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\begin{document}
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\preprint{
\begin{minipage}[t]{3in}
\begin{flushright}
IASSNS--HEP--98/74\\OSU-HEP-98-5\\
 \\
\end{flushright}
\end{minipage}
}

\title{\Large\bf A Reexamination of Proton Decay \\
in Supersymmetric Grand Unified Theories}
\author{{\bf
K.S. Babu$^{1,2}$} and
{\bf
Matthew J. Strassler$^1$}}
\address{\ \\ $~^1$School of Natural Sciences,
Institute for Advanced Study,
Olden Lane,
Princeton, NJ 08540, USA\\}
\address{\ \\ $~^2$Department of Physics, Oklahoma State
University, Stillwater, OK 74078, USA.\footnote{Permanent address
as of August 1, 1998.}
\\ ~ \\
{\tt babu@osuunx.ucc.okstate.edu, strasslr@ias.edu}}
\maketitle
\begin{abstract}

We reconsider dimension--five proton decay operators, making
semi-quantitative remarks which apply to a large class of
supersymmetric GUTs in which the short-distance operators are
correlated with the fermion Yukawa couplings. In these GUTs, which
include minimal $SU(5)$, the operators
$(u^cd_i^c)^\dagger(d_j\nu_\tau)$, induced by charged Higgsino
dressing, completely dominate for moderate to large $\tanb$.  The rate
for $p \rarr (K^+,\pi^+) \overline{\nu}_\tau$ grows rapidly, as
$\tanbt4$, and the $K^+$ to $\pi^+$ branching ratio can often be
precisely predicted.  At small $\tanb$ the operators $(u d_i)(d_j
\nu)$ are dominant, while the operators $(u d_i)(u \ell^-)$, with
left-handed charged leptons, are comparable to the neutrino operators
in the generic GUT and suppressed in minimal GUTs.  Charged-lepton
branching fractions are always small at large $\tanb$.  The electron
to muon ratio is small in minimal GUTs but can be larger, even of
order one, in other models.  All other operators are very small.  At
small $\tanb$ in non-minimal GUTs, gluino and neutralino dressing
effects on neutrino rates are not negligible.
\end{abstract}
}

%\pacs{???}


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\vskip 0.4 in

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Unification of the three gauge forces of the standard model into a
single symmetry group is an attractive idea with a long history
\cite{patisalamA,patisalamB,ggGUT,gqwGUT,sakaisGUT,dgsGUT}.  Grand
Unified Theories (GUTs) provide a beautiful explanation of the
multiplet structure of the standard model fermions, and predict the
value of one of the gauge couplings at the weak scale in terms of the
other two.  The success of this prediction in the context of the
minimal supersymmetric extension of the standard model (MSSM) has
given new impetus to the study of SUSY GUTs over the past decade; for
a review see \cite{susyunifrev}.

 GUTs place quarks, leptons and their antiparticles in the same
multiplet, thereby making the nucleon unstable.  The momentum scale of
grand unification, as inferred from the extrapolation of the standard
model gauge couplings, is $M_{\rm GUT} \sim 2 \times 10^{16}$ GeV, so
large that nucleon decay rates are compatible with present
experimental limits.  In supersymmetric GUTs there are two sources of
baryon number violation. Dimension--six (four--fermion) operators are
generated by the exchange of superheavy gauge bosons of the GUT
symmetry group.  With $M_{\rm GUT} \sim 2 \times 10^{16}$ GeV, the
expected proton lifetime from these operators is $\tau(p \rightarrow
\pi^0 e^+) \simeq 10^{36 \pm 2}$ yr.~\cite{susyunifrev,hmy,nathC}; the
present experimental lower limit is $\tau(p \rightarrow \pi^0 e^+)\ge
2 \times 10^{33}$ yr.~\cite{superpi}.  The second source of proton
decay --- the subject of this paper --- is the dimension--five
(two--fermion/two--sfermion) operators induced by the exchange of
color--triplet Higgsinos, the GUT partners of the MSSM Higgs(ino)
doublets \cite{sakai,weinbergsusy}.  The associated amplitudes scale
as $M_{\rm GUT}^{-1}$, but are suppressed by light fermion Yukawa
couplings.  They are also suppressed by a loop factor, which arises
from the ``dressing'' of the operator by a gaugino or Higgsino that
converts the two sfermions into light fermions.  The size of the loop
integral is highly uncertain, due to the unknown masses of the
supersymmetric particles.  Additional uncertainty stems from the
unknown values of $\tanb \equiv \left\langle H_u
\right\rangle/\left\langle H_d\right \rangle$ and the triplet Higgsino
mass.  (In branching ratios, some of the unknown parameters cancel, so
they can be more reliably predicted than the overall rate.)  Since the
strength of these operators is proportional to Yukawa couplings, the
dominant modes are those with strange quarks -- the heaviest fermion
into which the proton can decay kinematically.  In minimal SUSY
$SU(5)$, including all the uncertainties, the lifetime may be
estimated as $\tau(p \rightarrow K^+ \overline{\nu}) =
(10^{28}-10^{34})$ yr.  This range is nearly eliminated by the
experimental limit $\tau(p \rightarrow K^+ \overline{\nu}) \ge 5
\times 10^{32}$ yr. \cite{superk}, suggesting that proton decay must
be discovered imminently if the idea of grand unification is on the
right track.

These issues have been studied extensively in minimal $SU(5)$
\cite{sakai,weinbergsusy,hmy,nathC,raby,rudaz,vysotskii,pal,nath,nathB,goto}
--- see \cite{hmy,nathC} for more references to the early literature
--- as well as in other GUTs \cite{bbtwo,bbsotenA,nathD,lucrab,macph}.
Most calculations have been specific to particular models.  In this
letter we attempt to make statements which, while more qualitative
than quantitative, apply to a wide class of theories.  Our purpose in
this paper is to summarize and clarify the expectations concerning
branching ratios in a large ensemble of supersymmetric GUTs, in which
the dimension-five operators are roughly correlated with the fermion
Yukawa couplings.  We will make this more definite below.  Our
approach is somewhat similar in spirit to that of \cite{kaplan}.


General statements of this type require a thorough analysis.
To this end a systematic method %, applicable to a wide range of models,
has been developed in which rough upper bounds on
baryon-number-violating operators may be established; this will be
presented elsewhere \cite{estimates}.  These bounds apply under the
following conditions.  The dimension-five operators take the form
%
\be{Lfiveo}
{\cal L}_5 = \int d^2\theta
\left\{\lambda_{ijkr}Q_i Q_j Q_k L_r
+\bar\lambda_{ijkr} U^c_iD^c_r U^c_j E^c_k\right\}+h.c.
\ee
%
(Capital letters denote superfields; $U_i$ contains the squark $\tilde
u_i$ and the quark $u_i$, {\it etc.})  We take as our ansatz the
relations $\lambda_{ijkr}\sim\bar\lambda_{ijkr}\sim{1\over
M}\eps_i\eps_j\eps_k$, independent of the index $r$, where $M$ is an
overall mass scale independent of $i,j,k,r$ and $\eps_3\sim 1,
\eps_2\equiv \eps \sim 1/25$ and $\eps_1\sim\eps_2^2$.  (The motivation
\cite{babubarr,msfermion} for the ansatz is associated with the
fact that, near the GUT scale, most of the flavor structure of the
theory can be roughly encoded in powers of $\eps$ \cite{tenc}.)
In \eref{Lfiveo} the $\lambda_{ijkr}$ are expressed in the gauge
basis, but it will be more convenient to work in the ``supersymmetric
basis'', in which the matter fermions are expressed as mass
eigenstates and the sfermions are expressed as the
supersymmetric partners of those eigenstates.  To convert from the
gauge $(g)$ to the supersymmetric $(m)$ basis, we must rotate the
superfields as $U_i^{(g)} =
R^u_{ij}U_j^{(m)},~D_i^{(g)}=R^d_{ij}D_j^{(m)},$ {\it etc.} so
that the fermion mass matrices are diagonal. 
%Since we maintain supersymmetry, the neutral
%gaugino interactions remain flavor diagonal.  From the original
%superpotential \eref{Lfiveo}, we obtain, 
In the supersymmetric basis,
%
\begin{equation}\label{Wfivesusy}
W_{dim-5}=(\hat{\lambda}^\ell_{ijkr} U_i^\alpha D_j^\beta
U_k^\gamma E_r
-\hat{\lambda}^\nu_{ijkr}
U_i^\alpha D_j^\beta D_k^\gamma
\NN_r
+\hat{\bar{\lambda}}_{ijkr} \bar{U}_i^\alpha \bar{D}_r^\beta
\bar{U}_j^\gamma \bar{E}_k) \epsilon_{\alpha\beta\gamma}~.
\end{equation}
%
where $\alpha,\beta,\gamma$ are color indices,  we have dropped the
superscript $(m)$, and we have defined
%
\begin{eqnarray}\label{lambdahat}
\hat{\lambda}^\ell_{ijkr} &=&\sum_{i',j',k',r'=1}^3
\lambda_{i'j'k'r'}R^u_{ii'}R^d_{jj'}R^u_{kk'}R^e_{rr'} \ ; %\nonumber\\
%\hat{\lambda}^\nu_{ijkr} &=&\sum_{i',j',k',r'=1}^3
%\lambda_{i'j'k'r'}R^u_{ii'}R^d_{jj'}R^d_{kk'}R^\nu_{rr'} \nonumber\\
%\hat{\bar{\lambda}}_{ijkr} &=&\sum_{i',j',k',r'=1}^3
%\lambda_{i'j'k'r'}\bar{U}^u_{ii'}\bar{U}^u_{jj'}\bar{U}^e_{kk'}
%\bar{U}^d_{rr'} ~;
\end{eqnarray}
%
the coefficients $\hat\lambda^\nu_{ijkr}$ and
$\hat{\bar{\lambda}}_{ijkr}$ are defined analogously.
As part of our ansatz, we assume the matrices $R^u$, $R^d$,
$R^{\bar u}$, $R^{\bar e}$ --- those associated with the ${\bf 10}$
representations of $SU(5)$ --- have the same texture as
$V_{CKM}$; we do not assume this for the other $R$ matrices.  If
$V_{CKM}$ satisfied $V_{ij} \sim \min\{\eps_i/\eps_j,\eps_j/\eps_i\}$, then
%the rotation would have no effect; 
we would have $\hat{\lambda}^\ell_{ijkr} \sim \hat{\lambda}^\nu_{ijkr}
\sim \hat{\bar{\lambda}}_{ijkr} \sim {\lambda}_{ijkr} \sim
\eps_i\eps_j\eps_k $.  However, the Cabibbo angle $\tc$ is of order
$5\eps_1/\eps_2$ and can enter into the $\hat\lambda$. An enhancement
by a factor of $\theta_c/({\eps_1\over\eps_2})\sim 5$ may occur for
each index $i,j,k$ taking value 1 \cite{tenc}; thus
%
\be{ansatz}
\hat{\lambda}^\ell_{ijkr} \sim \hat{\lambda}^\nu_{ijkr}
\sim \hat{\bar{\lambda}}_{ijkr}
\sim \eps_i\eps_j\eps_k
\Big[\theta_c/ (\eps_1/\eps_2)
\Big]^{(\delta_{i1}+\delta_{j1}+\delta_{k1})} \ .
\ee
%
Note $\delta_{i1}+\delta_{j1}+\delta_{k1}\leq 2$ because of the
antisymmetry properties of the $\hat\lambda$ coefficients.  More
details of this ansatz are given in \cite{estimates} and \cite{tenc}.

