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\newcommand{\mev}{\hspace{1pt}{\rm M}^{}{\rm e}\hspace{0.4mm}\!{\rm V}}
\newcommand{\gev}{\,{\rm G}^{}{\rm e}\hspace{0.4mm}\!{\rm V}}
\newcommand{\plus}{{\raisebox{1pt}{\tiny$+$}}}
\newcommand{\minus}{{\raisebox{1pt}{\tiny$-$}}}
\newcommand{\coeff}[2]{\raisebox{-1pt}{\Large$\frac{#1}{#2}$}}
\newcommand{\Gf}{G_{\!\mbox{\scriptsize$F$}}}
\newcommand{\Hw}{H_{\mbox{\scriptsize$W$}}}
\newcommand{\Mw}{M_{\mbox{\scriptsize$W$}}}
\newcommand{\mw}{m_{\mbox{\scriptsize$W$}}}
\newcommand{\MA}{M_{\!\mbox{\scriptsize$A$}}}
\newcommand{\Mp}{M_{\!\mbox{\scriptsize$P$}}}
\newcommand{\Mxi}{M_{\mbox{\scriptsize$\Xi$}}}
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\newcommand{\bsR}{\bar s_{\!\mbox{\scriptsize$R$}}}
\newcommand{\dL}{d_{\mbox{\scriptsize$L$}}}
\newcommand{\uSigma}{u_{\mbox{\scriptsize$\Sigma$}}}
\newcommand{\Xim}{\Xi^\minus}
\newcommand{\Xip}{\Xi^\plus}
\newcommand{\Sim}{\Sigma^\minus}
\newcommand{\Sip}{\Sigma^\plus}
\newcommand{\Omm}{\Omega^\minus}
\newcommand{\pim}{\pi^\minus}
\newcommand{\kE}{\varkappa_{\mbox{\scriptsize$\Sigma$}}}
\newcommand{\kL}{\varkappa_{\mbox{\scriptsize$\Lambda$}}}
%\figurebox{2cm}{3cm}{} %to have a box alone

\markboth{\quad \small \sl I.~I.~Kogan and M.~A.~Shifman \hfill}
         {\hfill\small \sl Once More on Weak Radiative Decays \quad }
\setcounter{page}{2199}
\begin{document}

\title{Once more on weak radiative decays}

\author{I.\,I.~Kogan \lowercase{\,and\,} M.\,A.~Shifman}

\address{Institute of Theoretical and Experimental Physics,\\
State Atomic Energy Commission\\(Submitted 24 January 1983)}

\maketitle

\centerline{\small Yad.\ Fiz.\ {\bf 38}, 1045 (1983)~~~(Submitted 24 January 1983) }

\centerline{\small [Sov.\ J.\ Nucl.\ Phys.\ {\bf 38}, 628 (1983),
            translated by L.~J.~Reinders\,]}
%\centerline{\small(Submitted 24 January 1983)}

\bigskip

\abstracts{
We consider weak radiative decays of hyperons. It is shown that there exists
an exact unitary lower bound for the decay probability, for example,
\mbox{
BR$(\Xi^\minus\!\to\!\Sigma^\minus\gamma)\!\geq\!1.0\cdot 10^{\minus 4}$.
}
We show
that the real part of the amplitude is singular in the chiral limit, i.e.\ it
contains terms $\sim \ln m_{q}$, where $m_{q}$ is the current quark mass.
The coefficient of the logarithmic terms is fixed uniquely. In the case of
the decays $\Xi^\minus\!\to\!\Sigma^\minus\gamma$ and
$\Omega^\minus\!\to\!\Xi^\minus \gamma$ the nonsingular
(model-dependent) terms are relatively small, and it is possible to obtain 
a reasonably accurate estimate for the real part of the amplitudes
$\Xi^\minus \!\to\! \Sigma^\minus\gamma$
and~\mbox{$\Omega^\minus\!\to\!\Xi^\minus\gamma$.}
Taking the real part into account,
BR$(\Xi^\minus\!\to\!\Sigma^\minus\gamma)\approx 1.7\cdot 10^{\minus4}$.}

\section{Introduction and Formulation of the Problem}

In the last ten years weak radiative decays of hyperons have repeatedly
been discussed in the literature \cite{one,two}. So far there exists no
adequate theoretical description of these processes. This is one of the
reasons which stimulate theorists to return continually to this problem.
Another reason for so much steady attention is the hope of obtaining
nontrivial information on the structure of the weak interaction and on the
structure of hadrons (hyperons).

In this note we will show that for the decays
\begin{equation}\label{eq:1}
\Xi^\minus\!\to \Sigma^\minus \gamma
\qquad{\rm and}\qquad
\Omega^\minus\!\to \Xi^\minus \gamma\,,
\end{equation}
there exist reliable and fairly accurate theoretical predictions. With some
reservations one can say that there exists a theory of the decays in framework
of which the amplitudes \eqref{eq:1} can be expressed in terms of
quantities which are either empirically or theoretically well known. In
fairness one has to note that the analysis of the decays
$\Xi^\minus\!\!\to\!\Sigma^\minus\gamma$ and
$\Omega^\minus\!\!\to\!\Xi^\minus\gamma$ adds
practically nothing new to what is already known about the weak interactions.

Before formulating the predictions and giving their derivation, we recall the
general situation with respect to weak radiative decays. A short historical 
digression will serve as an introduction and will also contain a series of 
critical remarks on the literature.

The amplitude of the weak radiative decay has the form
\begin{equation}\label{eq:2}
   M\bigl(B_i\to B_f\gamma\bigl)
= -^{}ie\!\int\!d^4x^{}e^{iqx}
   \bigl\langle B_f|^{}T\{\Hw(0),J_\mu^{\rm em}(x)\}|B_i
   \bigr\rangle^{}\varepsilon_\mu\,,
\end{equation}
where
$J^{\rm em}_\mu\!= \!\frac{2}{3}\,\bar u \gamma_\mu u\!
                  -\!\frac{1}{3}\,\bar d \gamma_\mu d^{\!}+\!\ldots\,$
is the electromagnetic current, $B_i$ and~$B_f$ are the initial and final 
baryon states, $q_\mu$ and $\varepsilon_\mu$ are the momentum and polarization
of the emitted photon, and finally $\Hw$ is
the Hamiltonian of the weak interaction.

There are two contributions to the amplitude \eqref{eq:2} whose nature is
essentially different. The first of these is connected with the emission of the
photon at short distances $x\sim 1/\Mw$
($\Mw$ is the mass of the $W$ boson) 
and can be expressed in terms of the local quark operator
\begin{equation}\label{eq:3}
T= i^{} \bsR^{\,}\sigma_{\!\mu\nu}^{\,}\dL F_{\mu\nu}\,,
\end{equation}
which describes the transition $s\!\to\!d\gamma$ (the notation is
self-evident). The coefficient of this operator is determined by the graphs
of Fig.\,\ref{fig:one} and can be calculated reliably in the framework of QCD.
\begin{figure}
\centerline{\epsfxsize=4.5in\epsfbox{fig1.eps}}
\caption{One- and two-loop diagrams which determine the local transition
         $s\!\to\! d\gamma\,$. The solid lines denote quarks, the wavy lines
         photons, and the dashed lines gluons.}
\label{fig:one}
\end{figure}

