\documentclass[12pt]{article}
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\textwidth 6.2in
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\font\er=cmr10 scaled\magstep0
\def \b{{\cal B}}
\def \Bbar{\bar B}
\def \beq{\begin{equation}}
\def \beqn{\begin{eqnarray}}
\def \bo{B^0}
\def \eeq{\end{equation}}
\def \eeqn{\end{eqnarray}}
\def \ca{{\cal A}}
\def \cn{Collaboration}
\def \ite{{\it et al.}}
\def \ob{\overline{B}^0}
\def \obs{\overline{B}^0_s}
\def \od{\overline{D}^0}
\def \ok{\overline{K}^0}
\def \s{\sqrt{2}}
\def \sf{\hbox{$\scriptstyle{1\over\sqrt2}$}}
\def \st{\sqrt{3}}
\def \sx{\sqrt{6}}
\def \bu{{\bar u}}
\def \bd{{\bar d}}
\def \bs{{\bar s}}
\def \tG{\tilde{\Gamma}}
\def \v#1#2{V_{#1#2}}
\def \vc#1#2{V^*_{#1#2}}
\begin{document}
\renewcommand{\thetable}{\Roman{table}}
\rightline{EFI-03-03}
\rightline
\rightline{January 2003}
\bigskip
\bigskip
\centerline{\bf FINAL-STATE PHASES IN $B \to $ BARYON-ANTIBARYON DECAYS
\footnote{To be submitted to Phys.~Rev.~D.}}
\bigskip
\centerline{\it Zumin Luo and Jonathan L. Rosner}
\centerline{\it Enrico Fermi Institute and Department of Physics}
\centerline{\it University of Chicago, Chicago, IL 60637}
\bigskip
\centerline{\bf ABSTRACT}
\medskip
\begin{quote}
The recent observation of the decay $\ob \to \Lambda_c^+ \bar p$
suggests that related decays may soon be visible at $e^+ e^-$
colliders.  It is shown how these decays can shed light on strong
final-state phases and amplitudes involving the spectator quark,
both of which are normally expected to be small in $B$ decays.
\end{quote}
\medskip
\leftline{\qquad PACS codes: 13.25.Hw, 11.30.Hv, 14.40.Nd, 13.75.Lb}
\bigskip

\centerline{\bf I. INTRODUCTION}
\bigskip

Phases in $B$ decays arising from final-state interactions are an
important gateway to the observation of direct CP violation.  The
pattern of decays to $D \pi$, $D^* \pi$, $D \rho$, and related states
has been elaborated recently by the CLEO \cite{PedlarD,Ahmed}, BaBar
\cite{BaDsK}, and Belle \cite{BeDro,BeDK,BeDsK} Collaborations.  Some
amplitudes for decays involving the weak subprocess $b \to c \bar u d$
obey isospin triangle relations.  In certain cases these triangles have
non-zero area, indicating non-zero final-state phases between different
contributing amplitudes \cite{CR}.  Some decays governed by the
Cabibbo-suppressed subprocess $b \to c \bar u s$ also involve amplitude
triangles with apparently non-zero area, though not yet at a statistically
significant level \cite{CR,Xing}.  One would expect this behavior if
flavor SU(3) is a good symmetry for $B$ decays.

The decays of $B$ mesons to baryon-antibaryon pairs also obey simple isospin
relations and flavor-SU(3) regularities \cite{LW,SW}.  The recent observation
of the decay $\ob \to \Lambda_c^+ \bar p$ by the Belle Collaboration
\cite{BeLp} with a branching ratio $\b(\ob \to \Lambda_c^+ \bar p) = (2.19
^{+0.56}_{-0.49} \pm 0.32 \pm 0.57) \times 10^{-5}$
indicates that such processes are within experimental reach at
existing $e^+ e^-$ colliders.  The present paper indicates how
these data may be useful in elaborating final-state phases among
different amplitudes contributing to the decays. It also indicates
how one can test for suppression of decay amplitudes involving the
spectator quark.

We shall discuss the decomposition of $\ob \to \Lambda_c^+ \bar p$
and related decays into invariant amplitudes of flavor SU(3) in
Sec.\ II.  The triangles formed by these amplitudes, and their
significance for final-state interactions, are discussed in Sec.\
III.  We conclude with some experimental prospects in Sec.\ IV.
Conventions for the quark composition of baryons are given in the
Appendix.
\newpage

\centerline{\bf II.  INVARIANT AMPLITUDES OF FLAVOR SU(3)}
\bigskip

The weak Hamiltonian giving rise to the subprocess $b \to c \bar u
d$ transforms as the $I = 1, I_3 = -1$ member of an octet of
flavor SU(3).  The $\overline B$ mesons $b \bar q$ ($\bar q=-\bar
u,\bar d,\bar s$) form a $3^*$. (Recall that $(-\bar u, \bar d)$
is an isodoublet.) Thus the SU(3) representations of the initial
state are those in the product
%
\beq \label{eqn:To}
3^* \times 8 = 3^* + 6 + 15^*~~~.
\eeq
%
The $\Lambda_c^+ = c[ud]$ belongs to a flavor-SU(3) antitriplet
($3^*$) along with the $\Xi_c^+ = c[su]$ and the $\Xi_c^0 =
c[sd]$. The brackets indicate antisymmetry with respect to flavor.
For decays to a final state of a $3^*$ charmed baryon and an octet
antibaryon, all three representations in Eq.\ (\ref{eqn:To})
occur.  Hence there must be three independent invariant amplitudes
of flavor SU(3) characterizing such decays. Similarly, in the
Cabibbo-suppressed decays governed by $b \to c \bar u s$, the weak
Hamiltonian transforms as the strange charged isodoublet member of
a flavor octet, so the invariant amplitudes are the same.