If the coefficients $\hat\lambda$ in a given GUT are all of order or
less than those appearing in \Eref{ansatz}, then we can apply the
results of \cite{estimates} to the model.  Although we will not
classify them here, many GUTs (including minimal $SU(5)$ and $SO(10)$
and their more realistic variants, as well as the ten-centered models
of \cite{babubarr,msfermion,tenc}) are compatible with the ansatz
\eref{ansatz}.  We will now demonstrate that the $\hat\lambda$
coefficients in the minimal $SU(5)$ GUTs are consistent with
\eref{ansatz}.  In the supersymmetric basis, the superpotential induced
by the color--triplet Higgsinos in minimal $SU(5)$ is
%
\be{GUTops}
W_C =  (U_i D_j e^{i \sigma_i} f_i V_{ij} + U_i^c E_j^c f_i V_{ij})H_C
+ (D_i \NN_i h_i - U_i E_j V_{ij}^* h_j + U_i^c D_j^c e^{-i
\sigma_i}V_{ij}^* h_j)\bar H_C~.
\ee
%
Here $H_C$ and $\bar H_C$ are the color triplet and anti--triplet from
the ${\bf 5}_H$ and ${\bf \overline{5}}_H$ of the Higgs multiplet, $V$
is the CKM matrix, $f_i$ ($h_i$) is the diagonal Yukawa coupling
matrix of the up--quarks (down-quarks and leptons), and the $\sigma_i$
are phase factors with $\sum_{i=1}^3 \sigma_i = 0$.  Exchange of the
Higgs color triplets, of mass $M$, leads to the dimension-five operators
%
\begin{eqnarray}\label{dimfive}
W_{d=5} &=& M^{-1}(U_i D_j D_k \NN_r e^{i \sigma_i} f_i V_{ij}
\delta_{kr} h_r - U_i D_j U_k E_r e^{i \sigma_i} f_i V_{ij} V_{kr}^*
h_r + U_i^c D_r^c U_j^c E_k^c e^{-i\sigma_i} f_j V_{jk} V_{ir}^* h_r)
\end{eqnarray}
%
along with other terms irrelevant for proton decay.

If it were the case that $\tc\sim\eps_1/\eps_2$, so
that $V_{ij}\sim \min\{\eps_i/\eps_j,\eps_j/\eps_i\}$, along with 
$f_i\sim \eps_i^2$, $h_i\sim \eps_i\zeta$, where $\zeta=\tanb/60$, 
we would have
%
\begin{eqnarray}\label{GUTmatches}
  f_i V_{ij} \delta_{kr} h_r \sim \eps_i^2 \eps_r \delta_{kr}
\min\{\eps_i/\eps_j,\eps_j/\eps_i\}\zeta \alt \eps_i\eps_j\eps_k\zeta
\nonumber \\
f_i V_{ij} V_{kr} h_r \sim \eps_i^2 \eps_r
\min\{\eps_i/\eps_j,\eps_j/\eps_i\}\min\{\eps_k/\eps_r,\eps_r/\eps_k\}\zeta
\alt \eps_i\eps_j\eps_k\zeta
\nonumber \\
 f_j V_{jk} V_{ir}^* h_r \sim \eps_j^2 \eps_r
\min\{\eps_j/\eps_k,\eps_k/\eps_j\}\min\{\eps_i/\eps_r,\eps_r/\eps_i\}\zeta
\alt \eps_i\eps_j\eps_k\zeta
\end{eqnarray}
%
All coefficients in \eref{GUTmatches} would then be equal to or
smaller than those in \Eref{ansatz}; note the overall factor of
$\zeta$ can be absorbed into the overall mass scale which affects all
$\hat\lambda$ equally.  Now let us account for the fact that $\tc\sim
5\eps_1/\eps_2$.  Inspection of \Eref{GUTmatches} shows that factors
of the Cabibbo angle can only enhance a coupling above
$\eps_i\eps_j\eps_k$ if one of the indices $i,j,k$ takes value 1;
$r=1$ cannot cause such an enhancement because of the factor $h_1$,
which reduces $\hat\lambda_{ijk1}^{(e,\nu)},\hat{\bar\lambda_{ijk1}}$
below its ansatz value.  If two such indices take value 1, then one
may get a double enhancement.  The resulting coefficients are the same
as or less than those appearing in \Eref{ansatz}, and thus the upper
bounds of \cite{estimates} are applicable.



Rough upper bounds from \cite{estimates} on various four-fermion
baryon-number-violating operators are given in the Table.  All
operators not shown are negligibly small. The Table shows the
different contributions from various gauginos and Higgsinos.
Wino-Higgsino mixing is not listed; in each case the upper bound on
such contributions lies between or below the pure Wino and the pure
Higgsino bounds.  Explicitly indicated are factors of $\eps$,
$\theta_c$, gauge coupling constants $g_i$ and third-generation Yukawa
couplings $y_b,y_\tau$ which stem from left-right mixing or Higgsino
couplings; note $y_b$ and $y_\tau$ are roughly of order $\zeta$.  Also
appearing are parameters $\gamma$ and $\delta$.  The first measures
the extent to which the $\tilde b,\tilde t$ are split in mass from and
mixed with the other left-handed squarks; if the messenger scale of
supersymmetry breaking is high, then $\gamma\sim 1$, while if it is
near the weak scale $\gamma$ may be small.  The factor $\delta =
(A\cos\beta+\mu\sin\beta)(A\sin\beta+\mu\cos\beta)v/\tilde m^2$
parameterizes the size of left-right squark mixing; here $v,\mu,\tilde
m,A$ are $246$ GeV, the Higgsino mass, the universal squark mass and
the coefficient of the trilinear scalar terms.  An overall factor of
$M^{-1}\zeta$ is omitted from every entry.  Neutrino flavors have not
been distinguished, and the flavor of a charged lepton is only
indicated when the bounds depend on the lepton flavor.  For additional
details see \cite{estimates}.