Let us forget for a moment about gluon exchanges and concentrate on
Fig.\,\ref{fig:one}a.
A structure of the type \eqref{eq:3} can only arise from the region of
virtual momenta $p^2\!\sim\!\Mw^2\,$. It is easy to
convince oneself of this if one takes into 
account: a) gauge invariance with respect to the photon; b) the fact that the 
$W$ boson interacts only with left-handed currents. It is obvious that the 
unitarity of the Kobayashi-Maskawa matrix leads to a cancellation of the $u$,
$c$, and $t$ contributions in leading orders, i.e.\ to order 
$\Gf^{}e/\pi^2$, where
$\Gf\!=\!10^{\minus 5}\Mp^{\minus 2}$ and
$e^2\!/4\pi\!=\!1/^{\!}137\,$.
This results in an extra power suppression of the type
$(m_{c}^{2}\!-m_{u}^{2})/\mw^{2}$, and the graph of Fig.\,\ref{fig:one}a is
negligibly small. Taking gluons into account (Fig.\,\ref{fig:one}b) removes
the power suppression, since the factors
$(m_{c}^{2}\!-m_{u}^{2})/\mw^{2}$
are replaced by $\ln(m_{c}^{2}/m_{u}^{2})$ (for more details
see Ref.\cite{two},
where an accurate calculation of the coefficient of the operator $T$ has been
performed in the leading logarithmic approximation in the four-quark model).
Of course, there remains a very strong numerical suppression, since the 
important diagrams have two or more loops. A generalization of the results of
Ref.\cite{two} to the modern Kobayashi-Maskawa six-quark model does not exist
in the literature, as far as we know. The contribution of the $t$ quark,
however, cannot be dominant, since there exist rather strong bounds on the 
corresponding mixing angles. The estimate presented in the Appendix shows 
that it is not larger than the $c$ and $u$ contributions. Thus, the result of 
Ref.\cite{two} gives a good idea of the order of magnitude of the effect;
the local $s\!\to\!d\gamma$ transition is
\begin{equation}\label{eq:4}
  M(s \to d\gamma)
\simeq
  i\,\coeff{e\,\Gf}{16\pi^2}\;
  0.4\,m_s\, \bar \sR^{\,}\sigma_{\!\mu\nu}^{\,}\dL F_{\mu\nu}\;.
\end{equation}
It is natural to assume that the matrix element of the operator $T$ for baryon
states is of order of unity, for instance,
\begin{equation}\label{eq:5}
  \bigl\langle^{}p,\gamma|^{}T|\Sigma^\plus\bigr\rangle
\sim
  \coeff12\,\bar u_p ^{} \sigma_{\!\mu\nu} (1+\gamma_{^{}5})
  \uSigma F_{\mu\nu}\;.
\end{equation} 
Combining \eqref{eq:4} and \eqref{eq:5}, we get for the
contribution of the local transition~\mbox{$s\!\to\!d\gamma$} to the
decay $\Sigma^\plus\!\to\!p\gamma$
\begin{equation}\label{eq:6}
{\rm BR}\bigl(\Sigma^\plus\!\to p\gamma\bigr)\big|_{s\to d\gamma}\!
\sim 2\cdot10^{\minus 6}\,,
\end{equation}
where the current $s$ quark mass $m_s$ in formula \eqref{eq:4} is
about $150\mev$. We recall that experimentally 
\begin{equation}\label{eq:7}
{\rm BR}\bigl(\Sigma^\plus\!\to p\gamma\bigr)\big|_{\rm exp}\!
=1.2\cdot10^{\minus 3}\,.
\end{equation}
Even if for some unknown reason the estimate \eqref{eq:5}
has been underestimated by an order of magnitude (which is extremely
unlikely), the contribution of the local transition $s\!\to\!d\gamma$ is
still an order of magnitude less than we see experimentally.

Thus, purely theoretical arguments show that the mechanism of
Fig.\,\ref{fig:one} gives a negligibly small contribution to the amplitude
of weak radiative decays. One can also give an independent phenomenological
argument. Indeed, let us assume for a moment (although this is absolutely
inconceivable) that the local $s\!\to^{\!}d\gamma$ transition gives the whole
amplitude of the decay $\Sigma^\plus\!\to^{\!}p\gamma$ measured experimentally.
Since the coefficient in front of the operator $T$ is fixed (see
Eq.\,\eqref{eq:4}), this means that we fix the matrix element
$\langle^{}p\gamma^{}|^{}T^{}|^{}\Sigma^\plus\rangle^{}$. By means of $SU(3)$
symmetry the latter can be connected to the matrix elements of other decays,
for instance
$\bigl\langle\Sigma^\minus\gamma^{}|^{}T^{}|^{}\Xi^\minus\bigr\rangle$.
A detailed analysis has been carried out in Ref.\cite{three}, where the
following prediction has been obtained:
$$
{\rm BR}\bigl(\Xi^\minus\!\!\to\Sigma^\minus\gamma\bigr) \sim 10^{\minus 2}\,,
$$
which exceeds by an order of magnitude the experimental upper bound
\begin{equation}\label{eq:8}
{\rm BR}\bigl(\Xi^\minus\!\!\to\Sigma^\minus\gamma\bigr)_{\rm exp}\!
< 1.2\cdot10^{\minus 3}\,.
\end{equation}

Thus, the mechanism of Fig.\,\ref{fig:one} cannot play an important role
in weak radiative decays. This conclusion is unambiguous, and consequently
assuming dominance of photon emission at short distances does not even
remotely correspond to the real situation.

We will turn to another mechanism which therefore should play the main role in
weak radiative decays. Let us consider photon emission at large distances, 
i.e.\ $x \sim R_c$ in the amplitude \eqref{eq:2}, where $R_c$ is the
confinement radius. In this case the contribution from weak and
electromagnetic interactions can be separated. The amplitude \eqref{eq:2}
can be presented in the form of a sum over intermediate hadronic states
\begin{equation}\label{eq:9}
\begin{split}
\hspace{-2mm}
     M\bigl( B_i \!\to\! B_{\!f}\gamma \bigr)\!
=\! -ie^{}\varepsilon_\mu\!\!\int\!\!d^{\,4}\!x^{}e^{iqx}
     \Bigl(
&           \sum_X\theta(\minus x_0) \langle      B_f|\Hw(0)|X       \rangle
                               \langle X|J_\mu^{\rm em}(x)|B_i \rangle
\hspace{-2mm}
\\
+&          \sum_X^{\!}\theta( x_0) \langle B_f|J_\mu^{\rm em}(x)|X \rangle
                               \langle      X|\Hw(0)|B_i       \rangle
     \Bigr)\,.\hspace{-3mm}
\end{split}
\end{equation}

Usually one considers the pole approximation, in which only one-baryon
intermediate states are kept in \eqref{eq:2}. The corresponding diagrams
are given in Fig.\,\ref{fig:two}\,.
\begin{figure}
\centerline{\epsfxsize=3.8in\epsfbox{fig2.eps}}
\caption{Pole diagrams. The solid lines denote baryons.}
\label{fig:two}
\end{figure}
This approximation is apparently adequate for rough
estimates of decays like $\Sigma^\plus\!\!\to^{\!}p\gamma\,$, but is completely
unacceptable for the decays \eqref{eq:1}\,;
see below.\footnote{\,In a
recent paper \cite{four}, the decays \eqref{eq:1} are considered in the
pole approximation. The probabilities predicted are much smaller than ours.}
Moreover, even in the favorable case of amplitudes like
$\Sigma^\plus\!\!\to\!p\gamma$ one cannot calculate with an
accuracy better than a factor of 2, which gives an uncertainty of
a factor of 4 in the decay probability. 