Charmed baryons belonging to a flavor-SU(3) sextet ($6$) also have
been seen, consisting of an isotriplet $\Sigma_c^{++} =
cuu,~\Sigma_c^+ = c(ud), \Sigma_c^0 = cdd$, an isodoublet
${\Xi'}_c^+ = c(us),~{\Xi'}_c^0 = c(ds)$, and an isosinglet
$\Omega_c^0 = css$.  The parentheses indicate symmetry with
respect to flavor. Similarly, one can consider not only octet but
also (anti)decuplet antibaryons. In Table \ref{tab:reps} we
summarize the SU(3) representations that can contribute to each
class of decays.

\begin{table}
\caption{Invariant amplitudes in the direct channel contributing
to $\overline B \to$ (charmed baryon) + (antibaryon) decays via
the subprocess $b \to c \bar u d$ or $b \to c \bar u s$.
\label{tab:reps}}
\begin{center}
\begin{tabular}{l c c} \hline \hline
\quad Charmed baryon & $3^*$ & 6 \\
Antibaryon           &       &   \\ \hline
8                    & $3^* + 6 + 15^*$ & $3^* + 6 + 15^*$ \\
$10^*$               &        6         & $3^* + 15^*$ \\ \hline \hline
\end{tabular}
\end{center}
\end{table}

An economical tensor notation was utilized by Savage and Wise to
describe these processes \cite{SW}.  We illustrate with the $3^* +
8$ final state.  We use subscripts to denote the components of a
$3^*$ representation of $SU(3)_f$ and use superscripts to denote
the components of a $3$ representation. The $\overline B$ mesons,
in a $3^*$ representation as mentioned, can then be written as
$(-B^-, \ob, \obs) \equiv B_i$. The charmed baryons in a $3^*$
representation can be expressed as $(-\Xi_c^0, \Xi_c^+,
\Lambda_c^+) \equiv (\Xi_c)_i$. The octet of charmless baryons, on
the other hand, can be represented by a two-index tensor:
%
\beq
 N^i_j \equiv \left( \begin{array}{ccc}
-\Sigma^0/\sqrt{2}+\Lambda/\sqrt{6} & \Sigma^+ & p
\\ -\Sigma^- & \Sigma^0/\sqrt{2}+\Lambda/\sqrt{6} & n \\ -\Xi^- & \Xi^0 &
 -2\Lambda/\sqrt{6} \end{array} \right)
\eeq
%
The weak Hamiltonian responsible for the Cabibbo-favored quark
subprocess $b \to c d {\bar u}$ and the Cabibbo-suppressed $b \to
c s {\bar u}$, belonging to an $SU(3)_f$ octet as mentioned above, can
similarly be written as
%
\beq
H^i_j \sim (d{\bar u})V_{ud} + (s{\bar u})V_{us} = \left(
\begin{array}{ccc} 0 & 0 & 0
\\ V_{ud} & 0 & 0 \\ V_{us} & 0 & 0 \end{array} \right) ,
\eeq
%
where $V_{ud}$ and $V_{us}$ are the Cabibbo-Kobayashi-Maskawa
(CKM) matrix elements: \\ $V_{us}/V_{ud} \simeq  \lambda \simeq 0.2256$.
The effective Hamiltonian for the
decays $B \to \Xi_c {\overline N}$ can be written in terms of
invariant amplitudes $\alpha, \beta$ and $\gamma$ \cite{SW}:
%
\beq \label{eqn:SW} H_{\rm eff} = \alpha \Xi_c^i N^j_i H^k_j B_k
+\beta \Xi_c^i H^j_i N^k_j B_k +\gamma \Xi_c^i B_i H^j_k N^k_j ,
\eeq
%
where we sum over repeated indices. Expanding the sum would give
us the amplitudes for the relevant processes. (Remember to
multiply each amplitude by $(-1)^{n_{\bar u}}$, where $n_{\bar u}$
is the number of $\bar u$ quarks in the antibaryon.)

Two equivalent notations are helpful to visualize possible
relations among invariant amplitudes.  The second is particularly
relevant when certain dynamical assumptions are made.

(1) The process $3^* \times 8 \to 3^* \times 8$ in the crossed channel reads
\beq
3^* \times 3 \to 1 + 8_D + 8_F \to 8 \times 8~~~,
\eeq
where $D$ and $F$ denote the two ways of coupling an octet to a pair of octets.
The singlet $S$ and octet amplitudes $D$ and $F$ (suitably normalized) are
related to $\alpha$, $\beta$, and $\gamma$ by
\beq
\alpha = D + F~~,~~~\beta = D - F~~,~~~\gamma = S - \frac{2}{3} D~~~.
\eeq
The $S,~D,~F$ notation is that (aside from normalization) used by
Li and Wu \cite{LW}.  In Table \ref{tab:xreps} we summarize the SU(3)
representations that contribute to each class of decays, including also
sextet charmed baryons and antidecuplet antibaryons.  We see, of course,
that the number of invariant amplitudes is the same as in the direct
channel.

\begin{table}
\caption{Invariant amplitudes in the crossed channel contributing
to $\overline B \to$ (charmed baryon) + (antibaryon) decays via
the subprocess $b \to c \bar u d$ or $b \to c \bar u s$.
\label{tab:xreps}}
\begin{center}
\begin{tabular}{l c c} \hline \hline
\quad Charmed baryon & $3^*$ & 6 \\
Antibaryon           &       &   \\ \hline
8                    & $1 + 8_D + 8_F$ & $8_D + 8_F + 10$ \\
$10^*$               &        8        & $8 + 10$ \\ \hline \hline
\end{tabular}
\end{center}
\end{table}