\bigskip




\def\udsn{$uds\nu$}%_r
\def\uusl{$uus\ell$}%_r
\def\uddn{$udd\nu$}%_r
\def\uudl{$uud\ell$}%_r
\def\ucdcsn{$(u^c d^c)^\dagger s\nu$}%_\tau
\def\ucscdn{$(u^c s^c)^\dagger d\nu$}%_\tau
\def\ucdcdn{$(u^c d^c)^\dagger d\nu$}%{$u^\cd d^\cd s\nu_\mu$}
%\def\{$u^\cd s^\cd d\nu_\mu
\def\ucmucsu{$ (u^c \mu^c)^\dagger su$}
\def\ucmucdu{$(u^c \mu^c)^\dagger du$}
\def\ucecsu{$(u^c e^c)^\dagger su$}
\def\ucecdu{$ (u^c e^c)^\dagger du$}

\begin{tabular}{|l|c|c|c|c|c|}
\hline
&&$\tilde W^{\pm}$&$\tilde H^{\pm}$&$\tilde g$&$\tilde W^0,\tilde B$
\\ \hline
\udsn,\uddn&&$g_2^2(\eps^3\tc^2,\eps^3\tc^3)
$&$\delta y_b^2(\eps^4\tc,\eps^5\tc)
$&$g_3^2\gamma(\eps^3\tc^2,\eps^4\tc^2)
$&$g_2^2(\eps^3\tc^2,\gamma\eps^4\tc^2)$
\\ \hline
\uusl,\uudl&&$g_2^2(\eps^3\tc^2,\eps^3\tc^3)
$&$\delta y_b^2(\eps^4\tc,\eps^5\tc)
$&$g_3^2\gamma y_b^2(\eps^4\tc,\eps^4\tc^2)
$&$g_2^2\gamma y_b^2(\eps^4\tc,\eps^4\tc^2)$
\\ \hline
\ucdcsn&&$g_2^2\delta y_\tau\eps^2\tc$&$y_\tau\eps^2\tc$&$0$&$0$
\\ \hline
\ucscdn,\ucdcdn&&$g_2^2\delta y_\tau\eps^3\tc$&$y_\tau\eps^3\tc$&$0$&$0$
\\ \hline
\ucmucsu,\ucmucdu&&$g_2^2\delta y_b(\eps^4\tc^2,\eps^5\tc^2)
$&$y_b(\eps^4\tc^2,\eps^5\tc^2)
$&$g_3^2\gamma \delta y_b^3(\eps^5\tc,\eps^6\tc)
$&$g_2^2\gamma \delta y_b^3(\eps^5\tc,\eps^6\tc)$\\ \hline
\ucecsu,\ucecdu&&$g_2^2\delta y_b(\eps^4\tc^3,\eps^5\tc^3)
$&$y_b(\eps^4\tc^3,\eps^5\tc^3)
$&$g_3^2\gamma \delta y_b^3(\eps^5\tc^2,\eps^6\tc^2)
$&$g_2^2\gamma \delta y_b^3(\eps^5\tc^2,\eps^6\tc^2)$
\\ \hline
\end{tabular}


\vskip .2 in  {\it Rule-of-thumb upper bounds on coefficients
of four-fermion baryon-number-violating operators, as found in
\cite{estimates} using the ansatz in \Eref{ansatz}.   Important loop
factors, matrix elements and overall coefficients are omitted and must
be accounted for when using this Table.  See the text for further
explanation.}
\vskip .2 in


At small $\tanb$ the operators $uds\nu$, $uus\mu$ and $uuse$ have the
largest bounds, while at large $\tanb$ the operator
$(u^cd^c)^\dagger(s\nu)$ potentially exceeds all others.  The
replacement of the $s$ quark by a $d$ quark engenders a small
suppression \cite{raby,rudaz}.  As we will see, it is always true that
many of these operators saturate or nearly saturate their bounds.  It
follows from the Table that operators with right-handed leptons, even
$u^\cd \mu^\cd s u$, are negligible at all values of $\tanb$, as are
operators with a left-handed charged lepton and two right-handed
quarks.  We will therefore restrict out attention to the first four
rows in the Table.

 At small $\tanb$ the largest four-fermion operator is $uds\nu$,
induced through dressing of the $CDS\NN_\mu$, $TDB\NN_\mu$ term in the
superpotential by the charged Wino, as in \FFig{udsn}
\cite{rudaz,nath}.  In minimal GUTs the second-generation amplitude is
proportional to $y_c y_s \theta_c^2$ ($y_c$ is the charm quark Yukawa
coupling, {\it etc.}), while the third-generation effect is roughly of
the same order.  Since $y_s y_c\theta_c^2\sim
(\eps_2\zeta)(\eps_2^2)\tc^2 \sim \eps^3\tc^2\zeta$, the coefficient
of \udsn\ saturates its upper bound in the Table.  Similar statements
apply to $udd\nu$, with an additional factor of $\tc$.  Interference
effects between second- and third-generation diagrams [which
unfortunately depend on the otherwise-unmeasurable phases $\sigma_i$
in \Eref{dimfive}] might suppress either $p\rarr K^+\bar\nu$ or
$p\rarr \pi^+\bar\nu$ \cite{nath} but not both \cite{hmy}. Accounting
for this and for the different hadronic matrix elements and phase
space factors, one finds the branching ratio $\Gamma(p\rarr K^+
\overline{\nu})/\Gamma(p\rarr \pi^+ \overline{\nu})$ tends to lie
between .1 and 1. (Throughout this letter we assume that sfermions of
the same charge are degenerate at the messenger scale of supersymmetry
breaking.) In more general GUTs satisfying \Eref{ansatz}, even this
predictivity may be lost, though the tendency is still the same.  The
rates for these processes grow as $\tanbt2$.


At large $\tanb$, the largest operator in these GUTs is
$(u^cd^c)^\dagger(s\nu)$, first discussed in \cite{nathB,lucrab}.
This operator is obtained through Higgsino exchange or through
left-right sfermion mixing, which always entails a factor of the
associated right-handed fermion mass.  The largest effect therefore
comes from changing a third-generation {\it right-handed} squark or
slepton to a light {\it left-handed} quark or lepton. Specifically,
begin with the operator $U^cD^cT^c\tau^c$ in the superpotential, and
then convert the sfermions $\tilde t^c\tilde \tau^c$ to the fermions
$s\nu_{\tau}$.  The Higgsino and Wino dressing diagrams are given in
\FFig{ucdcsn}; there are also diagrams with Higgsino-Wino mixing whose
size is intermediate between them.  The diagrams are proportional to
the coefficient $\hat{\bar{\lambda}}_{1331}$. For the Higgsino
dressing one has the couplings $y_tV_{ts}^*\sim y_t\eps_2$ for the
$\tilde t^c - s$ vertex, $y_\tau$ for the $\tilde\tau^c-\nu_\tau$
vertex.  A similar structure emerges for the Wino diagram but
suppressed by gauge couplings and mixing factors; the Higgsino diagram
usually dominates.  The coefficient of this operator in minimal
$SU(5)$ is proportional to $y_d y_t^2 y_\tau V_{ts}
\sim\eps^3\zeta^2$.  This lies $\eps/\tc$ below its upper bound,
because, as is also the case in many other GUTs,
$\hat{\bar{\lambda}}_{1331}=y_dy_t$ is a factor of $\eps/\tc$ below
the ansatz in \Eref{ansatz}.  This will not be true in non-minimal
GUTs where the matrix $R^{\bar d}$ is non-hierarchical and $R^{\bar
u}_{12}\sim \tc$, as in certain ten-centered models \cite{tenc} where
the bound on this coefficient can be saturated.

The same set of graphs, with only the external down-type quark flavors
changed, gives $(u^c d^c)^\dagger(s \nu_\tau)$, $(u^c s^c)^\dagger(d
\nu_\tau)$ and $(u^c d^c)^\dagger(d \nu_\tau)$.  In minimal $SU(5)$
the three amplitudes are in the ratio $y_dV_{ts},y_sV_{us}V_{td},y_d
V_{td}$; note these ratios are independent of the phases $\sigma_i$,
unlike the $ud_id_j\nu$ case.  The coefficients of $(u^c
d^c)^\dagger(s \nu_\tau)$ and $(u^c s^c)^\dagger(d \nu_\tau)$ are of
opposite sign, and may be of the same order.  (This is slightly
inconsistent with the Table; however, the estimates therein are rough,
and, as noted above, the $(u^c d^c)^\dagger(s \nu_\tau)$ amplitude
does not saturate its bound.)  However, the hadronic matrix elements
of these operators have opposite signs (to see this use
\cite{chw,chadha}) so interference in $p \rightarrow K^+
\overline{\nu}_\tau$ is constructive; also, since $(u^c d^c)^\dagger(s
\nu_\tau)$ has a significantly larger matrix element, its contribution
dominates.  The operator $(u^c d^c)^\dagger(d \nu_\tau)$ leads to $p
\rightarrow \pi^+ \overline{\nu}_\tau$; since its coefficient is in a
known relationship to the other two, a precise prediction for the
branching ratio $\Gamma(p \rightarrow \pi^+ \overline{\nu}_\tau)/
\Gamma(p \rightarrow K^+ \overline{\nu}_\tau)$ is possible,
independent of the supersymmetric spectrum. This predictivity is
retained in models where
$\hat{\bar{\lambda}}_{1331}/\hat{\bar{\lambda}}_{1332}$ is determined.
Even this information is unnecessary in GUTs where
$\hat{\bar{\lambda}}_{1331}$ is as large as allowed in \Eref{ansatz},
since in this case the $(u^c s^c)^\dagger(d \nu_\tau)$ operator is
negligible and only $\hat{\bar{\lambda}}_{1331}$ appears in the
amplitudes.