In this paper we wish to draw attention to two facts. First, one can 
determine exactly a lower bound for the probability of weak radiative decays
from the imaginary part of the amplitude.\footnote{\,The existence of real
intermediate states which generate imaginary parts of the amplitudes has
apparently first been noted in Ref.\cite{five}, which is devoted to the decay
$\Sigma^\plus\!\!\to\!p\gamma\,$. In the same paper a unitary bound for the
decay $\Xi^\minus\!\to\Sigma^\minus\gamma$ is quoted which practically
coincides with \eqref{eq:10}. Unfortunately, we learned about this only after
our paper was sent to the Publisher.} We will show that in the case of the
decays \eqref{eq:1} this lower bound is of the same order of magnitude
as the total probability of the decay, and amounts to
\begin{equation}\label{eq:10}
\begin{split}
    {\rm BR}\bigl( \Xim\!\to\Sim\gamma \bigr)
    _{\genfrac{}{}{0pt}{2}{\!\rm unitary}{\!\!\!\!\!\rm limit}}\!
&>  1\cdot10^{\minus 4}\,,
\\
    {\rm BR}\bigl( \Omm\!\to\Xim\gamma \bigr)
    _{\genfrac{}{}{0pt}{2}{\!\rm unitary}{\!\!\!\!\!\rm limit}}\!
&>  0.8\cdot10^{\minus 5}\,.
\end{split}
\end{equation}

Second, for the decays \eqref{eq:1} one can estimate fairly reliably the
real part of the amplitude, on the basis of an exact low-energy theorem,
which states that in the chiral limit two-particle baryon-pion states
dominate in the amplitude~\eqref{eq:9}. The contribution of the latter
contains a logarithmic singularity in the pion mass, and the coefficient in
front of $\ln\!\mu^2$ is fixed uniquely ($\mu$ is the pion mass). So far,
these two aspects have not been discussed in the literature, at least not
on the level required by the experimental  investigation of the decays
\eqref{eq:1}. For the decay $\Xim\!\!\to\!\Sim\gamma$ we
have obtained the following value of the relative decay probability:
BR$(\Xim\!\!\to\!\Sim\gamma)\!\approx\!1.7\cdot10^{\minus 4}$. To avoid
misunderstanding we emphasize that these two points are not specific for the 
decays \eqref{eq:1}. The amplitudes of all weak radiative transitions have
an imaginary part, determined by unitarity, as well as a factor $\ln\!^{}\mu^2$
in the real part. However, in most cases these contributions are relatively
small and the real part, which corresponds to Fig.\,\ref{fig:two} and does
not contain $\ln\!^{}\mu^2$, dominates in the amplitudes. The decays
\eqref{eq:1} are distinguished by the fact that for them the diagrams
of Fig.\,\ref{fig:two} are forbidden (more accurately, strongly 
suppressed), which is due to the specific form of $\Hw$. On these decays loop 
diagrams, which are usually very small against the background of the tree 
diagrams of Fig.\,\ref{fig:two}, play the major role. 

The layout of the paper is as follows. In Section 2 we discuss the amplitude
of radiative decay and calculate its imaginary part in the case
$\Xim\!\!\to\!\Sim\gamma$. In Section 3 we calculate the real part,
which dominates in the chiral limit, and we discuss the estimate of the
relative probability of the decay $\Xim\!\!\to\!\Sim\gamma$.
In Sections 4 and 5 we estimate the contribution of the mechanism suggested
to the relative probability of the decays $\Omm\!\!\to\!\Xim\gamma$ and
$\Sip\!\!\to\!p\gamma$, and we discuss the pole contributions, which
dominate in the latter case and are absent in the cases
$\Xim\!\!\to\!\Sim\gamma$ and $\Omm\!\!\to\!\Xim\gamma$. In the Appendix
we give an estimate of the $t$ quark contribution to the local transition 
$s^{\!}\to d\gamma$. 

\section{Calculation of the Imaginary Part
         of the $\Xi^- \!\to\Sigma^- \gamma$ Amplitude}