(2) A topological expansion of amplitudes \cite{Chau,GHLR} yields
three invariant amplitudes of which two are associated with the
subprocess $b \to c d \bar u$ or $b \to c s \bar u$, with an
additional light quark-antiquark pair produced from the vacuum
[Fig.~\ref{fig:decay}(a)], and one is associated with the exchange
process $b \bar d \to c \bar u$ or $b \bar s \to c \bar u$, in
which two such pairs are produced from the vacuum
[Fig.~\ref{fig:decay}(b)].  We call the first two amplitudes $a_1$
and $a_2$ and the third amplitude $a_E$ (to denote exchange).
Explicit definitions of these amplitudes are given below. Consider
the amplitudes for $\overline B$ to decay to 6 quarks
($c~q_{w'}~q_v$ and $\bar q_s$~$\bar q_w$~$\bar q_v$) via
color-suppressed processes as shown in Fig.~\ref{fig:decay}(a).
With the $c$ quark staying at the top, there are 2 permutations
for \{$q_{w'}~q_v$\} and 6 permutations for \{$\bar q_s$~$\bar
q_w$~$\bar q_v$\}. Thus there are 12 possible color-suppressed
diagrams contributing to a specific amplitude. The amplitudes of
the 12 diagrams are denoted by $A_{i j k}^{l m}$, where $lm$ is a
permutation of \{$w' ~v$\} and $ijk$ is a permutation of
\{$s~w~v$\}. The color-suppressed amplitude for $\overline B$ to
decay to a charmed baryon and an antibaryon is then a weighted sum
of the 12 amplitudes, with the weights being the products of the
coefficient of $cq_lq_m$ in the quark composition of the charmed
baryon and that of $\bar q_i\bar q_i\bar q_k$ in the quark
composition of the antibaryon. It turns out that each
color-suppressed amplitude is a linear combination of $a_1$ and
$a_2$, with
%
\beqn a_1 & = & \frac{1}{2}(A_{[sw]v}^{[w'v]} + A_{[sv]w}^{[w'v]}) \\
a_2 & = & \frac{1}{2}(A_{[ws]v}^{[w'v]} + A_{[wv]s}^{[w'v]}) \eeqn
%
Here $A_{[ij]k}^{[lm]} \equiv (A_{ijk}^{lm}-A_{jik}^{lm}) -
(A_{ijk}^{ml}-A_{jik}^{ml})$ and $1/2$ is merely a normalization
factor. Similarly, $E_{ijk}^{lm}$ is used to denote the amplitude
for ${\overline B}$ to decay to 6 quarks via an exchange process
as shown in Fig.~\ref{fig:decay}(b). Here $lm$ is a permutation of
\{$v_1 ~v_2$\} and $ijk$ is a permutation of \{$w~v_1~v_2$\}.
Since the two quark-antiquark pairs ($q_{v_1}\bar q_{v_1}$ and
$q_{v_2}\bar q_{v_2}$) are both produced from the vacuum, $a_E$
should not depend on the ordering of $v_1$ and $v_2$. One finds
that all exchange amplitudes for $\overline B$ to decay to a
charmed baryon and an antibaryon are multiples of
%
\beq a_E = \frac{1}{2}(E_{[v_1v_2]w}^{[v_1v_2]} +
E_{[wv_2]v_1}^{[v_1v_2]} - E_{[wv_1]v_2}^{[v_1v_2]}) \eeq
%
The topological decompositions of the amplitudes are presented in
Table~\ref{tab:To}. They are in agreement with those obtained from
Eq.~(\ref{eqn:SW}) if we set
%
\beq a_1 =-\gamma,~~~a_2=-\beta,~~~a_E = \alpha+\gamma~~~. \eeq
%
In particular, if processes involving the spectator quark are
suppressed, as has been argued for heavy-quark decays (see, e.g.,
the discussion in \cite{GHLR}), one expects $|a_E| \ll
|a_1|,~|a_2|$, and hence an approximate symmetry \beq \alpha = -
\gamma~~~. \eeq We shall explore the consequences of this relation
in the next Section.
%
\begin{table}
\caption{$SU(3)_f$ predictions of the amplitudes for $\overline B
\to$ an SU(3) $3^*$ charmed baryon and an octet antibaryon. CF =
Cabibbo-favored, CS = Cabibbo-suppressed. \label{tab:To}}
\begin{center}
\begin{tabular}{l c l c}
\hline \hline
CF Decay & Amplitude & CS Decay & Amplitude \\
\hline ${\overline B}^0 \to \Lambda_c^+ {\bar p}$ & $a_1+a_E$ &
${\overline B}_s^0 \to \Xi_c^+ {\overline \Sigma}^-$ & $\lambda (a_1+a_E)$ \\
${\overline B}_s^0 \to \Xi_c^0 {\overline \Xi}^0$ & $-a_2$ &
${\overline B}^0 \to \Xi_c^0 {\bar n}$ & $-\lambda a_2$ \\
${\overline B}_s^0 \to \Lambda_c^+ {\overline \Sigma}^-$ &
$-a_1$ & ${\overline B}^0 \to \Xi_c^+ {\bar p}$ & $-\lambda a_1$ \\
${\overline B}^0 \to \Xi_c^0 {\overline \Sigma}^0$ &
$-(a_1+a_2+a_E)/\sqrt{2}$ & ${\overline
B}_s^0 \to \Xi_c^0 {\overline \Sigma}^0$ & $-\lambda (a_1+a_E)/\sqrt{2}$ \\
${\overline B}^0 \to \Xi_c^0 {\overline \Lambda}$ &
$(a_1-a_2+a_E)/\sqrt{6}$ & ${\overline
B}_s^0 \to \Xi_c^0 {\overline \Lambda}$ & $\lambda (a_1+2a_2+a_E)/\sqrt{6}$ \\
${\overline B}^0 \to \Xi_c^+ {\overline \Sigma}^-$ & $a_E$ &
${\overline
B}_s^0 \to \Lambda_c^+ {\bar p}$ & $\lambda a_E$ \\
$B^- \to \Xi_c^0 {\overline \Sigma}^-$ & $-(a_1+a_2)$ & $B^-
\to \Xi_c^0 {\bar p}$ & $-\lambda (a_1+a_2)$ \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
%

