group. To avoid breaking $R_p$ in the process, it is necessary
and sufficient that all of the order parameters
have even $\suPS$ quadrality, since
$$
SU(4)_{\ssc PS} \> {\rm quadrality} \> = \> 3(B-L) \qquad [{\rm mod }\> 4]
\>\>\> .
\eqno(quadrality)
$$
The Higgs superfields $H$ and $\H$
have zero $\suPS$ quadrality and thus do not break $R_p$ when they
acquire vacuum expectation values. Since any other order parameters must
also be color singlets, they may transform under $\suPS$ as
${\bf 1}, {\bf 10}, {\bf 15}, {\bf 35} \ldots$
and their conjugates, which I refer to as ``safe" reps. The order parameters
should not transform in ``unsafe" reps ${\bf 4}, {\bf 20^{\prime\prime}},
{\bf 36}, {\bf 56} \ldots$ if we want to ensure that $R_p$ survives.
In particular, the $\suL$-singlet
order parameters may transform in the ``safe" reps
$( {\bf 1}, {\bf 1}, 0)$; $( {\bf 10}, {\bf 1}, 1)$; $( {\bf 15}, {\bf 1}, 0)$;
$( {\bf 35}, {\bf 1}, 2)\ldots$ of $ \suPS \times \suL \times \uR$
and their conjugates, but not in the ``unsafe" reps
$( {\bf 4}, {\bf 1}, \half)$;
$( {\bf 20^{\prime\prime}}, {\bf 1}, -\frac 3/2)$; $\ldots$ and their
conjugates. As long as we arrange for the theory to only have order parameters
in ``safe" reps, then $R_p$ is automatic and cannot be be broken.

Actually, $3(B-L)$ is always an integer multiple of 6 for safe order
parameters in $\suPS$,
since they must also be color singlets. Thus the surviving discrete subgroup
of $\uBL$ is a $Z_6$; however, a $Z_3$ subgroup of this is just the
discrete center of $\suc$, which is already taken into account.
So the remaining $Z_2 = R_p$ is what really counts. Also note that if
all order parameters in the theory had zero $\suPS$ quadrality, then because
of Eq.~\(quadrality) we would be left with a $Z_4$ which contains $R_p$ as a
subgroup and eliminates certain operators of dimension $\geq 5$
which are allowed by $R_p$. However, such a situation is very unlikely,
since obtaining a realistic neutrino mass spectrum via the seesaw
mechanism[\cite{seesaw}] requires a Majorana mass term for $\nu$, which in turn
requires an order parameter with quadrality $2$.

The grand unified theory based on $SO(10)$ contains $B-L$ as a subgroup,
and so we may expect to obtain a nice criterion for this case also.
The standard model quark and lepton superfields all transform as components of
the 16-dimensional spinor rep of $SO(10)$: $Q,L,d,e,u,\nu \sim {\bf 16}$.
Now the absence of the couplings $\lambda_1$,
$\lambda_2$, and $\lambda_3$ follows in the language of
unbroken $SO(10)$ from the group theory fact
${\bf 16} \times {\bf 16} \times {\bf 16}
\not\supset {\bf  1}$. Furthermore, since ${\bf 16} \times {\bf 16} =
{\bf 10}_{\ssc S} + {\bf 120}_{\ssc A} + {\bf 126}_{\ssc S}$, it must be that
$H$ and $\H$ are each linear combinations of appropriate
components of ${\bf 10}$, ${\bf \overline {126}}$, and ${\bf 120}$
(which couples families antisymmetrically) in order to allow the
Yukawa couplings in $W_0$. Then from
${\bf 16} \times {\bf 10} \not\supset {\bf 1}$,
${\bf 16} \times {\bf 120} \not\supset {\bf 1}$,
and ${\bf 16} \times {\bf \overline {126}} \not\supset {\bf 1}$ it follows
that $\mu^\prime = 0$  also for unbroken $SO(10)$.