Since these amplitudes go as $\zeta^2$, the rates for $p\rarr K^+\bar
\nu_\tau,\pi^+\bar\nu_\tau$ grow like $\tanbt4$, leading to short
lifetimes and corresponding strong constraints at large $\tanb$.  We
find that the amplitudes for $uds\nu$ and $(u^cd^c)^\dagger(s\nu)$
become comparable in most GUTs for $\tanb$ somewhere between 3 and 15;
in a minimal supergravity $SU(5)$ GUT (accounting for the different
short-distance renormalizations and hadronic matrix elements of the
two operators) we find this number is of order $9 (m_{\tilde W}/\mu)$,
with large uncertainties from the ratio $y_d/y_s$, the sfermion
spectrum, poorly measured CKM angles, and the third-generation
contribution to the $uds\nu$ operator.\footnote{While this letter was
in preparation, a preprint appeared which studies the
$(u^cd_i^c)^\dagger(d_j\nu)$ operators in the minimal $SU(5)$ GUT
\cite{gotonihei}.}  For $\tanb\sim60$, the rate from
$(u^cd^c)^\dagger(s\nu_\tau)$ dominates that from $uds\nu$ by
$10-1000$.  In those non-minimal GUTs where the bound on
$(u^cd^c)^\dagger(s\nu_\tau)$ is saturated, the amplitude is enhanced
by another factor of $\tc$ and can dominate for even smaller values of
$\tanb$.  The constraints on large $\tanb$ models from these effects,
although discussed in \cite{lucrab}, do not appear to have been fully
incorporated in the literature.





The bounds on $uus\ell$ and $uud\ell$ are comparable to those on
$uds\nu$ and $udd\nu$.  However, in minimal (and some non-minimal)
GUTs, these bounds are not saturated.  From the Table, we see that at
small $\tanb$ we need only consider Wino exchange.  The argument that
$uus\ell$ is highly suppressed \cite{raby,rudaz} is that all
contributions are proportional to $f_1=y_u\sim\eps^4$, which makes the
resulting amplitudes of order $f_1h_2\sim\eps^5$, smaller than the
$\eps^3\tc^2$ bound.  To see this, consider the contribution of $U_i
D_j D_k \NN_r$ in \Eref{dimfive}; the sneutrino must couple to the
Wino, implying $i=1$ and giving a factor of $f_1$.  If instead we use
$U_i D_j U_k E_r$, either $i=1$ or $k=1$.  The former case gives $f_1$
directly, and the latter gives $f_1$ through a unitarity cancellation:
the diagram in \FFig{uusl} is proportional to
%
\be{CKMcancel}
h_2 V^*_{12} \sum_{i,j=1}^3  V^*_{1j} 
 V_{ij} f_i V^*_{ip} = h_2  f_1 V^*_{12} V_{1p} \ .
\ee
% 
%
However, the unitarity cancellation in \Eref{CKMcancel} partly fails
due to subtle renormalization group effects.  The operator $U_i D_j
U_k E_r$ is proportional to $V_{ij}$ at the GUT scale, but after
renormalization to low-energy it is no longer proportional to $V_{ij}$
at the weak scale.  This effect is of order $(1-y_t^2/y_f^2)^{1/24}$
\cite{theisenCKM,babuCKM} (here $y_f\sim 1.1$ is the fixed-point value
of $y_t$) or about $0.1$ for $\tanb\sim1.4-3$.  Specifically, consider
the $U_iD_jH_C$ coupling $\hat{F}_{ij}$ (we use the notation of
\cite{babubarr};) naively $\hat{F}_{ij}=f_iV_{ij}$, but in fact
$\hat{F}_{31},\hat{F}_{32}$ differ from this at the weak scale by
$\sim 10 \%$.  The sum over $j$ in \eref{CKMcancel} becomes $h_2
V_{12}^*\sum_j V_{1j}^* \hat{F}_{ij} V_{ip}\sim (0.1) h_2 V_{12}^*
V_{13}^* V_{33} V_{3p}f_3$, which is about a tenth of $h_2f_1$. The
loop factor for this effect can be enhanced if the third-generation
squarks are lighter than those of the first two generations; still,
even if the CKM angles, $f_1$ and the spectrum are at the edge of
their ranges, it can never be as large as the leading contribution.  A
second effect of the same order (if the messenger scale of
supersymmetry breaking scale is high) comes from the mixing and mass
splittings between the down-type squarks.  Both of these effects,
while interesting, are most likely lost in the uncertainties
surrounding $y_u$.


 In many non-minimal GUTs, the unitarity cancellation in
\eref{CKMcancel} simply does not occur.  A sufficient condition for
this is that $\hat{F}_{ij}$ be hierarchical but not precisely equal to
$f_iV_{ij}$. If $\hat{F}_{ij}\sim f_i
\min\{\eps_i/\eps_j,\eps_j/\eps_i\}$ as occurs in many realistic
models of flavor (including those in which higher-dimension operators
or non-minimal Higgs bosons contribute to the up-type quark masses),
the sum $V_{1j}^*\hat{F}_{ij}$ does not equal $f_1\delta_{1i}$;
instead it gives $\tc f_2,\eps\tc f_3$ for $i=2,3$. In this case the
diagrams in \FFig{uusl} with $i=2,3$ are proportional to $h_2 V^*_{12}
(\tc f_2) V^*_{22},h_2 V^*_{12} (\eps\tc f_3) V^*_{32}\sim
\eps^3\tc^2$.  This saturates the bound in the Table, and thus in
these theories the neutrino to charged-lepton branching ratio is order
one at small $\tanb$ (although the hadronic matrix elements favor
neutrinos.)  Enhancements of $F_{ij}$ by factors of
$\tc^{\delta_{i1}+\delta_{j1}}$ do not change this conclusion.

The charged lepton rates from these operators increase only as
$\tanbt2$.  As suggested in the Table, at large $\tanb$ a bigger
contribution to these operators may come from up-type squark mixing in
gluino dressing \cite{goto,bbtwo}; however the branching fraction to
charged leptons remains small due to the large $(u^cd^c)^\dagger
s\nu_\tau$ operator.

 It is clear from the Table that all observed muons should be
left-handed, in contrast to proton decays mediated by dimension-six
operators \cite{weinbergbl,wilczek}.  Although the Table suggests
branching fractions to electrons can be of the same order as those to
muons, in minimal and many non-minimal GUTs they are much smaller
\cite{raby,rudaz}.  In such GUTs the coefficients
$\hat\lambda^e_{ijkr}$ are not roughly independent of $r$, in contrast
to the ansatz \eref{ansatz}. However, the suppression factor is model
dependent, and there are theories (such as ten-centered models
\cite{babubarr,msfermion,tenc}) in which electron and muon decays do
have comparable rates.  The electron-to-muon branching ratios thus are
good probes of flavor physics \cite{kaplan}.



Finally, as evident in the Table, dressings involving gluinos and
neutralinos can be important at small $\tanb$ in their contribution to
the $uds\nu,udd\nu$ operators.  In minimal GUTs these contributions
are suppressed by $y_u$, but in non-minimal GUTs (as in \cite{lucrab})
they may become important.  Gluino dressing is naively subleading due
to the symmetry structure of the dimension-five operators
\cite{raby,vysotskii,pal}, and can only play a role when there is
significant flavor violation \cite{goto,bbtwo}.  In the supersymmetric
basis, flavor violation appears as intergenerational squark mixing.
At small values of tan$\beta$, $\tilde{d_i}-\tilde{d_j}$ mixings are
induced proportional to $y_t^2 V_{3i}V_{3j}^*\sim\eps_i\eps_j$ times
the factor $\gamma$ in the Table.  In minimal $SU(5)$, because neutral
gauginos do not change flavor and intergenerational mixing is small in
the up-squark sector if $\tanb\ll 60$, the $uds\nu$ and $udd\nu$
operators must come from $U_iD_jD_k\NN_r$ with $i=1$; from
\Eref{dimfive} this implies a factor of $f_1=y_u\sim \eps^4$, giving
effects proportional to $y_u y_s \sim \eps^5\ll\eps^3\tc^2$.  But in a
non-minimal GUT, if $\hat{F}_{ij}\neq f_i V_{ij}$, and both $R^u_{12}$
and $R^d_{21}$ are of order $\tc$, then it is possible that
$\hat{F}_{11}\sim {f_2}\tc^2$, giving effects of order $\eps^3\tc^2$,
comparable to the leading Wino dressing contributions \cite{lucrab}.
Dressing by neutralinos ($W_3$-ino and $B$--ino) is similar to the
gluino dressing; however, squark mixing is not required for
neutralinos to contribute to $uds\nu$, so their effects may overshadow
the gluino if $\gamma$ is small \cite{estimates}.  Gluino dressing
also can contribute without mixing to $uds\nu$ if there is substantial
D-term splitting of squark masses \cite{pal}.