% This is Figure 1
\begin{figure}
\begin{center}
\includegraphics[height=2.2in]{dec.eps}
\caption{Diagrams for $\overline B$ $\to$ a charmed  baryon and an
antibaryon. (a) Color-suppressed diagram. $q_{w'} = d$ for
Cabibbo-favored decays and $q_{w'} = s$ for Cabibbo-suppressed
decays; $\bar q_w=\bar u$. (b) Exchange diagram. $\bar q_s = \bar
d$ for  Cabibbo-favored decays and $\bar q_s = \bar s$ for
Cabibbo-suppressed decays; $\bar q_w=\bar u$. \label{fig:decay}}
\end{center}
\end{figure}

More generally, the topological amplitudes contributing to each
type of process are summarized in Table \ref{tab:gphs}. For decays
to $6 + 8$, both $q_{w'}q_v$ and $q_{v_1}q_{v_2}$ are symmetrized
and therefore
%
\beqn b_1 & = & \frac{1}{2}(A_{[sw]v}^{(w'v)} + A_{[sv]w}^{(w'v)}) \\
b_2 & = & \frac{1}{2}(A_{[ws]v}^{(w'v)} + A_{[wv]s}^{(w'v)}) \\
b_E & = & \frac{1}{2}(E_{[wv_1]v_2}^{(v_1v_2)} +
E_{[wv_2]v_1}^{(v_1v_2)} )~~~ \label{eqn:bE}, \eeqn
%
where $A_{[ij]k}^{(lm)} \equiv (A_{ijk}^{lm}-A_{jik}^{lm}) +
(A_{ijk}^{ml}-A_{jik}^{ml})$ and $E_{[ij]k}^{(lm)}$ is defined in
a similar way. Note that if $q_{v_1}$ and $q_{v_2}$ are identical,
only one term in Eq.~(\ref{eqn:bE}) contributes. For decays to
$3^* + 10^*$, there is no exchange diagram since $q_{v_1}$ and
$q_{v_2}$ are antisymmetrized in a $3^*$ charmed baryon but $\bar
q_{v_1}$ and $\bar q_{v_2}$ are symmetrized in a $10^*$
antibaryon; and
%
\beq c = A_{(swv)}^{[w'v]}/\sqrt{2} \equiv \sum_{\sigma}
(A_{\sigma\{swv\}}^{w'v} - A_{\sigma\{swv\}}^{vw'})/\sqrt{2}~~~,
\eeq
%
where the sum runs over all permutations $\sigma$ of \{$s~w~v$\}.
For decays to $6 + 10^*$,
%
\beq d = A_{(swv)}^{(w'v)} \equiv \sum_{\sigma}
(A_{\sigma\{swv\}}^{w'v} + A_{\sigma\{swv\}}^{vw'})~~~, \eeq
%
and $d_E = E_{(wv_1v_2)}^{(v_1v_2)}$ is defined in a similar
fashion. In Tables \ref{tab:So}--\ref{tab:Sd} we summarize the
corresponding amplitudes for decays to $6 + 8$, $3^* + 10^*$, and
$6 + 10^*$, respectively.  These are equivalent to the
decompositions presented in Ref.\ \cite{SW}, but we find the
present notation convenient for seeing what happens when we assume
that the exchange amplitudes are small. We do not show the
amplitudes for Cabibbo-suppressed decays (which can be looked up
in \cite{SW}), since these decays generally involve $\Xi'_c$,
$\Omega_c$ or ${\overline B}_s^0$, none of which is easy to
observe or produce in experiments. Furthermore, the branching
ratios for these decays are expected to be only a few percent of
those for the Cabibbo-favored ones.