The matrix element of a weak radiative hyperon decay is usually
para\-metrized\cite{six} in the form
\begin{equation}\label{eq:11}
   M\bigl( B_i\to B_f\gamma \bigr)
=  \Gf\sqrt{4\pi\alpha} \,\Delta\, \bar B_f (a\gamma_{^{}5}+b)
   \sigma_{\!\mu\nu}q_\nu B_i\,\varepsilon_\mu\,,
\end{equation}
where $q_v$ and $\varepsilon_\mu$ are the momentum and polarization of the 
photon, and~\mbox{$\Delta\!=\!140\mev$}, which coincides numerically
with the pion mass $\mu^{}$. We note however that we will consider $\mu$
as a variable parameter, and in particular~$\mu\!\to\!0$ in the chiral limit.
The parameter $\Delta$ is a fixed number introduced for convenience.
Generally speaking, the amplitudes $a$ and $b$ have real as well as
imaginary parts, which do not interfere in the decay probability. Therefore,
the imaginary parts of the amplitudes determine the theoretical lower bound
of the decay width. By virtue of unitarity the imaginary part of the
amplitude is determined by the contribution of real intermediate states.
In the case of the decay $\Xim\!\to\!\Sim\gamma$ there is only one real
intermediate state $\Lambda\pi^\minus$ in first order in the weak and the
electromagnetic interactions. Thus, the imaginary part is determined by the
diagram of Fig.\,\ref{fig:three}.
\begin{figure}
\centerline{\epsfxsize=3.8in\epsfbox{fig3.eps}}
\caption{The diagram which determines the imaginary part of the amplitude
of weak radiative decay. The dashed line denotes a pion.}
\label{fig:three}
\end{figure}
The amplitude of the weak decay
$\Xim\!\!\to\!\Lambda\pi^\minus$ is known from experiment. It is usually 
presented \cite{six} in the form
\vspace{-2mm}
\begin{equation}\label{eq:12}
   M \bigl( \Xim\!\to\Lambda\pim \bigr)
= -i \Gf \Delta^2 \bar\Lambda \bigl( A+B\gamma_{^{}5} \bigr) \Xi\,,
\end{equation}
where
\vspace{-2mm}
\begin{equation}\label{eq:13}
A=2.04\pm0.01\,, \qquad B=-7.49\pm0.28\,,
\end{equation}
and we denote bispinors by the same symbols as particles. The right-hand 
block in Fig.\,\ref{fig:three} is the amplitude $T_\mu$ for photoproduction
of the pion on the hyperon. Since the values of the pion momentum $k$ and
the photon energy~$q_{\,0}$ in the c.m.s. are small ($k\!=\!139\mev$ and
$q_0\!=\!118\mev$), the amplitude $T_\mu$ is accurately fixed in the
framework of PCAC.
\begin{figure}
\centerline{\epsfxsize=5in\epsfbox{fig4.eps}}
\caption{The diagrams which determine in leading order the amplitude
         of pion photoproduction on a baryon.}
\label{fig:four}
\end{figure}
More accurately, this amplitude can be found theoretically
up to terms linear in the momentum~$q_\mu\,$\cite{seven}.
In this approximation $T_\mu$ is determined by the pole and contact diagrams
of Fig.\,\ref{fig:four} and $T_\mu=T_{\mu_1}+\ldots+T_{\mu_4}$. The explicit
expressions for the amplitudes~$T_{\mu_1},\ldots,T_{\mu_4}$
have the following form:
\begin{align}\label{eq:14}
\nonumber
    T_{\mu_1}
&= -i\sqrt{4\pi\alpha}^{}f^{}\bar\Sigma\,
    \bigl( \minus \gamma_\mu
           +\coeff{\varkappa_{\mbox{\tiny$\Sigma$}}}
                             {2M_{\mbox{\tiny$\Sigma$}}}\,
            \sigma_{\!\mu\nu}q_\nu
    \bigr)^{}
    \coeff{1}{\hat p-M_{\mbox{\tiny$\Sigma$}}}\,
    \hat k \gamma_{^{}5} \Lambda\,,
\\[1.5mm]
\nonumber
    T_{\mu_2}
&= -i\sqrt{4\pi\alpha}^{}f^{}\bar\Sigma\,
    \coeff{\hat k \gamma_{^{}5}}
          {\hat p - \hat k - \hat q - M_{\mbox{\tiny$\Lambda$}}}\,
    \coeff{\varkappa_{\mbox{\tiny$\Lambda$}}}
          {\overset{}{2M_{\mbox{\tiny$\Lambda$}}}}\;
    \sigma_{\!\mu\nu}q_\nu\Lambda\,,
\\[1.5mm]
\nonumber
    T_{\mu_3}
&=  i\sqrt{4\pi\alpha}^{}f^{}\bar\Sigma\,
    \coeff{(2k-q)_\mu}{(k-q)^2-\mu^2}^{}
    \bigl(\hat k - \hat q\bigr)\gamma_{^{}5}\Lambda\,,
\\[1.5mm]
    T_{\mu_4}
&= -i\sqrt{4\pi\alpha}^{}f^{}\bar\Sigma\gamma_\mu\gamma_{^{}5}\Lambda\,,
\end{align}
where the momenta $p$, $k$, and $q$ are arranged in Fig.\,\ref{fig:four} and 
$\kE$ and $\kL$ are the anomalous magnetic moments of the $\Sim$ and $\Lambda$
hyperons, respectively ($\kE\!=\!-0.48\!\pm\!0.37$ and
$\kL\!=\!-0.67\!\pm\!0.06$). The PCAC coupling constant is
\begin{equation}\label{eq:15}
f=\subsup{g}{A}{\,\Lambda\Sigma}/f_\pi\,,
\end{equation}
where $f_{\!\pi}\!\!=\!^{\!}133\mev$ and $\subsup{g}{A}{\,\Lambda\Sigma}$ is
the axial constant of the
transition~\mbox{$\Sim\!\!\to\!\Lambda e^\minus\nu^{}$}.
We note that, since we use the PCAC technique, the $\Sigma \Lambda \pi$
vertex has the axial-vector form, and the corresponding constant
$\subsup{g}{A}{\,\Lambda\Sigma}/f_\pi$ is connected with the standard
pseudoscalar constant $\subsup{g}{\Sigma\Lambda\pi}{}$ via the
Goldberger-Treiman relation. We note that exactly the same approximation
has been verified empirically in an analogous process, pion photoproduction on
a nucleon, where it has an accuracy of the order of 10\% in the amplitude.
This accuracy is in general characteristics for the PCAC technique. One can 
show (see Ref.\cite{seven}, Sec. 4.1.1) that the terms linear and of higher
orders in $q_{\mu}$ which we have not kept, give contributions of order
$(k/M)^2\!\sim 10^{\minus 2}$ to the imaginary part
of~$M(\Xim\!\!\to\!\Sim\gamma)$,
where $M$ is the hyperon mass.\footnote{\,More accurately, to ${\rm Im}a$,
where the invariant amplitude $a$ is defined in Eq.\ref{eq:11}.}
These contributions are negligibly small. Thus, the diagram of
Fig.\,\ref{fig:three} is completely known. It is essential to note the
following. The contributions of the amplitudes $T_{\mu_1}$ and~$T_{\mu_2}$
which describe photon emission by $\Sigma^\minus $ and $\Lambda$ hyperons
(see Figs.\,\ref{fig:four}$^{}$a,b\,; \ref{fig:five}$^{}$a,b) is suppressed
relative to the contribution of the amplitudes $T_{\mu_3}$ and $T_{\mu_4}$
(see Figs.\,\ref{fig:four}$^{}$c,d\,; \ref{fig:five}$^{}$c,d\,) by
$k/M$ and $(k/M)^2$ in the charged and magnetic parts, respectively.
\begin{figure}
\centerline{\epsfxsize=5in\epsfbox{fig5.eps}}
\caption{The diagrams which determine in leading order the imaginary part
\         of weak radiative decay in the case $\Xi^-\!\!\to\Sigma ^m \gamma$.}
\label{fig:five}
\end{figure}
But since we have neglected terms of the order of $q_\mu$ in the
photoproduction amplitude, which correspond to corrections of order $(k/M)^2$,
it is not legitimate to keep the terms with anomalous magnetic moments. Thus,
within our approximation one can take $T_{\mu_2}\!\!=\!0$, and keep only the
charged part in $T_{\mu_1}\,$.

Using the standard formula for the imaginary part
\begin{equation}\label{eq:16}
\begin{split}
   {\rm Im}^{\,}M\bigl( \Xim\!\!\to\!\Sim\gamma \bigr)
=  \coeff12\!\int\!
&  \coeff{d^{\,4}k}{(2\pi)^2}\;
   \delta \bigl( k^2\!-\mu^2 \bigr)\,
   \delta \bigr( (p-k)^2\!-\subsup{M}{\!\Lambda}{2} \bigr)
\\[1mm]
   \times
&\,\bar\Sigma{\,} \varepsilon_\mu T_\mu
   \bigl( \hat p - \hat k + M\subs{\!\Lambda} \bigr)
   M \bigl( \Xim\!\!\to\Lambda\pi^\minus \bigr) \Xi\,,
\end{split}
\end{equation}
where $p$ is the momentum of the $\Xim$ hyperon, we get
\begin{align}
\label{eq:17}
\begin{split}
   {\rm Im}^{\,}a
&= \coeff{\Delta fA}{8\pi}\,
   \Bigl( \coeff{M\subs{\Xi}+M\subs{\Sigma}}{2M\subs{\Xi}}\Bigr)
   \coeff{|{\bf k}|}{M\subs{\Xi}}
   \Bigl[ -\coeff{M\subs{\Xi}+M\subs{\!\Lambda}-2^{}\varepsilon}{q_0}
          +\coeff{|{\bf k}|(M\subs{\!\Lambda}^{\!}+M\subs{\Sigma})}{2q_0^2}\,
           \gamma^{}
   \Bigr]^{\!}
\\[1mm]
&= -5.7\cdot10^{\minus 2}\,,
\end{split}
\\[2mm]
\label{eq:18}
    {\rm Im}^{\,}b
&= -\coeff{\Delta fB}{8\pi}\,
    \coeff{|{\bf k}|}{M\subs{\Xi}}\,
    \coeff{|{\bf k}|}{2q_0}\,\gamma
  = 6.2\cdot10^{\minus 3}\,,
\end{align}
in which $\varepsilon\!=\!\sqrt{{\bf k}^2+\mu^2}$ is the energy of the pion
in the c.m.s. and
$$
\gamma = \Bigl( 1 - \coeff{\mu^2}{\varepsilon k}\,
                    \ln\coeff{\varepsilon+|{\bf k}|}{\mu}
         \Bigr)\,
         \coeff{\varepsilon}{|{\bf k}|}
\approx  0.53\,.
$$
To obtain these numbers we substituted
$$
\subsup{g}{\!A}{\Sigma\Lambda}\!=0.6
\quad{\rm and}\quad
f\!=
\subsup{g}{\!A}{\Sigma\Lambda\!}\!/^{\!}f_{\!\pi}^{\!}\simeq 4.6\gev^{\minus 1}\,,
$$
which follows directly from the data on the decay
$\Sim\!\!\to\!\Lambda e^\minus \nu$ \cite{eight}. A near value
$g\subs{\!A}\!=\!0.65$ is obtained by using $SU(3)$ symmetry and 
the well known value of the nucleon axial coupling constant.