\rem{For supersymmetry breaking scenarios in which the first two
generations of sfermions are much heavier than the third
\cite{twoplusone,decoupling,moreminimal} only the
third generation contributes.  The $ud_id_j\nu$ operators
may be slightly reduced, the $(u^cd_i^c)d_j\nu_\tau$ operators are
unchanged, and the $uud_i\ell$ operators saturate their bounds since
the unitarity cancellation in \eref{CKMcancel} fails.  
}

Further details will appear in \cite{estimates}.

\medskip

We thank S. Barr for a preliminary reading of the manuscript.
M.J.S. thanks the Aspen Center for Physics where part of this work was
done.  K.S.B.~was supported in part by DOE grant No. DE-FG02-
90ER-40542 and by funds from the Oklahoma State University.
M.J.S.~was supported in part by National Science Foundation grant NSF
 and by the W.M.~Keck Foundation.



%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\epsfxsize=3.2in
\hspace*{0in}\vspace*{.2in}
\epsffile{udsn.eps}
\caption{Wino-dressing diagrams contributing to $uds\nu$; there are
also diagrams with $s,d$ exchanged.}
\label{fig:udsn}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\epsfxsize=3.2in
\hspace*{0in}\vspace*{.2in}
\epsffile{ucdcsn.eps}
\caption{The Higgsino- and Wino-dressing diagrams leading
to $u^\cd d^\cd s\nu_{\tau}.$}
\label{fig:ucdcsn}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\epsfxsize=1.5in
\hspace*{0in}\vspace*{.2in}
\epsffile{uusl.eps}
\caption{The important 
diagrams contributing to $uus\mu$.}
\label{fig:uusl}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%


%   \nocite{*}                %this uses *everything* in the .bib file
   \bibliography{gut}        %or whatever your .bib file is
%\bibliographystyle{utphys}   %if you use utphys.bst
\bibliographystyle{h-physrev}
\end{document}


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\end{document}





The usual GUT calculation of the $uds\nu$ coefficient from
Wino dressing saturates
its bound.  However, according to standard literature of the minimal
$SU(5)$ GUT, the other operators do not saturate their
bounds.  Furthermore, the upper
bounds on contributions to the $uds\nu$ coefficient
from other gauginos, including the gluino,  are not saturated.
The fact that an operator coefficient lies below its upper bound
does not present a contradiction.  It means simply that
there are special symmetries in the coefficients $\lambda_{ijkr}$
or in the sfermion-sfermion-gaugino/Higgsino loop factors
which reduce some contributions below their estimated bounds.








\noindent{\bf Charged lepton mode at small tan$\beta$:}

A general superpotential operator
${1 \over 2M_C}\hat{F}_{ij}{G}_{kl} Q_iQ_j Q_k L_l$, after dressing by
the charged Wino, results in the following baryon number violating
four--fermion Lagrangain:
\begin{eqnarray}
{\cal L}_{\Delta B \ne 0} &=&
{1 \over M_C} {\alpha_2 \over 4 \pi} \hat{F}_{ij}
\hat{G}_{kl} \epsilon_{\alpha \beta \gamma} \nonumber \\
& \times & [(u_i^\alpha d_j^{\prime\beta})(d_k^{\prime\gamma} \nu_l')
\left(f(d'_i, u_j) + f(u_k, \ell_l^-) \right) \nonumber \\
& + & (d_i^{\prime \alpha} u_j^\beta)(u_k^\gamma \ell_l^-)
\left(f(u_i, d'_j)+f(d'_k, \nu_l')\right) \nonumber \\
& + & (d_i^{\prime\alpha} u_k^\beta)(d_j^{\prime\gamma} \nu'_l) \left(
f(u_i, d'_k)+f(u_j, \ell^-_l)\right) \nonumber \\
& + & (u_i^\alpha d_k^{\prime\beta})(u_j^\gamma \ell_l^-) \left(
f(d'_i, u_k) + f(d'_j,\nu'_l)\right) ]~.
\end{eqnarray}
Here we have defined the gauge interaction eigenbasis for the down
quarks, $d_i'=V_{ij}d_j$, so that the charged Wino coupling in the
supersymmetric basis is flavor diagonal: $u_i \tilde{d}_i'^*\tilde{W}^-$.
The color indices $(\alpha, \beta, \gamma)$ and the
flavor indices $(i,j,k,l)$ are understood to be summed over,
and the fermion fields paired together in parentheses are
spin--contracted to singlets.  $f$ is a loop integral, with
magnitude of $M_{\rm SUSY}^{-1}$, defined as
%\be{fab}
%f(a,b) = {M_{\tilde{W}} \over m_{\tilde{a}}^2-m_{\tilde{b}}^2 }
%\left( {m_{\tilde{a}}^2 \over m_{\tilde{a}}^2 - m_{\tilde{W}}^2 }
%{\rm ln} {m^2_{\tilde{a}} \over m^2_{\tilde{W}} } - [a \rightarrow b]
%\right)~.
%\ee
For minimal SUSY $SU(5)$, $\hat{F}_{ij} = f_i \delta_{ij}e^{i\sigma_i}$ and
$\hat{G}_{ij} = V_{ij}^*h_j$ (see Eq. x).

The usual argument that the charged lepton modes are suppressed in
$SU(5)$ can be
reproduced from Eq. (x) as follows.  In the second term the index $j$
has to be 1 since proton must have an up--quark, but then
$\hat{F}_{i1}$ is proportional
to the up--quark mass and is very small.  Similarly the fourth term of
Eq. (x) becomes proportional to $m_u$.

However, this argument is incorrect.  It would be true if all three
of $(\tilde{d}_L, \tilde{s}_L, \tilde{b}_L)$ squarks are degenerate in
mass at the weak scale.  This is however not the case in any realistic
supersymmetric model because the large top--quark Yukawa coupling will
tend to lower the mass of $\tilde{b}_L$ relative to the other two.
If all three were degenerate at the weak scale, then $\tilde{d}_i'$
would be mass eigenstate squarks with no inter--generational mixings
and the argument in the previous paragraph would apply.


Next we consider the neutral gaugino contributions to
the four-fermion operators.  Wino and Bino;then gluino.

As we mentioned earlier, the actual computation of branching
fractions is subject to large uncertainties from the unknown
sfermion and gaugino/Higgsino spectrum.  Overall rates
are even less certain.  However,  it is useful to consider
what may occur in a couple of simple cases.  CASE STUDIES.

%%
%Grand Unification of the standard model gauge groups, aside from its
%inherent beauty as an idea, provides a possible explanation of the
%size of the weak mixing angle $\sin^2\theta_W$ and of the matter
%content of each generation of standard model fermions.  Since many
%Grand Unified Theories (GUTs) predict baryon number violation, the
%scale $M_{GUT}$ involved in the breaking of the GUT gauge group must be
%very high, of order $10^16$ GeV to evade experimental bounds.  This is
%consistent with the running of the standard model coupling constants
%if supersymmetric partners of the standard model particles are
%present, with masses of order 1 TeV.  However, in supersymmetric GUTs
%there are two sources of baryon number violating operators at low
%energy.  One involves dimension-six (four-fermion) operators generated
%by heavy gauge bosons of the GUT group.  These are negligible if
%$M_{GUT} > ???$ The other involves dimension-five
%(two-fermion/two-sfermion) operators, which can be induced by colored
%Higgs exchange.\footnote{Actually there is no obvious reason these
%operators should not be present in the Planck scale Lagrangian, and if
%suppressed only by the Planck scale, they will far exceed experimental
%bounds.  This embarrassment will not be addressed here, but models are
%known \cite{anmsone,tenc} in which this difficulty is evaded through
%the physics of the fermion masses.}  It is the effects of these
%operators in minimal $SU(5)$ GUTs which we will study in this letter.
%
%It is difficult to predict the rate for dimension-five-mediated
%nucleon decay.  The reason is that the actual decay amplitude
%involves a loop, as shown in FIGURE, by which the dimension-five
%operator is converted below the \susy\ breaking scale to
%a four-fermion operator.  This loop involves two sfermions
%and a gaugino or Higgsino, but since the masses of these
%particles are not known, the value of the loop factor is quite
%uncertain.
%Furthermore, the overall rate depends both on the colored
%Higgs mass and on $\tanb \equiv \vev{H}/\vev{\bar H}$, where
%$H$ and $\bar H$ are the Higgs bosons which give mass to
%the up-type and down-type quarks.  Neither of these parameters
%is determined in GUT models.  However, these
%parameters completely or partly cancel out of most branching ratios
%for nucleon decay, making them somewhat more reliable than
%overall rates.