\begin{table}
\caption{Invariant amplitudes in a topological expansion for $B \to$
(charmed baryon) + (antibaryon) decays via the subprocess $b \to c \bar u d$
or $b \to c \bar u s$. \label{tab:gphs}}
\begin{center}
\begin{tabular}{l c c} \hline \hline
\quad Charmed baryon & $3^*$ & 6 \\
Antibaryon           &       &   \\ \hline
8    & $a_1,~a_2,~a_E$ & $b_1,~b_2,~b_E$ \\
$10^*$ &   $c$     & $d,~d_E$ \\
\hline \hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{$SU(3)_f$ predictions of the amplitudes for $\overline B
\to (6$ charmed baryon $+$ octet antibaryon). Only Cabibbo-favored
decays are shown. \label{tab:So}}
\begin{center}
\begin{tabular}{l c l c} \hline \hline
Decay & Amplitude & Decay & Amplitude \\ \hline $\ob \to
\Sigma_c^0 \bar n$ & $-(b_2 + b_E)$ &
  $B^- \to \Sigma_c^0 \bar p$ & $-(b_1+b_2)$ \\
$\ob \to \Sigma_c^+ \bar p$ & $(b_E-b_1)/\s$ &
  $B^- \to {\Xi'}_c^0 \overline \Sigma^-$ & $(b_1+b_2)/\s$ \\
$\ob \to {\Xi'}_c^0 \overline \Lambda$ & $(b_2-b_1+3 b_E)/(2 \st)$
&
  $\overline{B}_s^0 \to {\Xi'}_c^0 \overline \Xi^0$ & $b_2/\s$ \\
$\ob \to {\Xi'}_c^0 \overline \Sigma^0$ & $(b_1+b_2+b_E)/2$ &
  $\overline{B}_s^0 \to \Sigma_c^+ \overline \Sigma^-$ & $b_1/\s$ \\
$\ob \to {\Xi'}_c^+ \overline \Sigma^-$ & $-b_E/\s$ &
  $\overline{B}_s^0 \to \Sigma_c^0 \overline \Sigma^0$ & $-b_1/\s$ \\
$\ob \to \Omega_c^0 \overline \Xi^0$ & $b_E$ &
  $\overline{B}_s^0 \to \Sigma_c^0 \overline \Lambda$ & $(b_1+2b_2)/\sx$ \\
% $\overline{B}_s^0 \to \Sigma_c^0 \overline \Lambda$ & $-(b_1+2b_2)/\sx$ \\
\hline \hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{$SU(3)_f$ predictions of the amplitudes for $\overline B
\to (3^*$ charmed baryon $+$ antidecuplet antibaryon)
(Cabibbo-favored decays). Note that $\ca(\ob \to \Xi_c^+ \overline
\Sigma^{*-}) = 0$. \label{tab:Td}}
\begin{center}
\begin{tabular}{l c l c} \hline \hline
Decay & Amplitude & Decay & Amplitude \\ \hline $\ob \to
\Lambda_c^+ \overline \Delta^-$ & $- c/\st$ &
  $\ob \to \Xi_c^0 \overline \Sigma^{*0}$ & $c/\sx$ \\
$B^- \to \Lambda_c^+ \overline \Delta^{--}$ & $- c$ &
  $B^- \to \Xi_c^0 \overline \Sigma^{*-}$ & $c/\st$ \\
$\overline{B}_s^0 \to \Lambda_c^+ \overline \Sigma^{*-}$ &
$-c/\st$ & $\overline{B}_s^0 \to \Xi_c^0 \overline \Xi^{*0}$ & $c/\st$ \\
\hline \hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{$SU(3)_f$ predictions of the amplitudes for $\overline B
\to (6$ charmed baryon $+$ antidecuplet antibaryon)
(Cabibbo-favored decays). \label{tab:Sd}}
\begin{center}
\begin{tabular}{l c l c} \hline \hline
Decay & Amplitude & Decay & Amplitude \\ \hline $\ob \to
\Sigma_c^{++} \overline \Delta^{--}$ & $-d_E$ &
   $B^- \to \Sigma_c^+ \overline \Delta^{--}$ & $d/\s$ \\
$\ob \to \Sigma_c^+ \overline \Delta^{-}$ & $(2d_E+d)/\sx$ &
  $B^- \to \Sigma_c^0 \overline \Delta^-$ & $-d/\st$ \\
$\ob \to \Sigma_c^0 \overline \Delta^0$ & $-(d_E + d)/\st$ &
  $B^- \to {\Xi'}_c^0 \overline \Sigma^{*-}$ & $-d/\sx$ \\
$\ob \to {\Xi'}_c^+ \overline \Sigma^{*-}$ & $(2/3)^{1/2}d_E$ &
  $\overline{B}_s^0 \to \Sigma_c^+ \overline \Sigma^{*-}$ & $d/\sx$ \\
$\ob \to {\Xi'}_c^0 \overline \Sigma^{*0}$ & $-(2d_E+d)/(2 \st)$ &
  $\overline{B}_s^0 \to \Sigma_c^0 \overline \Sigma^{*0}$ & $-d/\sx$ \\
$\ob \to \Omega_c^0 \overline \Xi^{*0}$ & $-d_E/\st$ &
  $\overline{B}_s^0 \to {\Xi'}_c^0 \overline \Xi^{*0}$ & $-d/\sx$ \\
\hline \hline
\end{tabular}
\end{center}
\end{table}

\bigskip
%\newpage
\centerline{\bf III. TRIANGLE RELATIONS}
\bigskip

\leftline{\bf A.  $3^* + 8$ final states.}
\bigskip

The Cabibbo-favored amplitudes of Table~\ref{tab:To} are denoted
by arrows in Fig.~\ref{fig:amps}. Three independent complex
amplitudes will be specified completely, up to an irrelevant
overall phase, by five lengths of these vectors, leaving two
predictions for rates. There will be a discrete ambiguity
corresponding to the folding of two adjacent triangles about their
common side.  (We do not show the corresponding figure for
Cabibbo-suppressed decays.)  We now discuss some individual
triangle relations associated with this construction.  These
triangles, if shown to have non-zero area, will indicate non-zero
relative final-state phases between their contributing amplitudes.

% This is Figure 2
\begin{figure}
\begin{center}
\includegraphics[height=3.5in]{amps.eps}
\caption{Triangles for $\ca(\overline{B} \to \Lambda_c^+ \bar p) =
\alpha$ and related amplitudes described in Table~\ref{tab:To}.
Note that $\alpha = a_1 + a_E$, $\beta = - a_2$ and $\gamma = -
a_1$. \label{fig:amps}}
\end{center}
\end{figure}

As a consequence of the isospin of the weak Hamiltonian for $b \to
c \bar u d$, two invariant isospin amplitudes, with $I = 1/2$ and
$I = 3/2$, govern $\overline{B} \to \Xi_c \overline \Sigma$.  The
three decay processes then obey a triangle relation:
%
\beq \ca(B^- \to \Xi^0_c \overline \Sigma^-) = \s \ca(\ob \to
\Xi^0_c \overline \Sigma^0) + \ca(\ob \to \Xi^+_c \overline
\Sigma^-)~~~. \eeq
%
This relation is somewhat challenging in view of the need to
reconstruct the $\bar \Sigma^0$ through its $\bar \Lambda \gamma$
decay.  However, it involves only non-strange $B$ mesons, which
are the focus of current studies at $e^+ e^-$ colliders.