We emphasize that the amplitude Im$a$ of zeroth order in the momentum
$(|{\bf k}|/q_0 \sim 1)$ dominates. The fact that the amplitude does not
vanish at threshold leads also to terms logarithmic in $\mu$ in the real
part. We will consider these terms in the next section. The amplitude Im$b$
is parametrically smaller since it tends to zero for $|{\bf k}|\!\to\!0\,$.

Using the standard formula for the decay width,
\begin{equation}\label{eq:19}
\begin{split}
   \Gamma\bigl( \Xim\!\!\to\Sim\gamma \bigr)
&= 4\alpha\subsup{G}{\!F}{2\,}\Delta^{\!2}
   \bigl(^{} |a|^2\!+|b|^2 \bigr)\!
   \left(\!  \coeff{\subsup{M}{\Xi}{2}\,-\,\subsup{M}{\Sigma}{2}}
                   {2M\subs{\Xi}}
   \right)^{\!\!3}
\\
&= 1.84\cdot10^8 \bigl(^{} |a|^2\!+|b|^{2\,} \bigr){\rm sec}^{\minus 1}\,,
\end{split}
\end{equation}
we get automatically that
\begin{equation}\label{eq:20}
   {\rm BR}\bigl( \Xim\!\!\to\Sim\gamma \bigr)
=  3.04\cdot10^{\minus 2} \bigl(^{} |a|^2\!+|b|^2 \bigr)\,.
\end{equation}
Since we know the imaginary parts of the amplitudes $a$ and $b$
[see \eqref{eq:17} and~\eqref{eq:18}\,], we get the exact lower bound 
\begin{equation}\label{eq:21}
{\rm BR}\bigl( \Xim\!\!\to\Sim\gamma \bigr)\geq1.0\cdot10^{\minus 4}\,.
\end{equation}

This unitary bound is more than an order of magnitude higher than the estimate
of the relative decay probability due to the transition $s\!\to\!d\gamma$.
This demonstrates once more the negligibly small contribution of this
mechanism to the amplitude of weak radiative decay.

\section {Calculation of the Real Part
          of the $^{}\Xi^- \!\to \Sigma^- \gamma$ Amplitude}

In this section we will show that the amplitude of weak radiative decay has a 
logarithmic singularity $\ln\!\mu^2$, and we will determine the coefficient in 
front of the logarithm by means of a low-energy theorem.

The fact that the imaginary part is constant at threshold (see Sec.\,2)
suggests a logarithmic singularity in the real part of the amplitude. Indeed, 
let us write down a dispersion representation for the amplitude
$M(\Xim\!\!\to\!\Sim\gamma)$ in the square $p^2$ of the momentum of the
$^{}\Xim$ hyperon,
\begin{equation}\label{eq:22}
   M \bigl( \Xim\!\!\to\Sim\gamma \bigr)
=  \frac{1}{2\pi}\!
   \int\limits_{\subsup{p}{\rm thr}{2}}^\infty\!
   \frac{{\rm Im}^{\,}M^{\!}(p^2)^{\,}dp^2}
        {p^{2\!}-\subsup{M}{\Xi}{2}-i\varepsilon}\;\;.
\end{equation}
Since the imaginary part contains terms which are constant at threshold 
(Im$M\!\!\to${\sl const} for $p^2\!\!\to\!p^2_{\rm thr}$), the dispersion
integral \eqref{eq:22} has an infrared logarithmic divergence in the chiral
limit ($\mu\!=\!0,\,p^2_{\rm thr}\!=\!\subsup{M}{\Xi}{2}\!=\!M^2$). In the
real world this divergence will be cut off by the pion mass or by the mass
difference $(\Mxi-M\subs{\!\Lambda})\,$. Thus, if in the chiral limit the
amplitude of weak radiative decays contains $\ln(M^2\!/0)$ in the real
part, then in the the real world it has the form
\begin{equation}
{\rm Re}^{\,}M =
\alpha^{} \bigl( ^{}\ln(M^2\!/\delta^2) + \beta \bigr)\,,
\end{equation}
as one can easily convince oneself. Here,
$\delta\!\!=\!\!{\rm max}\{\mu,\Delta M\}$, $\Delta M$ is the mass difference
of the initial baryon and the baryon in the intermediate state,
and~$\alpha$ and $\beta$ are constants which are nonsingular in the chiral
limit. [The coefficient $\alpha$ is determined uniquely, and $\beta$ is
model-dependent. It is important that in the decays \eqref{eq:1} the
coefficient $|\beta|\lesssim\pi^{}$. In the decay
$\Sip\!\!\to p\gamma$ the value of $\beta$ is about 
$\pi^2\!\gg\ln(M^2\!/\delta^2)$, see below.]

In the case of the decay $\Xim\!\!\to\!\Sim\gamma$
\begin{equation}\label{eq:d}
\nonumber
\Delta M=M\subs{\Xi}-M\subs{\!\Lambda}\simeq206\mev\,.
\end{equation}

This number is of the same order of magnitude as the pion mass. Therefore, 
after calculating the coefficient $\alpha$ we will assume
that it is multiplied by $\ln(M^2\!/\mu^2)$. The difference between
$\ln(M^2\!/\mu^2)$ and $\ln\bigl(M^2\!/(\Delta M)^2\bigr)$ comes down to
a definite constant term $\beta$ 
which is anyway not fixed theoretically. We note that the logarithmic term 
occurs only in the $P$-odd amplitude $a$ of weak radiative decay. Similar 
$P$-odd logarithmic terms have been discussed recently in connection with the 
dipole moment of the neutron \cite{nine}. 

In calculating the amplitude by means of the dispersion integral
\eqref{eq:22} one has to take account of the physical as well as the
nonphysical imaginary parts which arise from all cuts of the diagrams of
Fig.\,\ref{fig:five} with any two-particle intermediate states.
To avoid unnecessary work it is more convenient to 
determine the coefficient of the logarithmic term by calculating directly the 
uncut Feynman diagrams of Fig.\,\ref{fig:five} in the infrared region. All
one has to do is to extract from these diagrams the part which (in the
chiral limit) is proportional to $\int\!d^{\,4\!}k/^{\!}k^4$, where $k$
is the momentum of the virtual pion. Of course, one has to take into account
all possible two-particle states which contain a pion. In the case of
$\Xim\!\!\to\!\Sim\gamma$, there are three such
states:~\mbox{$\Lambda\pi^\minus$, $\Sigma^0\pi^\minus$}
and~$^{}\Xi^{\,0}\pi^\minus$, while in the latter case 
the weak transition occurs at the end.