%In non-minimal GUTs, there is an simpler way that the cancellation may
%totally fail.  Suppose the couplings $\hat\lambda^\ell_{ijkr}$ are not
%proportional to $V_{ij}$, as will happen in any model in which the
%Higgs triplet couplings are not exactly aligned with those of the
%Higgs doublet, or in any model in which the dimension-five operators
%in \Eref{GUTops} are partly generated by a particle in a different
%multiplet from that of the Higgs doublet. In this case, the
%cancellation in \Eref{CKMcancel}, which depends critically on the
%appearance of $V_{ij}$ in the sum over $j$, will not occur.  Even if
%$V_{ij}$ is replaced by a matrix $W_{ij}$ with the same rough
%hierarchy, $W_{ij}\sim \min\{\eps_i/\eps_j,\eps_j/\eps_i\}$, the
%cancellation will completely fail.  In particular, the sums
%$V_{1j}^*W_{ij}$ will be of order $1, \theta_c, \eps\theta_c$ instead
%of $1,0,0$ as would occur for $W_{ij}=V_{ij}$.  As a result, the
%diagram FIGURE is proportional to $g_2 f_2 |V_{12}|^2 \sim y_\mu y_c
%\tc^2 \sim \eps^3\theta_c^2$.  This saturates the upper bound on the
%$uusl$ amplitude given in Table 1.  Again, the rate for this process
%grows only as $\tanbt2$.  We find, therefore, that in this
%circumstance the rate for $p\rarr K^0\mu^+$ will be {\it comparable}
%to that of $p\rarr K^+\bar\nu$ for small $\tanb$, in contrast to minimal
%GUTs.  As before, there should be no right-handed leptons observed
%(since the upper bounds on these processes are much smaller.)




The usual argument why $uus\ell$ is highly suppressed is that all
contributions are proportional to $y_u\sim\eps^4$. Combined with other
suppression factors, this makes the resulting amplitude of order
$\eps^5\tc$, unobservably small. Let us see why a factor of $y_u$
always appears, using \Eref{dimfive}.  From the Table we need only
consider Wino exchange at small $\tanb$.  If we begin with the
superpotential term $U_i D_j D_k \NN_r$, then the sneutrino must
couple to the Wino; therefore $i=1$, and so we obtain a factor of
$f_1=y_u$.  If instead we use $U_i D_j U_k E_r$, we must take
$k=1,i\neq 1$ to avoid a factor of $y_u$, and convert the charm or top
squark $\tilde u_i$ to $s$ via a Wino exchange.  The largest effect
comes from $r=2$, with the Wino coupling to both $\tilde u_i$ and
$\tilde d_j$,  as in
\FFig{uusl}. But this diagram, although naively of size $h_2
V^*_{12} f_i V_{i1}$, which would saturate the bound in the Table for
$i=2$, participates in a cancellation.  Accounting carefully for all
of the CKM factors, one finds the diagram is proportional to
%
\be{CKMcancel}
\sum_{i,j=1}^3 h_2 V^*_{12} (V^*_{1j}  V_{ij} f_i) V^*_{i2} \ .
\ee
%
The sum over $j$, by unitarity of the CKM matrix, projects onto
$\delta_{i1}$, giving a factor of $f_1=y_u$.

However, the cancellation in \Eref{CKMcancel} is actually incomplete.
In many non-minimal GUTs the $U_iD_jH_C$ coupling $W_{ij}$ is not
precisely equal to $f_i V_{ij}$, in which case the sum over $j$ does
not yield zero.  It is natural in many realistic models of flavor for
$W_{ij}$ also to be hierarchical, roughly satisfying $W_{ij}\sim f_i
\min\{\eps_i/\eps_j,\eps_j/\eps_i\}$.  With or without a $\tc$
enhancement of $W_{12}$, $W_{21}$, the sum over $j$ in
\eref{CKMcancel} will be of order $f_1,\tc f_2,\eps\tc f_3$ for
$i=1,2,3$.  In this case the diagrams with $i=2,3$ dominate and are
proportional to $h_2 V^*_{12} (\tc f_2) V^*_{22},h_2 V^*_{12} (\eps\tc
f_3) V^*_{32}\sim \eps^3\tc^2$, which saturates the bound in the Table.

In minimal GUTs, by contrast, the expression \eref{CKMcancel} would in
fact be zero, except for a subtle effect from squark mixing.  At first
glance squark mixing does not hinder the cancellation.  Loop effects
involving the top-quark Yukawa coupling cause the mass of the $\tilde
b'\equiv V_{3j}\tilde d_j$, the isospin partner of the $\tilde t$, to
be different from that of the $\tilde s'$ and $\tilde d'$ \cite{}.
This leads to a squark mass-squared
matrix of the form $\left[\tilde m^2\delta_{ij}-Cy_t^2
V^*_{3i}V_{3j}\tilde m^2\right]\tilde d_i^*\tilde d_j$ in the supersymmetric
basis.  Here $C$ is an unknown coefficient of order four times
$(1/16\pi^2)\log (\tilde m^2/M_m^2)$, where $M_m$ is the messenger
scale for supersymmetry breaking and $\tilde m$ is the universal
squark mass.  (GUT SCALE???)  The term proportional to $C$ can more
simply be written $Cy_t^2 \tilde m^2\tilde b'^*\tilde b'$.
Including the $\tilde d_j$  propagator in \Eref{CKMcancel} gives
%
\be{stillcancels}
\sum_{i,j,k=1}^3 h_2 V_{12}V^*_{1k}{ 1
\over (p^2+\tilde m^2)\delta_{jk}-Cy_t^2 V_{3j}^* V_{3k}\tilde m^2 }
                 V_{ij}f_i V^*_{i2}
\ee
%
The inverse of $A\delta_{jk}-B V^*_{3j}V_{3k}$ takes the form
$(1/A)\delta_{jk}-bV^*_{3j}V_{3k}$ for some constant $b$; inserting
this into the above expression gives $\sum_{j,k}(1/A)V^*_{1k}\delta_{jk} V_{ij}
=\delta_{i1}/A$ from the first term and $\sum_{j,k}bV^*_{1k}
V^*_{3j}V_{3k}V_{ij}=0$ from the second term.  The squark mass
splitting thus seems to have has no impact on the cancellation.

%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\epsfxsize=1.5in
\hspace*{0in}\vspace*{.2in}
\epsffile{uusl.eps}
\caption{The important
diagrams contributing to $uus\mu$.}
\label{fig:uusl}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%


A slick, and equivalent, way to understand this cancellation is to
note that the external $u$ quark couples via the Wino to the linear
combination $\tilde d' \equiv V_{uj}\tilde d_j$.  The $\tilde d'$ in
turn couples via the color triplet Higgs only to a $\tilde u$ squark,
with a coupling $y_u$.  Thus, the $y_u$ coupling is ineviTable because
of the alignment of the Wino and triplet Higgs couplings.  Since the
$\tilde b'$ is orthogonal to the $\tilde d'$, its mass is irrelevant.

However, this argument is not quite right.  The reason is that the CKM
matrix elements run as a function of scale, and therefore the
definition of the $\tilde b'$ is scale-dependent.  Consequently, the
lightest squark mass eigenstate is not $\tilde b'$ as defined at the
scale $\tilde m$; it has an admixture of $\tilde d'$. The above
cancellation is therefore incomplete, although it still reduces the
amplitude substantially. To estimate the size of the amplitude, we
note the following.  The squarks receive their universal masses
$\tilde m$ at the scale $M_m$. At this scale the mass of the $\tilde
b'$ begins to shift; the down-squark mass matrix receives corrections
proportional to $(1/ 16\pi^2) y_t^2 V_{3i}V^*_{3j}$.  Meanwhile,
$V_{32}$ and $V_{31}$ both run, while $V_{32}/V_{31}$ and $V_{33}$ are
essentially constant with scale.  Let $1+2\rho_i \equiv V_{3i}(\tilde
m)/V_{3i}(M_m)$; if $M_m\sim M_{GUT}$ then $\rho_1\approx \rho_2\sim
.1,\rho_3\approx 0$ for a wide range of parameters ($\rho =
(1-y_t^2/y_f^2)^ {1/12}$, where $y_f \simeq 1.1$ is the fixed point
value of the top quark Yukawa coupling.  When $y_t$ is close to its
fixed point value $\rho$ is larger.  For $tanb = 1.5-10$, $\rho =
0.8-0.9$.)  \cite{}.  The $\tilde d_i$ mass matrix is then
approximately $\left[\tilde m^2\delta_{ij}- Cy_t^2 V^*_{3i}(\tilde
m)V_{3j}(\tilde m)(1-\rho_i-\rho_j)\tilde m^2\right]$.  As a result,
the sum over $k$ in \Eref{stillcancels} gives
%
\be{nocancel}
\sum_{k=1}^3 V^*_{1k}(\tilde m) V_{3k}(\tilde m)(1-\rho_k) \sim
-\rho_1 V^*_{13}(\tilde m)V_{33}(\tilde m)
\ee
%
If $M_m$ is close to $M_{GUT}$, as in supergravity or high-scale
gauge-mediated models of supersymmetry breaking, then \eref{nocancel}
will be of order $.1 V_{ub}$; also, the coefficient $C$ is of order 1,
and the masses of $\tilde b,\tilde t$ will be significantly less than
the other left-handed squarks.  Because the loop integral from
\FFig{uusl}
%
\be{fab} f(\tilde q_i,\tilde q_j) = {M_{\tilde{W}} \over
m_{\tilde{i}}^2-m_{\tilde{j}}^2 } \left( {m_{\tilde{i}}^2 \over
m_{\tilde{i}}^2 - m_{\tilde{W}}^2 } {\rm ln} {m^2_{\tilde{i}} \over
m^2_{\tilde{W}} } - [i \rightarrow j] \right)~.  \ee
%
is proportional to $1/\tilde m_i^{2}$ if the squarks are much heavier
than the Winos, the smaller third-generation squark masses may lead to
an enhancement.  Altogether, we estimate the ratio of the amplitude
for the $uus\mu$ operator to that of the $uds\nu$ operator to be
%
\be{ampratio}
{y_\mu y_t
\rho_1 V_{33}V^*_{13}
V_{32}^*
f(\tilde t,\tilde b)\over y_\mu y_c
|V_{12}|^2 f(\tilde u,\tilde d)}
\alt
\rho_1
 {y_t V_{33}V_{13}^* V_{32}^*\over y_c V_{12} }
{m_{\tilde d}^2\over m_{\tilde b}^2}
\ee
%
which could be as large as $1/10$ under favorable circumstances.  The
usual computation in minimal GUTs gives a ratio $y_u/y_c V_{12}\sim
1/60$; thus the failed cancellation increases the rate by
up to a factor of thirty.