Three triangle relations involve the observed $\ob \to \Lambda_c^+
\bar p$ decay:
%
\beq \s \ca(\ob \to \Lambda^+_c \bar p) + \ca(\ob \to \Xi^0_c
\overline \Sigma^0) = \st \ca(\ob \to \Xi^0_c \overline
\Lambda)~~~, \eeq \beq \ca(\ob \to \Lambda_c^+ \bar p) + \ca(\obs
\to \Xi_c^0 \overline \Xi^0) = \sx \ca(\ob \to \Xi_c^0 \overline
\Lambda)~~~, \eeq \beq \ca(\ob \to \Lambda_c^+ \bar p) + \ca(\obs
\to \Lambda_c^+ \overline \Sigma^-) = \ca(\ob \to \Xi_c^+
\overline \Sigma^-)~~~. \eeq
%
The first one is particularly useful since it involves only $\ob$
decays. The last two relations involve the detection of a $\obs$
decay, requiring either a dedicated run at KEKB or PEP-II
(currently running below $B^0_s \obs$ threshold) or an experiment at
a hadron collider.

In the Cabibbo-suppressed sector two isospin relations stem from
the $I = 1/2$, $I_3 = -1/2$ nature of the weak Hamiltonian:
%
\beq \ca(B^- \to \Xi_c^0 \bar p) = \ca(\ob \to \Xi_c^0 \bar n) +
\ca(\ob \to \Xi_c^+ \bar p)~~~, \eeq \beq \ca(\obs \to \Xi_c^+
\overline \Sigma^-) = - \s \ca(\obs \to \Xi_c^0 \overline
\Sigma^0)~~~. \eeq
%
The first of these involves only non-strange $B$'s and no
$\overline \Sigma^0$'s. Two additional triangle relations may be
written, both involving $\obs$ decays.  Since these involve
Cabibbo-suppressed decays of the less easily produced $\obs$, the
corresponding triangles may not be so easy to construct.
\bigskip

\leftline{\bf B.  $6+8$ final states.}
\bigskip

The isospin triangles in these processes are
%
\beq \ca(B^- \to \Sigma_c^0 \bar p) = \s \ca(\ob \to \Sigma_c^+
\bar p) + \ca(\ob \to \Sigma_c^0 \bar n)~~~, \eeq
%
which involves an antineutron, and
%
\beq \ca(B^- \to {\Xi'}_c^0 \overline \Sigma^-) = \s \ca(\ob \to
{\Xi'}_c^0 \overline \Sigma^0) + \ca(\ob \to {\Xi'}_c^+ \overline
\Sigma^-)~~~, \eeq
%
which involves the ${\Xi'}_c$ states. These were not observed
until quite recently \cite{Jessop:1998wt} since they decay to
$\Xi_c \gamma$. A simple isospin relation \beq \ca(\obs \to
\Sigma_c^+ \overline \Sigma^-) = - \ca(\obs \to \Sigma_c^0
\overline \Sigma^0) \eeq involves $\obs$ decays.  Several
amplitude triangles not involving isospin can be formed from the
relations for $6 + 8$ decays, but they involve particles which are
not especially easy to produce ($\obs$) or detect (${\Xi'}_c$).

There are several ways to check whether the exchange amplitude
$b_E$ is much smaller than $b_1$ or $b_2$.  For example, the decay
$\ob \to {\Xi'}_c^+ \overline \Sigma^-$ occurs only via the
exchange amplitude, so it would be suppressed in comparison with
the other decays to ${\Xi'}_c \overline \Sigma$.  Similarly, the
decay $\ob \to \Omega_c^0 \overline \Xi^0$ would be suppressed.
If, indeed, $b_E$ is found to be suppressed, a useful amplitude
triangle based on the two independent amplitudes $b_1$ and $b_2$
could be formed:
%
\beq 2 \st \ca(\ob \to {\Xi'}_c^0 \overline \Lambda) + \ca(B^- \to
\Sigma_c^0 \bar p) = 2 \s \ca(\ob \to \Sigma_c^+ \bar p)~~~. \eeq
%
Other such triangles can also be formed, but they generally
involve $\obs$ decays.
\bigskip

\leftline{\bf C.  $3^* + 10^*$ final states.}
\bigskip

Here a single amplitude describes all decays.  The relation
%
\beq \ca(B^- \to \Lambda_c^+ \overline \Delta^{--}) = \st \ca(B^0
\to \Lambda_c^+ \overline \Delta^-) \eeq
%
is a consequence of the pure isospin $(I=3/2)$ of the final state.
The decays $\overline{B} \to \Xi_c \overline \Sigma^*$ involve
both $I = 1/2$ and $I = 3/2$, but these amplitudes are related to
one another since $\ob \to \Xi_c^+ \overline \Sigma^{*-}$ is
forbidden. This process could only have proceeded via an exchange
amplitude, but the final charmed baryon is antisymmetric in its
light quarks, which cannot couple to the symmetrized quarks in the
final antidecuplet antibaryon.  Thus the isospin relation
%
\beq \ca(B^- \to \Xi_c^0 \overline \Sigma^{*-}) = \s \ca(\ob \to
\Xi_c^0 \overline \Sigma^{*0}) + \ca(\ob \to \Xi_c^+ \overline
\Sigma^{*-})~~~ \eeq
%
is implemented as
%
\beq \ca(B^- \to \Xi_c^0 \overline \Sigma^{*-}) = \s \ca(\ob \to
\Xi_c^0 \overline \Sigma^{*0})~~~. \eeq
%
There are no triangle relations, and no tests for a vanishing
exchange amplitude since it never contributes in the first place.
\bigskip