It is easy to convince oneself that the logarithmic term occurs only in the 
diagram with photon emission from the pion. This can be seen from the fact 
that the diagrams with contact emission of the photon and with emission from 
the hyperon converge into the infrared region as $\int\!d^{\,4\!}k/^{\!}k^3$.
Thus, the logarithmic part of the amplitude of the decay
$\Xim\!\!\to\!\Sim\gamma$ is completely determined by the diagrams of
Fig.\,\ref{fig:six}, while the $P$-odd amplitude $B$ of the weak transition
(corresponding to the $P$-even amplitude $b$ of weak radiative decay) does not
lead to a logarithmic term.
\begin{figure}
\centerline{\epsfxsize=5in\epsfbox{fig6.eps}}
\caption{The diagrams which lead to a logarithmic singularity in the
         real part of the amplitude of the weak radiative decay
         $\Xi^- \!\!\to\Sigma^- \gamma$.}
\label{fig:six}
\end{figure}
The $P$-even amplitudes $A$ of the weak transitions
$\Xim\!\to\!\Sigma^0\pi^\minus$ and $\Xi^{\,0}\!\!\to\!\Sim\pi^\plus$
(corresponding to the $P$-odd amplitude $a$ of weak radiative decay) and
the axial constants
$\subsup{g}{A}{\Sim\Sigma^0}$ and~$\subsup{g}{A}{\Xi^{\,0}\Xim}$)
can by means of $SU(3)$ symmetry be connected with
$\subsup{g}{A}{\Sim\!\Lambda}\!=\!0.6$ and known $P$-even
amplitudes $A$ of weak nonleptonic decays \cite{ten}, which
leads to the following values of these quantities:
\begin{align}\label{eq:24}
\nonumber
    \subsup{g}{\!A}{\Sim\Sigma^0}\!\!
&=  \subsup{g}{\!A}{\Sigma\Lambda},
&
    \subsup{g}{\!A}{\Xi^{\,0}\Xim}\!\!
&=  0.54^{}\subsup{g}{\!A}{\Sigma\Lambda},
\\[-2mm]
\\[-2mm]
\nonumber
    A \bigl( \Xim\!\!\to\Sigma^0\pi^\minus \bigr)\!
&=  0.51\,,
&
    A \bigl( \Xi^{\,0}\!\!\to\Sim\pi^\plus \bigr)\!
&= -0.06\,.
\end{align}

Since the contribution of each diagram to the real part of the amplitude is
proportional to the product $g^{\!}\cdot\!A\,$, it is easy to see that the
dominant state is $\Lambda\pi^\minus$, which, as the reader will
remember, is also completely responsible for the imaginary part of the
amplitude. From \eqref{eq:24} it follows that the total contribution is
1.3 times the $\Lambda\pi^\minus$ contribution.
   
We note that
$
A \bigl( \Xi^{\,0}\!\!\to\!\Sim\pi^\plus \bigr) \!=\! 0
$
if the
$\Delta T \!=\! 1/2$ rule works. Therefore the graph of Fig.\,\ref{fig:six}c
is very small. The calculation of the diagrams of Fig.\,\ref{fig:six} is not
difficult and the expression for the real part of the amplitude has the form
\begin{align}\label{eq:25}
         {\rm Re}^{\,}M \bigl(\Xim\!\!\to\Sim\gamma \bigr)
&=  1.3\,{\rm Re}^{\,}M \bigl(\Xim\!\!\to\Sim\gamma \bigr)
    \subs{\Lambda\pi^\minus }
\\[1mm]
\nonumber
&= -1.3\,\coeff{\,\Gf\Delta^{\!2}f\sqrt{4\pi\alpha}}{8\pi^2}\,
    \ln^{\!}\coeff{M}{\mu}\,A
    \bigl( \Xim\!\!\to\!\Lambda\pi^\minus \bigr)
    \Sigma\sigma_{\!\mu\nu}\gamma_{^{}5}q_{\nu}\Xi\varepsilon_{\mu}\,.
\end{align}
Thus
\vspace{-2mm}
$$
   {\rm Re}^{\,}a
= -1.3\,\coeff{\Delta f}{8\pi^2}\,
   \ln^{\!}\coeff{M}{\mu}\,
   A\bigl( \Xim\!\!\to\!\Lambda\pi^\minus \bigr)
=  0.85^{\,}{\rm Im}^{}a\,,
$$
where we have assumed that the upper cutoff of the logarithmic is equal
to~$\Mxi\,$. Then, according to \eqref{eq:20}, the contribution of
\eqref{eq:25} to BR$\bigl(\Xim\!\!\to\!\Sim\gamma\bigr)$ amounts to
\vspace{-2mm}
\begin{equation}\label{eq:26}
{\rm BR} \bigl( \Xim\!\!\to\!\Sim\gamma \bigr)_{\rm log\;part}
\simeq
0.7\cdot10^{\minus 4}\,.
\end{equation}
If we assume that the contribution of the nonlogarithmic terms is small (we 
will give arguments for this in the next section), then we expect for the 
relative probability of the decay $\Xim\!\!\to\!\Sim\gamma$, according to
\eqref{eq:21} and \eqref{eq:26},
\begin{equation}\label{eq:27}
{\rm BR} \bigl( \Xim\!\!\to\!\Sim\gamma \bigr)
\simeq
1.7\cdot10^{\minus 4}\,,
\end{equation}
which makes this decay easily accessible for experimental
observation in present-day hyperon beams.

\section{The Decay $\Omega^-\!\!\to^{\!}\Xi^-\gamma$}

From a fundamental point of view the theoretical analysis of this process
does not differ from the analysis in the case $\Xim\!\!\to\!\Sim\gamma$.
All assertions on the imaginary part formulated above for $\Xi^\minus$ are
automatically applicable to the decay of $\Omm$. As far as the real
part is concerned, there are no logarithmic terms, which is due to the fact
that $\Xim$ and $\Omm$ belong to different multiplets
(see below). Moreover, there are very important differences in the
technical details, which finally lead to a much lower prediction for
BR($\Omm\!\!\to^{\!}\Xim\gamma$). First of all, the spin of $\Omm$ is equal
to {\small 3/2} and, consequently, for a description one needs in general four
invariant amplitudes instead of two:
\begin{equation}\label{eq:28}
\begin{split}
   M\bigl( \Omm\!\!\to^{\!}\Xim\gamma \bigr)
=
   \sqrt{4\pi\alpha^{}}^{}\Gf
   \big(
&        \alpha_1\bar\Xi^{}\sigma_{\!\alpha\beta}
         \gamma_{^{}5}^{}\Omega_{\mu}^{}q_{\nu}F_{\alpha\beta}
        +\tilde\Delta^{}\alpha_2^{}\bar\Xi^{}\gamma_{\alpha\!}
         \gamma_{^{}5}^{}\Omega_{\mu}F_{\alpha\mu}
\\[2mm]
&\!\!   +\beta_1\bar\Xi^{}\sigma_{\alpha\beta}^{}
         \Omega_{\mu}^{}q_{\mu}F_{\alpha\beta}
        +\tilde\Delta^{}\beta_2\bar\Xi
         \gamma_{\alpha}\Omega_{\mu}F_{\alpha\beta}
   \big)\,.
\end{split}
\end{equation}
Here, $\tilde \Delta\!=\!314\mev$ is a numerical constant introduced for
convenience (it coincides with the energy of the photon in the c.m.s.);
$\alpha_1$, $\alpha_2$, $\beta_1$, and $\beta_2$ are dimensionless constants. 
The first two ($\alpha_1$ and $\alpha_2$) correspond to $P$-waves, and 
$\beta_1$ and $\beta_2$ to $S$- and $D$-waves. We note that in the literature 
it is usually tacitly implied that $\alpha_2$ and $\beta_2$ are equal to zero.
In fact, as we see below, the unitary bound is essentially determined by the 
amplitude $\alpha_2\,$.