What happens after $SO(10)$ is broken?
Whether or not $R_p$ survives just depends
on how the order parameters transform under $SO(10)$.
The relevant property of the $SO(10)$ reps is the ``congruency
class" which is defined mod 4. In fact,
$$
SO(10) \> {\rm congruency}\>{\rm class} \>  = \>
3(B-L) \qquad [{\rm mod }\> 2],
\eqno(congruency)
$$
so that ``safe" reps for order parameters in $SO(10)$ are those with
congruency class 0 or 2. The safe reps are $\bf 10$,
$\bf 45$, $\bf 54$, $\bf 120$, $\bf 126$, $\bf 210$, $\bf 210^\prime$,
$\bf 320\, \ldots$ and their conjugates. The unsafe reps are
$\bf 16$, $\bf 144$, $\bf 560\, \ldots$ and their conjugates.
As long as only order parameters corresponding to even $SO(10)$ congruency
classes are used to break $SO(10)$, the resulting low energy gauge group
will inevitably include unbroken $R_p$.
The prospect of removing the unsafe reps for Higgs vacuum expectation
values from the model builder's palette seems unlamentable, since
$SO(10)$ can be broken down to
$\suc \times \suL \times \uY \times R_p$ with realistic Yukawa couplings and
then to $\suc \times \uEM \times R_p$ in a very wide variety of ways
using only safe order parameters.
Order parameters in unsafe reps are incapable of giving any masses to the
standard model states (or $\nu$) anyway. Note that the safe reps of $\suPS$
are precisely the ones which are embedded in safe reps of $SO(10)$, since
$\suPS$ quadrality $=SO(10)$ congruency class [mod 2].

(While chiral superfields in large reps may ruin the asymptotic freedom
of the unified gauge coupling, this need not concern us. The order
parameters associated with the large reps may find their place only in
a phenomenological description, and may not actually correspond to vacuum
expectation values for fundamental fields. Also, a Landau
singularity in the unified gauge coupling is presumeably irrelevant if it
occurs at a distance scale shorter than the Planck length.)

Note that gauged $\uBL$ does not occur in a pure $SU(5)$ grand unified theory.
With the standard $SU(5)$ assignments $L,d \sim {\bf \overline 5}$ and
$Q,u,e \sim {\bf 10}$, it is clear  that unbroken $SU(5)$ does allow
$\lambda_1$, $\lambda_2$ and $\lambda_3$, by looking at the standard
model content of the $SU(5)$ fact ${\bf \overline 5}\times
 {\bf \overline 5} \times {\bf 10} \supset {\bf 1}$.
Furthermore, $H$ may consist of some ${\bf \overline 5}$ and some
${\bf \overline {45}}$ and $\H$ of some ${\bf  5}$ and some
${\bf  45}$; so $\mu^\prime$ is also certainly allowed. There is
no reason for any of these couplings to vanish in pure $SU(5)$,
in sharp contrast to our other examples.
R-parity is never automatic in supersymmetric pure $SU(5)$ models.

The case of supersymmetric ``flipped" $SU(5)$[\cite{flipped}]
is quite different from pure $SU(5)$ since it
contains gauged $\uBL$. Under the gauge group $SU(5) \times U(1)_f$, the
standard model fermions transform as $Q,d,\nu \sim ({\bf 10},1)$;
$L,u \sim ({\bf \overline 5}, -3)$; and
$e \sim ({\bf 1}, 5)$. Naturally, $W_1$ and $W_2$ are absent as long as
$SU(5) \times U(1)_f$ is unbroken, which can be seen as a consequence of the
$\uBL$ subgroup. When
$SU(5) \times U(1)_f$ breaks, $R_p$ survives if the order parameters transform
as components of  ``safe" reps of
$SU(5) \times U(1)_f$. Since $U(1)_f$ charge $= 3(B-L)$ mod 2, the
safe reps are just those which have even integer
$U(1)_f$ charges in our normalization.
These include $({\bf 5},-2)$ and $({\bf \overline 5},2)$; the unsafe
reps include $({\bf 5},3)$ and $({\bf 10},1)$ and their conjugates.
One of the selling points of flipped $SU(5)$ is supposed to be that
the spontaneous symmetry breaking can be accomplished using only Higgs
fields in reps no larger than the $({\bf 10},1)$. However, this {\it cannot}
be accomplished if one insists on using only Higgs fields which are safe for
$R_p$. If unbroken $R_p$ exists in such models, it must come from an additional
structure (e.g. superstring theory).