\leftline{\bf D.  $6 + 10^*$ final states.}
\bigskip

There are a number of isospin triangles involving the charge
states of $\overline{B} \to \Sigma_c \overline \Delta$.  One
example for which detection of final states may be particularly
favorable is
%
\beq \ca(\ob \to \Sigma_c^{++} \overline \Delta^{--}) + \st
\ca(B^- \to \Sigma_c^0 \overline \Delta^-) = \st \ca(\ob \to
\Sigma_c^0 \overline \Delta^0)~~~. \eeq
%
Another useful relation involves the two charge states of $B^- \to
\Sigma_c \overline \Delta$:
%
\beq  \s \ca(\ob \to \Sigma_c^+ \overline \Delta^{--}) + \st
\ca(B^- \to \Sigma_c^0 \overline \Delta^-) = 0~~~. \eeq
%
In order that the isospin triangles have non-zero area, both $d$
and $d_E$ must be nonvanishing and have a nontrivial relative
phase. A good test for $d_E = 0$ is to check whether the decay
$\ob \to \Sigma_c^{++} \overline \Delta^{--}$ is suppressed in
comparison with other $\overline{B} \to \Sigma_c \overline \Delta$
decays. Another isospin triangle involving the charge states of
$\ob \to \Xi_c \overline \Sigma^*$ is
%
\beq \ca(B^- \to {\Xi'}_c^0 \overline \Sigma^{*-}) = \s \ca(\ob
\to {\Xi'}_c^0 \overline \Sigma^{*0}) + \ca(\ob \to {\Xi'}_c^+
\overline \Sigma^{*-})~~~. \eeq
%
However, experimentally it is not easy to construct.

\bigskip
\centerline{\bf IV.  DISCUSSION AND SUMMARY}
\bigskip

The recent observation of a two-body baryon-antibaryon $B$ decay \cite{BeLp}
is likely to be the first in a series of such decays.  We have shown that
these processes are capable of providing information on two main questions
which have been of interest in $B$ meson decays for some years:  (1) Are there
significant final-state interaction phases between different decay amplitudes
characterized by the same weak phases?  (2) Are processes involving the
spectator quark (such as the exchange amplitudes described here by the suffix
$E$) suppressed in comparison with other amplitudes in which the spectator
does not enter into the weak Hamiltonian?  We have described a number of tests
of both these questions which may be feasible in the near future.
Given the value of the observed branching ratio for $\ob \to
\Lambda_c^+ \bar p$ \cite{BeLp}, which was based on an integrated
luminosity of 78.2 fb$^{-1}$, several times the present data
sample may be needed to see some of the related decay modes, but
the triangle construction in Fig.~\ref{fig:amps} suggests that at
least some other decay modes to a charmed baryon and an octet
antibaryon may be observable with comparable branching ratios.

\bigskip
\centerline{\bf ACKNOWLEDGEMENTS}
\bigskip

This work was supported in part by the United States Department of Energy,
High Energy Physics Division, under Contract No.\ DE-FG02-90ER-40560.

\bigskip
\centerline{\bf APPENDIX: QUARK COMPOSITION OF BARYONS}
\bigskip

In our convention ($\Xi_c^0$, $\Xi_c^+$), ($\bar p$, $\bar n$),
($\overline \Sigma^-$, $\overline \Sigma^0$, $\overline
\Sigma^+$), ($\overline \Xi^0$, $\overline \Xi^+$), ($\overline
\Delta^{--}$, $\overline \Delta^-$, $\overline \Delta^0$,
$\overline \Delta^+$), ($\overline \Sigma^{*-}$, $\overline
\Sigma^{*0}$, $\overline \Sigma^{*+}$) and ($\overline \Xi^{*0}$,
$\overline \Xi^{*+}$) are in iso-multiplets. We recall that $I_- u
= d$, $I_- \bar d = - \bar u$. Our convention for the $\overline
B$ mesons is: $B^- = - b \bar u$, $\overline B^0 = b \bar d$,
$\overline B_s^0 = b \bar s$.
\bigskip

\noindent I) Antitriplet charmed baryons:
%
\begin{eqnarray*}
\Lambda_c^+ & = & (c u d - c d u)/\s \\
\Xi_c^+ & = & (c s u - c u s)/\s \\
\Xi_c^0 & = & (c s d - c d s)/\s
\end{eqnarray*}
%
II) Sextet charmed baryons:
%
\begin{eqnarray*}
\Sigma_c^{++} & = & c u u \\
\Sigma_c^+ & = & (c u d + c d u)/\s \\
\Sigma_c^0 & = & c d d \\
{\Xi'}_c^+ & = & (c u s + c s u)/\s \\
{\Xi'}_c^0 & = & (c d s + c s d)/\s \\
\Omega_c^0 & = & c s s
\end{eqnarray*}
%
III) Octet antibaryons:
%
\begin{eqnarray*}
\bar p & = & (\bu \bd \bu - \bd \bu \bu)/\s \\
\bar n & = & (\bd \bu \bd - \bu \bd \bd)/\s \\
\overline \Sigma^- & = & (\bs \bu \bu - \bu \bs \bu)/\s \\
\overline \Sigma^0 & = & (\bu \bs \bd - \bs \bu \bd + \bd \bs \bu - \bs \bd \bu)/2 \\
\overline \Sigma^+ & = & (\bs \bd \bd - \bd \bs \bd)/\s \\
\overline \Xi^0 & = & (\bu \bs \bs - \bs \bu \bs)/\s \\
\overline \Xi^+ & = & (\bs \bd \bs - \bd \bs \bs)/\s \\
\overline \Lambda & = & (2 \bu \bd \bs - 2 \bd \bu \bs - \bd \bs
\bu + \bs \bd \bu - \bs \bu \bd + \bu \bs \bd)/\sqrt{12}
\end{eqnarray*}
%
IV) Antidecuplet antibaryons:
%
\begin{eqnarray*}
\overline \Delta^{--}  & = & - \bu \bu \bu \\
\overline \Delta^- & = & (\bu \bu \bd + \bu \bd \bu + \bd \bu \bu)/\st \\
\overline \Delta^0 & = & -(\bu \bd \bd + \bd \bu \bd + \bd \bd \bu)/\st \\
\overline \Delta^+  & = & \bd \bd \bd \\
\overline \Sigma^{*-} & = & (\bu \bu \bs + \bu \bs \bu + \bs \bu \bu)/\st \\
\overline \Sigma^{*0} & = & -(\bu \bd \bs + \bu \bs \bd + \bd \bu \bs + \bd \bs \bu + \bs \bu \bd + \bs \bd \bu)/\sqrt{6} \\
\overline \Sigma^{*+} & = & (\bd \bd \bs + \bd \bs \bd + \bs \bd \bd)/\st \\
\overline \Xi^{*0} & = & -(\bu \bs \bs + \bs \bu \bs + \bs \bs \bu)/\st \\
\overline \Xi^{*+} & = & (\bd \bs \bs + \bs \bd \bs + \bs \bs \bd)/\st \\
\overline \Omega^+ & = & \bs \bs \bs
\end{eqnarray*}
%