The imaginary part is generated by the following intermediate states: 
$$
\Xi^{\,0}\pi^\minus,\quad
\Xim\pi^0,\quad
\Lambda K^\minus,\quad
\Xi^*(1530)\pi\,.
$$
One readily convinces oneself that $\Xi^{\,0}\pi^\minus$ gives the dominant 
contribution. Indeed, let us recall what happens in the decay 
$\Xim\!\!\to\!\Sim\gamma$. The left-hand block in the diagram of
Fig.\,\ref{fig:three} contains an $S$-wave which is converted into the
$P$-wave amplitude of the decay $\Xim\!\!\to\!\Sim\gamma$. The smallness
of the phase space ($\sim k/\Mxi$), corresponding to the $\Lambda\pi^\minus$
intermediate state is characteristic for $P$-wave amplitudes
compared to $S$-waves. There is no additional suppression, and this is due to 
the pole character of the diagrams of Fig.\,\ref{fig:four}. Formally, a
similar situation occurs in $\Omm$ decay. The weak block contains now a
$P$-wave and a $D$-wave, and it is clear that the main contribution to the
imaginary part comes from the $P$-wave, which in this case is converted
into the $P$-wave amplitudes $\alpha_1$ and $\alpha_2$ of the radiative
decay. It is clear that for the pole enhancement to occur (see above) the
pion ($K$ meson) has to be ultrarelativistic, since soft particles do not
easily emit photons. In the decay $\Omm\!\!\to\!\Lambda K^\minus$ the
$K^\minus$ meson is soft, $v_k\!\approx^{\!}0.39$, but the pion velocity
in~$\Omm\!\!\to^{\!}\Xi^{\,0}\pi^\minus$ is close to 1\,,
$v_\pi\!\approx^{\!}0.9$. One can easily see that the $\Lambda K^\minus$
contribution is, roughly speaking, suppressed by a factor
$\subsup{v}{\!K}{2\!}/v_\pi^2$, i.e.\ by one order of magnitude.

For the decay $\Omega\!\to^{\!}\Xi^\ast(1530)\pi$, although it is also 
$S$-wave, the pion velocity is very small, $\sim 10^{\minus2}$, so that the
corresponding contribution to the imaginary part is negligible.
In the intermediate state $\Xim_{\!}\pi^0$ the photon can only be emitted by
the~$\Xim$ hyperon, whose velocity is also small.

We see that the main contribution is due only to $\Xi^{\,0}\pi^\minus$.
The decay $\Omm\!\!\to^{\!}\Xi^{\,0}\pi^\minus$ is parametrized as
\begin{equation}\label{eq:29}
   M\bigl( \Omm\!\to\Xi^{\,0}\pi^\minus \bigr)
=  \Gf \tilde\Delta\, \bar\Xi
   \bigl(  A + B\gamma_{\,5} \bigr)
   \Omega_\mu k_\mu\,.
\end{equation}
If it is described by the mechanism of Ref.\cite{eleven}, then certainly
\begin{equation}\label{eq:30}
B/A \sim m_s/M \ll 1\,.
\end{equation}
Even if we assume that $B\!\sim\!A$ (and this seems inconceivable), the $B$ 
contribution to $\Gamma\bigl(\Omm\!\!\to^{\!}\Xi^{\,0}\pi^\minus\bigr)$ is
still negligibly small, of the order of 1\% of the $A$ contribution.
Thus, the constant $A$ is uniquely determined from the experimental data:
the $\Omm$ lifetime and BR$\bigl(\Omm\!\!\to^{\!}\Xi^{\,0}\pi^\minus$).
Concretely, 
\begin{equation}\label{eq:31}
   |A|^2\!=\coeff{24\pi}{|{\bf k}|^3}\,
   \Bigl( \coeff{M\subs{\Omega}}{M\subs{\Omega}+M\subs{\Xi}}\Bigr)^{\!2}
   \coeff{\Gamma_{\!{\rm tot}}{\rm BR}^{\,}(\Omm\to^{}\Xi^{\,0}\pi^\minus)}
        {\subsup{G}{\!F}{2}\,\tilde\Delta^2}\,
=  0.14\,.
\end{equation} 
On the other hand, one can also neglect the $B$ contribution to
\mbox{Im$^{}M\bigl(\Omm\!\!\to\!\Xim\gamma)$}, since also here it is suppressed
relative to the $A$ contribution and not larger than one percent.
Performing standard calculations, we get 
\begin{equation}\label{eq:32}
\begin{split}
    {\rm Im}^{}\alpha_1
&= -\coeff{M\subs{\Xi\,} |{\bf k}|^{2\!} f \tilde\Delta^{} A}
         {4\pi ( \subsup{M}{\Omega}{2} - \subsup{M}{\Xi}{2} )
               (     M\subs{\Omega}    -    M\subs{\Xi}     )}\,
    \Bigl[^{} \coeff12^{}\Bigl(1-\coeff{2\varepsilon}{q_0}\Bigr)\gamma
             +\coeff13\,\coeff{|{\bf k}|\,(3M\subs{\Omega}+M\subs{\Xi})}
                              {\subsup{M}{\Omega}{2} - \subsup{M}{\Xi}{2}}^{}
    \Bigr]
\\[2.5mm]
&=  -1.27\cdot10^{\minus 3}\,,
\\[2.5mm]
    {\rm Im}^{}\alpha_2
&= -\coeff{M\subs{\Xi\,} k^{2\!} f A}
         {2\pi ( \subsup{M}{\Omega}{2} - \subsup{M}{\Xi}{2} )}\,
    \Bigl[^{} \coeff12\Bigl(1-\coeff{\varepsilon}{q_0}\Bigr)\gamma
             +\coeff13\,\coeff{|{\bf k}|}{M\subs{\Omega} - M\subs{\Xi}}^{}
    \Bigr]
=  -^{}4.2\cdot10^{\minus 3}\,,
\\[4.5mm]
&\qquad\qquad\qquad\qquad
    \gamma
   =\frac{\varepsilon}{k}\,
    \Bigl( 1 - \coeff{^{}\mu^2}{k^{}\varepsilon}
               \ln^{\!}\coeff{\varepsilon+k}{\mu}
    \Bigr)\,,
\end{split}
\end{equation}
where the constant $\subsup{g}{\!A}{\Xim\Xi^{\,0}}\!\!=\!f_\pi^{\!}\cdot\!f$
is obtained by means of $SU(3)$ symmetry and is equal to
$\subsup{g}{\!A}{\Xim\Xi^{\,0}}\!\!=0.35$. In turn, these estimates give
\begin{equation}\label{eq:33}
       {\rm BR}_{\genfrac{}{}{0pt}{}{\rm unitary}{\rm limit}}
       \bigl(\Omm\!\!\to\Xim\gamma\bigr)
\geq\! 0.8\cdot10^{\minus 5}\,.
\end{equation} 
Taking into account the real part of the amplitude we can expect a
relative probability at the level $(1\!\div\!1.5)\cdot 10^{\minus 5}$.
Thus, the prediction is an order of magnitude less than in the case 
$\Xim\!\!\to\!\Sim\gamma$. (We note that the $\Lambda K^\minus$ intermediate 
state gives the unitary bound $\approx 1 \cdot 10^{\minus 6}$.)