In short, depending on the details of the supersymmetric spectrum, the
rate for charged leptons in minimal grand unification may be
observable if $\tanb$ is sufficiently small.  Under maximally
favorable circumstances the branching fraction to charged leptons
could actually be as much as a few percent of the neutrino rate.  In
non-minimal models the two rates may be comparable.  In both cases,
the branching fractions to leptons decreases as $\tanbt2$ due to the
effect in \FFig{ucdcsn}; up-type squark mixing in gluino dressing at large
$\tanb$ is therefore irrelevant.  Note also that we should see only
left-handed muons.  In minimal and in many non-minimal GUTs we expect
branching fractions to electrons to be much smaller than those to
muons.  However, there are classes of GUTs in which electron and muon
rates will be comparable, including the ten-centered models of flavor
\cite{babubarr} discussed in \cite{tenc}.


JARLSKOG MATRIX???

\rem{
The
momentum integral gives the loop factor
%
\be{fab} f(i,j;x ) = {M_{x} \over
m_{\tilde{i}}^2-m_{\tilde{j}}^2 } \left( {m_{\tilde{i}}^2 \over
m_{\tilde{i}}^2 - m_{\tilde{x}}^2 } {\rm ln} {m^2_{\tilde{i}} \over
m^2_{\tilde{x}} } - [i \rightarrow j] \right)~;
\ee
%
here $i$ and $j$ represent two sfermions and $x$ represents a gaugino
or Higgsino.  In addition there are long-distance matrix elements
which affect the amplitude; for a list see \cite{hmy}.  
} 

We begin with $uds\nu$.  The Wino dressing of the $CDS\NN_\mu$ term in
the superpotential, shown in \FFig{udsn}, leads to $uds\nu_\mu$ with a
coefficient proportional to $y_s y_c\theta_c^2\sim
(\eps_2\zeta)(\eps_2^2)\tc^2 \sim \eps^3\tc^2\zeta$; this saturates
the upper bound in the Table.  Contributions with $C,S$ replaced with
$T,B$ are roughly of the same size but of unknown relative 
phase \cite{hmy}.  (We 
will not discuss momentum integrals and hadronic matrix 
elements; for a review see
\cite{hmy}.)

Similar statements apply to $udd\nu$, with an additional factor of
$\tc$.  Possible interference cancellations between second- and
third-generation diagrams can suppress either $p\rarr K^+\bar\nu$ or
$p\rarr \pi^+\bar\nu$ but not both\cite{hmy}.



\rem{
 squark mixing effect.  Loops involving the top-quark Yukawa coupling
cause the mass of the $\tilde b'\equiv V_{3j}\tilde d_j$, the isospin
partner of the $\tilde t$, to be different from that of the $\tilde
s'$ and $\tilde d'$ \cite{}.  The squark mass-squared matrix is of the
form $\tilde d_i^* V_{ip}^* \tilde m^2 [1 -
Cy_t^2\delta_{p3}]V_{jp}\tilde d_j$ in the supersymmetric basis.  Here
$C$ is an unknown coefficient of order four times $(1/16\pi^2)\log
(\tilde m^2/M_m^2)$, where $M_m$ is the messenger scale for
supersymmetry breaking and $\tilde m$ is the universal squark mass.
Since the $\tilde b'$ and $\tilde d'$ are orthogonal, the former does
not appear in \Eref{CKMcancel} and so naively the squark mass
splitting has no effect.  However, this is not so.  The CKM matrix
elements $V_{32}$ and $V_{31}$ both run (while $V_{32}/V_{31}$ and the
other $V_{ij}$ are essentially constant with scale,) so the $\tilde
b'$ is a different state at different energies.  As a result, the
lightest squark mass eigenstate is not quite the $\tilde b'$; it mixes
slightly with $\tilde d'$ and $\tilde s'$. Let $1-2\rho_i \equiv
V_{3i}(M_m)/V_{3i}(\tilde m)$ where $\rho_1\approx
\rho_2,\rho_3\approx 0$.  Solving the renormalization group equations
for the squark masses to first order in $\rho_i$, with the initial
condition that they be equal at the scale $M_m$, shows that the
$\tilde d_i$ mass matrix at the scale $\tilde m$ is diagonalized by a
unitary matrix $W_{ij}$ with $W_{3j}\approx (1-\rho_j)V_{3j}(\tilde
m)$. The diagram in \FFig{uusl} then gives
%
\be{stillcancels}
h_2 V^*_{12}\sum_{i,j,k,p=1}^3 {V^*_{1k} W_{pk}W^*_{pj} V_{ij}f_i V^*_{i2}
\over[ p^2-\tilde m^2(1-Cy_t^2\delta_{p3})][ p^2-\tilde 
m^2(1-Cy_t^2\delta_{i3})]}
%\approx
%h_2 V^*_{12}\sum_{i,j,k}^3 {V^*_{1k} V_{3k}(1-\rho_k)(1-\rho_j)V^*_{3j} V_{ij}f_i V^*_{i2}
%\over[ p^2-\tilde m^2(1-Cy_t^2\delta_{i3})]}
%\left[{ 1\over p^2-\tilde m^2(1-Cy_t^2)} - { 1\over p^2-\tilde m^2}\right]
%V^*_{3j}(1-\rho_j) V_{ij}f_i{ 1\over p^2-\tilde m^2(1-Cy_t^2\delta_{i3})} V^*_{i2}
%\approx
%h_2 V^*_{12} V^*_{13} V_{33} \rho_1
%\left[{ 1\over p^2-\tilde m^2(1-Cy_t^2)} - { 1\over p^2-\tilde m^2}\right]
%f_3 V^*_{32}{ 1\over p^2-\tilde m^2(1-Cy_t^2)} \ .
\ee
%
After using unitarity relations, dropping terms of order $f_1$,
performing the momentum integrals, and accounting for the long
distance matrix elements NAME THEM \cite{hmy}, we estimate the
ratio of the amplitude for the $uus\mu$ operator to that of the
second-generation contribution to the $uds\nu$ operator to be
%
\be{ampratio}
{y_\mu y_t
\rho_1 V_{33}V^*_{13} 
V_{32}^*
[f(\tilde t,\tilde b)-f(\tilde t,\tilde d)]\over y_\mu y_c
|V_{12}|^2 [f(\tilde c,\tilde d)+f(\tilde c,\tilde \mu)]}
\alt  ??????
\ee
%
If $M_m$ is close to $M_{GUT}$, as in supergravity or high-scale
gauge-mediated models of supersymmetry breaking, then $C$ is of order
one and (for a wide range of other parameters) $\rho_1\sim .1$
\cite{}. Note that \eref{ampratio} could be as large as $1/10$ under
favorable circumstances.  The usual computation in minimal GUTs gives
a ratio $y_u/y_c V_{12}\sim 1/60$; thus the failed cancellation
increases the rate by up to a factor of thirty.  A detailed
calculation of the branching ratio of muons to neutrinos is needed.
%
%
In short, depending on the details of the supersymmetric spectrum, the
rate for charged leptons in minimal grand unification may be as much
as a few percent of the neutrino rate.  In non-minimal models the two
rates may be comparable } 


  However, in minimal GUTs subtle renormalization effects undue this
cancellation partly, so the rate \cite{hmy} is somewhat larger
than naively expected.  In many non-minimal GUTs the unitarity
cancellation completely fails and the coefficients for
$uus\ell,uud\ell$ can be comparable to those of $uds\nu,udd\nu$, so
that the rate for $p\rarr K^0\mu^+$ is comparable to that for $p\rarr
K^+\bar\nu$ at small $\tanb$.