The grand unified theory $E(6)$ also contains $\uBL$ as a subgroup.
However, reps of $E(6)$ cannot be classified as safe or unsafe for
$R_p$, because each irreducible rep contains components with both even and
odd values of $3(B-L)$. Since $R_p$ is an abelian discrete subgroup, it
suffices to classify superfields and possible order parameters in
$E(6)$ according to their transformation properties
under the subgroup $\suc \times \suL \times \uY \times \uBL \times \uA$.
(It does not concern us whether this subgroup is actually the unbroken gauge
group at any particular stage of symmetry breaking.)
The ${\bf 27}$ of $E(6)$ transforms under this subgroup as
$({\bf 3},{\bf 2}, \frac 1/6, \frac 1/3, 0) +
({\bf 1},{\bf 2}, -\half, -1, -1) +
({\bf {\overline 3}},{\bf 1}, -\frac 2/3, -\frac 1/3, 0) +
({\bf {\overline 3}},{\bf 1}, \frac 1/3, -\frac 1/3, -1) +
({\bf 1},{\bf 1}, 1, 1, 0) +
({\bf 1},{\bf 1}, 0, 1, 1) +
({\bf 1},{\bf 2}, \half, 0, 0) +
({\bf 1},{\bf 2}, -\half, 0, 1) +
({\bf 1},{\bf 1}, 0, 0, -1) +
({\bf {\overline 3}},{\bf 1}, \frac 1/3, \frac 2/3, 1) +
({\bf 3},{\bf 1}, -\frac 1/3, -\frac 2/3, 0) $.
This defines $\uA$, which may be
thought of as the extra $U(1)$ which lives in $E(6)$ but not $SO(10)$.
(A slightly clumsier choice for this $U(1)$ was made in [\cite{me}].)
The first five terms may be identified with $Q$, $L$, $u$, $d$, and $e$
respectively.
It then follows that $H$ and $\H$ transform as
$({\bf 1},{\bf 2}, -\half, 0, 1)$ and  $({\bf 1},{\bf 2}, \half, 0, 0)$.
With these assignments,
the Yukawa terms in $W_0$ all transform as
$({\bf 1},{\bf 1}, 0, 0, 0)$ and the terms $udd$, $QLd$ and $LLe$ in
$W_1$ and $W_2$ each transform as $({\bf 1},{\bf 1}, 0, -1, -2)$. The term
$L\H$ in $W_2$ transforms as $({\bf 1},{\bf 1}, 0, -1, -1)$. Note that
an order parameter $({\bf 1},{\bf 1}, 0, 0, -1)$ is necessary so that the
non-standard-model particles
$({\bf {\overline 3}},{\bf 1}, \frac 1/3, \frac 2/3, 1)$ and
$({\bf 3},{\bf 1}, -\frac 1/3, -\frac 2/3, 0)$ can get mass,
and to allow $H\H$ in $W_0$.
We now classify as safe or unsafe the possible $\Delta I=0$
and $\Delta I= \half$ order parameters which occur in the smallest few
reps of $E(6)$, namely the $\bf 27$, $\bf 78$, $\bf 351$, and
$\bf 351^\prime$. (Safe and unsafe reps in
${\bf \overline {27}}$, ${\bf \overline {351}}$ and
${\bf \overline {351^\prime}}$ can be found by conjugating the ones below.)