% Journal and other miscellaneous abbreviations for references
% Phys. Rev. D style
\def \ajp#1#2#3{Am.~J.~Phys.~{\bf#1}, #2 (#3)}
\def \apny#1#2#3{Ann.~Phys.~(N.Y.) {\bf#1}, #2 (#3)}
\def \app#1#2#3{Acta Phys.~Polonica {\bf#1}, #2 (#3)}
\def \arnps#1#2#3{Ann.~Rev.~Nucl.~Part.~Sci.~{\bf#1}, #2 (#3)}
\def \art{and references therein}
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Workshop on $B$-Physics at Hadron Machines, Los Angeles, October 13--17,
1997, edited by P. Schlein}
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M. L\'evy \ite, NATO ASI Series B:  Physics Vol.~363 (Plenum, New York, 1997)}
\def \cmp#1#2#3{Commun.~Math.~Phys.~{\bf#1}, #2 (#3)}
\def \cmts#1#2#3{Comments on Nucl.~Part.~Phys.~{\bf#1}, #2 (#3)}
\def \corn93{{\it Lepton and Photon Interactions:  XVI International
Symposium, Ithaca, NY August 1993}, AIP Conference Proceedings No.~302,
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\def \cp89{{\it CP Violation,} edited by C. Jarlskog (World Scientific,
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\def \dpff{{\it The Fermilab Meeting -- DPF 92} (7th Meeting of the
American Physical Society Division of Particles and Fields), 10--14
November 1992, ed. by C. H. Albright \ite~(World Scientific, Singapore,
1993)}
\def \dpf94{DPF 94 Meeting, Albuquerque, NM, Aug.~2--6, 1994}
\def \efi{Enrico Fermi Institute Report No. EFI}
\def \el#1#2#3{Europhys.~Lett.~{\bf#1}, #2 (#3)}
\def \epjc#1#2#3{Eur.~Phys.~J.~C {\bf#1}, #2 (#3)}
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\def \jmp#1#2#3{J.~Math.~Phys.~{\bf#1}, #2 (#3)}
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\def \mpla#1#2#3{Mod.~Phys.~Lett.~A {\bf#1}, #2 (#3)}
\def \nc#1#2#3{Nuovo Cim.~{\bf#1}, #2 (#3)}
\def \nima#1#2#3{Nucl.~Instr.~Meth.~A {\bf#1}, #2 (#3)}
\def \np#1#2#3{Nucl.~Phys.~{\bf#1}, #2 (#3)}
\def \npbps#1#2#3{Nucl.~Phys.~B (Proc.~Suppl.) {\bf#1}, #2 (#3)}
\def \pisma#1#2#3#4{Pis'ma Zh.~Eksp.~Teor.~Fiz.~{\bf#1}, #2 (#3) [JETP
Lett. {\bf#1}, #4 (#3)]}
\def \pl#1#2#3{Phys.~Lett.~{\bf#1}, #2 (#3)}
\def \plb#1#2#3{Phys.~Lett.~B {\bf#1}, #2 (#3)}
\def \pr#1#2#3{Phys.~Rev.~{\bf#1}, #2 (#3)}
\def \pra#1#2#3{Phys.~Rev.~A {\bf#1}, #2 (#3)}
\def \prd#1#2#3{Phys.~Rev.~D {\bf#1}, #2 (#3)}
\def \prl#1#2#3{Phys.~Rev.~Lett.~{\bf#1}, #2 (#3)}
\def \prp#1#2#3{Phys.~Rep.~{\bf#1}, #2 (#3)}
\def \ptp#1#2#3{Prog.~Theor.~Phys.~{\bf#1}, #2 (#3)}
\def \rmp#1#2#3{Rev.~Mod.~Phys.~{\bf#1}, #2 (#3)}
\def \rp#1{~~~~~\ldots\ldots{\rm rp~}{#1}~~~~~}
\def \si90{25th International Conference on High Energy Physics, Singapore,
Aug. 2-8, 1990}
\def \slc87{{\it Proceedings of the Salt Lake City Meeting} (Division of
Particles and Fields, American Physical Society, Salt Lake City, Utah,
1987), ed.~by C. DeTar and J. S. Ball (World Scientific, Singapore, 1987)}
\def \slac89{{\it Proceedings of the XIVth International Symposium on
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\def \smass82{{\it Proceedings of the 1982 DPF Summer Study on Elementary
Particle Physics and Future Facilities}, Snowmass, Colorado, edited by R.
Donaldson, R. Gustafson, and F. Paige (World Scientific, Singapore, 1982)}
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Theoretical Advanced Study Institute in Elementary Particle Physics,
Boulder, Colorado, 3--27 June, 1990), edited by M. Cveti\v{c} and P.
Langacker (World Scientific, Singapore, 1991)}
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23--31 July 1998}
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#4 (#3)]}
\def \zhetf#1#2#3#4#5#6{Zh.~Eksp.~Teor.~Fiz.~{\bf #1}, #2 (#3) [Sov.~Phys.
-- JETP {\bf #4}, #5 (#6)]}
\def \zpc#1#2#3{Zeit.~Phys.~C {\bf#1}, #2 (#3)}

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%\cite{Jessop:1998wt}
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%%CITATION = ;%%

\end{thebibliography}
\end{document}
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