 Nevertheless, as we
will show, the coefficients of these operators may differ somewhat
from the usual predictions, due to subtle renormalization effects
which disturb a unitarity relation.


\rem{
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|}
\hline
%Gauginos/
&&$uds\nu%_r
$&$uus\ell%_r
$&$udd\nu%_r
$&$uud\ell%_r
$&$u^\cd d^\cd s\nu%_\tau
$&$u^\cd s^\cd d\nu%_\tau
%$&$u^\cd d^\cd s\nu_\mu
%$&$u^\cd s^\cd d\nu_\mu
$&$u^\cd \mu^\cd su 
$&$u^\cd \mu^\cd du
$&$u^\cd e^\cd su  $
\\
%Higgsinos
&&
&
&
&
& \hfil
&$u^\cd d^\cd d\nu$
%_\tau $
%&
%& \hfil
& \hfil
& \hfil
& \hfil
\\ \hline  \hline
$\tilde W^{\pm}$
&&$ g^2_2\eps^3\tc^2
$&$ g^2_2\eps^3\tc^2
$&$ g^2_2\eps^3\tc^3
$&$ g^2_2\eps^3\tc^3
$&$y_t y_\tau  g^2_2\delta\eps^2\tc
$&$y_t y_\tau  g^2_2\delta\eps^3\tc
%$&$y_t y_\tau  g^2_2\eps^5
%$&$y_t y_\tau  g^2_2\eps^6
$&$y_t y_b  g^2_2\delta\eps^4\tc^2
$&$y_t y_b  g^2_2\delta\eps^5\tc^2
$&$y_t y_b  g^2_2\delta\eps^5\tc^2$
\\ \hline
$\tilde H^{\pm}$
&&$y_t^2y_b^2\delta\eps^3\tc^2
$&$y_t^2y_b^2\delta\eps^3\tc^2
$&$y_t^2y_b^2\delta\eps^3\tc^3
$&$y_t^2y_b^2\delta\eps^3\tc^3
$&$y_t y_\tau \eps^2\tc
$&$y_t y_\tau \eps^3\tc
%$&$y_t y_\tau \eps^5
%$&$y_t y_\tau \eps^6
$&$y_t y_b \eps^4\tc^2
$&$y_t y_b \eps^5\tc^2
$&$y_t y_b \eps^5\tc^2$
\\ \hline
$\tilde W^0,\tilde B$
&&$ g^2_2\eps^3\tc^2
$&$y_b^2 g^2_2 \gamma \eps^4\tc
$&$y_t^2  g^2_2\gamma \eps^4\tc^2
$&$y_b^2 g^2_2 \gamma \eps^4\tc^2$
&
&
%&
%&
&$y_t y_b g^2_2\delta\gamma\eps^5\tc$
&$y_t y_b g^2_2\delta\gamma\eps^5\tc^2$
&
\\ \hline
$\tilde g
$&&$y_t^2 g^2_3\gamma\eps^3\tc^2
$&$y_b^2 g^2_3\gamma \eps^4\tc
$&$y_t^2 g^2_3\gamma \eps^4\tc^2
$&$y_b^2 g^2_3\gamma \eps^4\tc^2$
&
&
%&
%&
&$y_t y_b g^2_3\gamma\delta\eps^5\tc$
&$y_t y_b g^2_3\gamma\delta\eps^5\tc^2$
&
\\
\hline
\end{tabular}
}


\rem{
As we will show, there are two important effects missing from this
lore.  The most important is a very large contribution from the
operators $(u^cd^c)^\dagger(s\nu_\tau)$ and
$(u^cs^c)^\dagger(d\nu_\tau)$, leading to the decay $p \rightarrow K^+
\overline{\nu}_\tau$.  These contributions are induced dominantly by
the charged Higgsinos, although there are also sizable contributions
from the charged Wino dressing in conjunction with left--right
sfermion mixings.  The rates from these operators grow as $\tanbt4$,
in contrast to the usual contribution to $uds\nu$ which grows like
$\tanbt2$.  This effect becomes the largest contributor to nucleon
decay for $\tanb\sim 2-4$, depending on the supersymmetric spectrum,
and completely dominates for moderate to large $\tanb$.  The second
interesting effect is that the operator $uus\mu$ is not as suppressed
for small tan$\beta$ as normally assumed.  This is due to a subtle
failure of a unitarity cancellation, as a result of renormalization
effects, leading to a potentially observable $p \rightarrow K^0
\mu^+$ rate in minimal GUTs.  In many non-minimal GUTs the
cancellation fails completely, and the $p \rightarrow K^0 \mu^+$ and
$p \rightarrow K^+ \overline{\nu}$ rates can be of the same order for
small $\tanb$.  In short, we find that modes with charged leptons can
be observable at small $\tanb$, while at large $\tanb$ one sees only
neutrinos --- opposite to the usual lore.  Furthermore, the rate for
proton decay is much higher at large $\tanb$ than previously
recognized. This implies much stronger constraints on GUTs with large
or even moderate values of $\tanb.$
}

\rem{
The way to use this Table in studing GUTs is as follows.  If an
operator receives a contribution which saturates its upper bound, then
one can be confident that one has not overlooked a much larger effect
on the operator.  On the other hand, if it appears that a given
operator does not saturate its upper bound, then one should understand
in detail why the various contributions to the operator are suppressed
and check carefully that no subtle effects bring its coefficient back
toward the bound.
}

\rem{An efficient way to understand this is to note that the external
$u$ quark couples via the Wino to the linear combination $\tilde d'
\equiv V_{uj}\tilde d_j$.  The $\tilde d'$ in turn couples via the
color-triplet Higgs only to a $\tilde u$ squark, with a coupling
$y_u$.  Thus, the $y_u$ coupling is inevitable because of the
alignment of the Wino and color-triplet Higgs couplings.}  

\rem{Our conclusions then are the following.  A very large contribution to
$p\rarr K^+\bar\nu_\tau$ has been neglected in the literature.  This
occurs through the operator $(u^cd^c)^\dagger(s\nu_\tau)$, induced through
Higgsino dressing of the operator $u^cd^c\tilde t^c\tilde \tau^c$.
The rate from the usual $uds\nu$ operator grows as $\tanbt{2}$, wheras
the rate from the new operator grows like $\tanbt{4}$ and dominates
for $\tanb\agt 2-4$.  The much larger rate puts much stronger
constraints on GUTs with large $\tanb$ than has previously been
recognized.  Similar results apply to $p\rarr \pi^+\bar\nu_\tau$, whose
rate should be comparable to or slightly smaller than the rate for
$p\rarr K^+\bar\nu_\tau$.  In addition, the rate for charged lepton
production at small $\tanb$ has been underestimated; a cancellation
involving the CKM matrix is actually incomplete.  In minimal GUTs it
is possible for the rate for $p\rarr K^0\mu^+$ to be of order a
percent of the $p\rarr K^+\bar\nu$ rate, if $\tanb$ is small enough that
the $uds\nu$ operator is the dominant one.  In many non-minimal GUTs
--- ones in which the coefficients of the dimension-five operators are
not precisely aligned with the up--quark fermion mass matrix --- the
rates for $p\rarr K^0\mu^+$ and $p\rarr K^+\bar\nu$ will generally be
comparable at small $\tanb$.  (Similar statements apply for $p\rarr
\pi^0\mu^+$ and $p\rarr \pi^+\bar\nu$.)  Only left-handed muons will be
observed.  In ordinary GUTs one expects the rate to electrons to be
much smaller than that to muons, although certain classes of GUTs
(including ten-centered models of flavor
\cite{babubarr,msfermion,tenc}) can have comparable electron and muon
branching fractions.  At moderate to large $\tanb$, the rate for
$p\rarr K^+\bar\nu_\tau$ is so large that all effects involving charged
leptons (including those induced through squark-mixing combined with
gluino dressing \cite{}) are negligible.  Finally, we note that gluino
and neutralino contributions at small $\tanb$ need not be negligible
in non-minimal GUTs.
}

\rem{
These results require further investigation and application.
Computations of branching fractions must be revisited, and the
constraints on minimal and non-minimal GUTs need to be reinterpreted.
We expect that many models formerly thought to be consistent with data
may now be severely restricted.  It will be especially interesting to
see which classes of models are still consistent with third-generation
Yukawa-coupling-constant unification ($\tanb\sim 60$).
}

%This follows from
%the approximate relations $m_d/m_s\sim m_s/m_b\sim m_\mu/m_\tau \sim
%\sqrt{m_c/m_t}\sim \sqrt{m_u/m_c}\sim V_{cb} \sim \sqrt{V_{ub}}\sim
%\eps$.  Note that $V_{us}\sim 5\eps$ and $m_e/m_\mu\sim .12 \eps$; the
%fact that the Cabibbo angle is greater than $\eps$ will play an
%important role below.  If the majority of the $\lambda_{ijkr}$ satisfy
%the above ansatz, with some of the coefficients possibly smaller, then
%the upper bounds of \cite{estimates} apply.