The $\Delta I=0$ order parameters for $E(6)$ which are safe for automatic
$R_p$ are: $({\bf 1},{\bf 1}, 0, 0, 0)$, which occurs three times in the
$\bf 78$ of $E(6)$;
$({\bf 1},{\bf 1}, 0, 0, 2)$, which occurs once  in the $\bf 351^\prime$;
$({\bf 1},{\bf 1}, 0, 0, -1)$, which occurs once in the ${\bf 27}$,
twice in the $\bf 351$, and once  in the $\bf 351^\prime$;
and $({\bf 1},{\bf 1}, 0, -2, 0)$, which occurs once  in the $\bf 351^\prime$.
The unsafe $\Delta I=0$ order parameters are:
$({\bf 1},{\bf 1}, 0, -1, -2)$ and $({\bf 1},{\bf 1}, 0, 1, 2)$, which each
occur once in the $\bf 78$; $({\bf 1},{\bf 1}, 0, 1, 1)$, which occurs once
in the ${\bf 27}$, twice in the $\bf 351$, and once  in the $\bf 351^\prime$;
and $({\bf 1},{\bf 1}, 0, -1, 0)$, which occurs once  in the $\bf 351$ and
the $\bf 351^\prime$.

Safe $\Delta I=\half$ order parameters for $E(6)$ transform as
$({\bf 1},{\bf 2}, -\half, 0, 1)$ and
$({\bf 1},{\bf 2}, \half, 0, 0)$, which
each occur once in the $\bf 27$, three times in the ${\bf 351}$ and twice
in the ${\bf 351^\prime}$. The unsafe $\Delta I= \half$ reps include
$({\bf 1},{\bf 2}, \half, 1, 0)$ and
$({\bf 1},{\bf 2}, -\half, -1,0)$ which each occur once in the $\bf 78$;
$({\bf 1},{\bf 2}, -\half, -1, -1)$ which occurs once in the ${\bf 27}$,
three times in the $\bf 351$, and twice in the $\bf 351^\prime$;
$({\bf 1},{\bf 2}, \half, 1, 2)$, $({\bf 1},{\bf 2}, \half, -1, -2)$ and
$({\bf 1},{\bf 2}, -\frac 3/2, -1, 0)$ which each occur once in the $\bf 351$
and in the $\bf 351^\prime$.

Now, using safe $\Delta I=0$ order parameters listed above for
$\bf 27$, $\bf 351$ and $\bf 351^\prime$, $E(6)$ can be broken
down to $\suc \times \suL \times \uY \times R_p$ and
the states other than $Q,L,u,d,e$ in the $\bf 27$ are all eligible to
obtain large masses. Also, safe order parameters
for $({\bf 1},{\bf 2}, \half, 0, 0)$ and  $({\bf 1},{\bf 2}, -\half, 0, 1)$
provide for standard model masses. So, the statement for $E(6)$ is that
the safe reps for order parameters correspond to those which are
necessary anyway to break $E(6)$ to the standard model with the correct mass
spectrum. Order parameters in the unsafe reps are not needed for anything
from this point of view, although demanding their absence would eliminate some
otherwise attractive symmetry breaking patterns.

Dimension-five operators which violate $B$ and $L$ can also contribute
to proton decay. The only such operators consistent with supersymmetry and
the standard model gauge group which are not already forbidden by $R_p$ are
$[uude]_F$, $[QQQL]_F$ and $[LL\H \H]_F$. The first two of these terms are
also allowed by any gauge invariance contained in $SO(10)$,
in view of the standard model content of the group theory fact
${\bf 16} \times {\bf 16} \times {\bf 16} \times {\bf 16} \supset {\bf 1}
+ {\bf 1}$. However, all three terms (and $H \H$ in $W_0$) are
prohibited by unbroken $E(6)$ because of ${\bf 27} \times {\bf 27} \times
{\bf 27} \times {\bf 27} \not\supset{\bf 1}$, etc.; this is a simple
consequence of $E(6)$ triality, since $Q,L,u,d,e,H,\H$ all
must have triality 1. Or, it may be viewed as a consequence of $\uA$
and $\uBL$ conservation.  The order parameter
$({\bf 1},{\bf 1}, 0, 0, 2)$ breaks $\uA$ to a $Z_{2}$ without allowing the
aforementioned  terms. An order parameter
$({\bf 1},{\bf 1}, 0, 0, 1)$, which is presumeably necessary as
noted earlier, will break this $Z_2$, thus allowing
$[uude]_F$, $[QQQL]_F$ and $[ H \H]_F$.
The order parameter
$({\bf 1},{\bf 1}, 0, -2, 0)$ breaks $\uBL$ to $R_p$ and allows $[LL\H \H]_F$.
The fact that these terms are forbidden by an
unbroken $E(6)$ gauge group [or any subgroup of $E(6)$ containing
$\uA$ and $\uBL$]
provides a means of suppressing them
in the  low energy theory, although the extent
to which this is true is model-dependent.

If an $R_p$-unsafe order parameter does occur in a model with gauged
$\uBL$, then $R_p$ is spontaneously broken. This is not necessarily
a disaster if the mass scales of the unsafe order parameters are
sufficiently small compared to the unification scale. One popular example
which has been explored in the literature[\cite{spon}] involves spontaneous
$R_p$ breaking due to an expectation value for the scalar partner
of a neutrino which transforms in the unsafe rep
$({\bf 1},{\bf 1},0,1,1) \subset {\bf 27}$ of
$\suc \times \suL \times \uY \times \uBL \times \uA \subset E(6)$, or the
unsafe rep $\bf 16$ of $SO(10)$. There are, of course, other possible
examples (of problematical phenomenological viability) for spontaneously
broken $R_p$ if other unsafe reps listed above obtain vacuum expectation
values. The dominant contributions to $B$ and $L$-violating terms in the low
energy effective superpotential come from tree graphs with unsafe order
parameters on external legs. Note that explicit R-parity
breaking is really a contradiction in terms for any model with gauged $\uBL$,
because of Eq.~\(rp); it must be either exact or spontaneously broken.

In superstring-inspired models based on remnants of $E(6)$, the existence of
$R_p$ depends on e.g.~the properties of the six-dimensional compactified
manifold. There are a bewildering plethora of possibilities for the vacuum,
some of which respect $R_p$ or other generalized matter parities, and some of
which do not. I only wish to note that it is generally
not possible to break the $\uBL$ subgroup to
$R_p$ in these models {\it after} compactification, because the necessary
safe order parameter resides in the $\bf 351^\prime$ (or larger) rep
of $E(6)$, and not in the $\bf 27$, $\bf\overline {27}$, or singlet reps
which are available for chiral superfields in the perturbative
field theory limit of superstring theory. In such models it is not possible
to understand in detail ``why" unbroken $R_p$ should exist without
understanding the compactification mechanism.

Here I have considered instead criteria which can be understood
using only effective $N=1$ supersymmetric field theories. These criteria
are reassuringly not too restrictive, and are
especially crisp in the languages of $\uBL$, $\suPS$ and $SO(10)$.
The case of $E(6)$ is a little more involved, as outlined above.
The situation for models based on arbitrary subgroups of $E(6)$
can be easily inferred from these
results. One might interpret the natural appearance of automatic R-parity
in these cases as circumstantial evidence against supersymmetric
models which do not possess a gauged $B-L$ subgroup
[such as the minimal supersymmetric standard model and
supersymmetric pure $SU(5)$].
It has been pointed out that besides R-parity there are other
possible discrete symmetries[\cite{FIQ},\cite{BHR}] which allow some terms
from either $W_1$ or $W_2$, as well as different
combinations of higher dimension operators violating $B$ and $L$.
However, from the point of view adopted here, those rival discrete
symmetries are disfavored because
it turns out that they cannot be realized as an automatic consequence of
the gauge invariance found in $E(6)$ or any of its subgroups.
Hopefully R-parity appears less {\it ad hoc} in the light of the facts
presented here.

I thank Pierre Ramond for helpful comments.
This work was supported in part by the National Science Foundation Grants
PHY-90-01439 and PHY-917809 and by the Department of Energy contract
DE-FG05-86-ER40272 and by the Institute for Fundamental Theory at the
University of Florida.

\references
\refis{Slansky}
R.~Slansky, \prpts 79, 1, 1981.

\refis{reviews}
For reviews, see H.~P.~Nilles,  \prpts 110, 1, 1984 and
H.~E.~Haber and G.~L.~Kane, \prpts 117, 75, 1985.

\refis{KW} L.~Krauss and  F.~Wilczek, \prl 62, 1221, 1989.

\refis{me} S.~P.~Martin \prd 46, 2769, 1992.

\refis{constraints}
F.~Zwirner, \pl B132, 103, 1983;
L.~J.~Hall and M.~Suzuki, {\it Nucl. Phys.} {\bf B231} (1984), 419;
I.~H.~Lee, \np B246, 120, 1984;
S.~Dawson, \np B261, 297, 1985;
R.~Barbieri and A.~Masiero, \np B267, 679, 1986;
S.~Dimopolous, L.~J.~Hall, \pl B207, 210, 1987;
V.~Barger, G.~F.~Giudice, and T.~Han, \prd 40, 2987, 1989;
S.~Dimopolous, R.~Esmailzadeh, L.~J.~Hall, J.-P.~Merlo, and
G.~D.~Starkman, \prd 41, 2099, 1990;
R.~Barbieri, D.~E.~Brahm, L.~J.~Hall and S.~D.~H.~Hsu, \pl B238, 86, 1990;
B.~A.~Campbell, S.~Davidson, J.~Ellis and K.~A.~Olive, \pl B256, 457, 1991;
H.~Dreiner and  G.~G.~Ross, \np B365, 597, 1991.

\refis{BHR}
M.~C.~Bento, L.~Hall and  G.~G.~Ross, \np B292, 400, 1987.

\refis{rp} G.~Farrar, P.~Fayet, \pl B76, 575, 1978;
S.~Dimopolous and H.~Georgi, \np B193, 150, 1981;
S.~Weinberg, \prd 26, 287, 1982;
N.~Sakai and T.~Yanagida, \np B197, 83, 1982.


\refis{seesaw} M.~Gell-Mann, P.~Ramond, and R.~Slansky, in Sanibel Talk,
CALT-68-709, Feb 1979, and in {\it Supergravity},
(North Holland, Amsterdam, 1979; T.~Yanagida, in {\it Proc. of the
Workshop on
Unified Theories and Baryon Number in the Universe}, Tsukuba, Japan, 1979,
edited by A. Sawada and A. Sugamoto (KEK Report No. 79-18, Tsukuba, 1979).

\refis{flipped}  A.~De R\'ujula, H.~Georgi, and S.~L.~Glashow,
\prl 45, 413, 1980;
S.~M.~Barr, \pl B112, 219, 1982;
I.~Antoniadis, J.~Ellis, J.~S.~Hagelin, and D.~V.~Nanopoulos,
\pl B199, 231, 1987.

\refis{FIQ}  A.~Font, L.~E.~Ib\'a\~nez and F.~Quevedo,
\pl B228, 79, 1989;
L.~E.~Ib\'a\~nez and  G.~G.~Ross, \pl B260, 291, 1991,
\np B368, 3, 1992.


\refis{spon} C.~S.~Aulakh and R.~N.~Mohapatra,
\pl B119, 136, 1982;
J.~Ellis, G.~Gelmini, C.~Jarlskog, G.~G.~Ross, and J.~W.~F.~Valle,
\pl B150, 142, 1985;
G.~G.~Ross and J.~W.~F.~Valle,
\pl B151, 325, 1985;
B.~Gato, J.~Leon, J.~Perez-Mercader, and M.~Quiros,
\np B260, 203, 1985;
A.~Masiero and J.~W.~F.~Valle,
\pl B251, 273, 1990.

\endreferences\endit\end