\section {The Decay $\Sigma^+\!\to p\gamma$} 

The decay $\Sip\!\!\to p \gamma$ is the only weak radiative decay which 
is observed experimentally (see Ref.\cite{six}). We will estimate the
contribution of the imaginary part and of the logarithmic terms in the real
part to the width of this decay. The imaginary part is determined by the
intermediate states $n\pi^\plus$ and $p\pi^0$ corresponding to the diagrams
of Fig.\,\ref{fig:seven}\,. The logarithmic terms in the real part of the
amplitude arise because of the diagrams of Fig.\,\ref{fig:eight}\,.
\begin{figure}
\centerline{\epsfxsize=4.5in\epsfbox{fig7.eps}}
\caption{The imaginary part of the amplitude of the decay
         $\Sigma^+\!\to\!\!p\gamma$ resulting from the two intermediate
         states $n\pi^+$ and $p\pi^0\,$.}
\label{fig:seven}
\end{figure}
We will first examine the imaginary part. Since
\begin{figure}
\centerline{\epsfxsize=5in\epsfbox{fig8.eps}}
\caption{The diagrams which lead to a logarithmic singularity in the real part
         of the amplitude in the decay $\Sigma^+\!\to\!p\gamma\,$.}
\label{fig:eight}
\end{figure}
the amplitudes with photon emission by the hyperons are suppressed relative
to the amplitudes with photon emission by the pion or with contact emission
(see Fig.\,\ref{fig:two}), the imaginary part will be determined mainly by
the intermediate state $n\pi^\plus$, since in the case~$p\pi^0$ the photon
is only emitted by the proton. The corresponding diagrams are completely
analogous to the diagrams of Fig.\,\ref{fig:five} for the decay
$\Xim\!\!\to\!\Sim\gamma$ which we have considered earlier. Using formulas
\eqref{eq:17} and \eqref{eq:18} with the obvious substitutions of masses and
pion-baryon coupling constants, it is easy to show that the corresponding
unitary bound of the relative decay probability is of the order of
$3\cdot10^{\minus 5}$.
In this case, because of the fact that the $P$-even amplitude
$A\bigl(\Sip\!\!\to\!n\pi^\plus\bigr)\!=0.06$ and the $P$-odd amplitude
$B\bigl(\Sip\!\!\to\!n\pi^\plus\bigr)\!=19.07$, the~unitary bound is
completely determined by the amplitude $B$. To estimate the real part of the
amplitude, whose logarithmic singularity is only determined by the amplitude
$A$ of the weak transition, we use
$SU(3)$ symmetry to determine the $P$-even amplitudes $A$ and the 
corresponding pion-baryon coupling constants \cite{six}.
In this case, the contributions of the intermediate states 
$\Sigma^0 \pi^\plus$, $n\pi^\plus$, and $\Lambda \pi^\plus$ strongly cancel
each other in the sum (apparently accidentally), which has, for instance,
been noted by Skovpen' \cite{one}. As a result we get for the real part
of the amplitude the following estimate
\begin{equation}\label{eq:34}
         {\rm Re}^{}M_{\rm log}\bigl(\Sip\!\!\to p\gamma\bigr)
\simeq   0.14\,{\rm Re}^{}M_{\rm log}\bigl(\Xim\!\!\to\!\Sim\gamma\bigr)\,.
\end{equation} 
Knowing that the contribution of
Re$M_{\rm log}\bigl(\Xim\!\!\to\!\Sim\gamma\bigr)$ to the relative
probability of the decay $\Xim\!\!\to\!\Sim\gamma$ is $\sim 10^{\minus 4}$,
and the ratio of the total widths is
$\Gamma(\Sip\!\!\to p\gamma)/\Gamma(\Xim\!\!\to\!\Sim\gamma)\!=\!2$, it is
easy to estimate the contribution of Re$^{}M_{\rm log}(\Sip\!\!\to p\gamma)$
to the relative probability of the decay $\Sip\!\!\to p\gamma$. It amounts
to $\sim 10^{\minus 6}$, which is negligibly small, even compared to the
unitary bound, whose contribution is equal to $3\cdot10^{\minus 5}$, not to
mention the experimental value $\sim\!1.2\cdot10^{\minus 3}$. What is going on?
The point is that there are tree pole diagrams, depicted in
Fig.\,\ref{fig:two}.
In contrast to the pole diagrams we have so far dealt with, they are not
suppressed by loop factors line $1/\pi^2$, i.e.\ roughly speaking, their
contribution should be a factor $\pi^2$ larger. The pole model has been
discussed repeatedly in the literature \cite{one}, but as we have already
noted it can only claim an order of magnitude accuracy. However, another
aspect which is more important for us here is the absence of pole
contributions in the decays \eqref{eq:1}\,. Why is the mechanism 
of Fig.\,\ref{fig:two} not important in these decays?

The point is that the weak block of the hyperon-hyperon transition is
determined by the matrix element $\langle B'|\Hw|B \rangle$,
where
$$
\Hw=\sqrt2\Gf\sin\theta\cos\theta\sum_{i=1}^6c_iO_i
$$
is the effective Hamiltonian of the weak interaction with $\Delta S\!=\!1$, 
the $O_i$ are the standard local four-quark operators introduced in
Ref.\cite{eleven}, and the $c_i$ are numerical coefficients. It is well
known \cite{eleven} that the operator
$$O_1 \!=\!
\bar s\subs{L} \gamma_\mu d\subs{L} \bar u\subs{L} \gamma_\mu u\subs{L}
\!-\!
\bar s\subs{L} \gamma_\mu u\subs{L} \bar u\subs{L} \gamma_\mu d\subs{L}
$$
appears with the largest coefficient $c_1\!=\!-2.5\,$, while the
coefficients of the other operators $O_2\,,\,\ldots,O_6$ are an order of
magnitude smaller. Since neither~$\Xim$ nor $\Sim$ contains valence $u$ quarks,
$\Xim$ cannot transform into $\Sim$ by means of the operator $O_1\,$,
but $^{}\Xim$ can transform into $\Sigma^0$ or $\Lambda^0$ with emission of a
$\pi^\minus$ meson. Thus, the leading operator in $\Hw$ in the case of the
decays \eqref{eq:1} works only in a loop diagram, while in the decay
$\Sip\!\!\to p\gamma$ it works already in a tree pole diagram. In other
words, in the case of the decay $\Xim\!\!\to\Sim\gamma$ large tree diagrams
of Fig.\,\ref{fig:two} can only be realized either by means of the quark sea
in the $\Xim$ hyperon, or through nonleading operators, for instance $O_5$ in
$\Hw\,$. In both cases one expects a suppression of one order of magnitude
in the amplitude, i.e.\ the corresponding ratio is
$\lesssim10^{\minus 5}$ \cite{four}. Thus, the mechanism of
Fig.\,\ref{fig:three}
dominates. On the contrary, in decays like $\Sip\!\!\to p\gamma$
the dominant mechanism is that of Fig.\,\ref{fig:two}, which induces a real
amplitude. This amplitude is, roughly speaking, a factor
\begin{equation}\label{eq:35}
\Bigl(  \coeff{\Delta\varkappa_p}{M\subs{\Sigma}-M_p}\,
        \coeff{f_\pi}{\sqrt2M_p}                         \Bigr)
\Big/
\Bigl(  \coeff{\Delta f}{4\pi^2}\ln^{\!}\coeff{M}{\mu}   \Bigr)
\approx 4
\end{equation}
larger than the estimate \eqref{eq:25} for
Re$^{}M\bigl(\Xim\!\!\to\!\Sim\gamma\bigr)$. We see that everything is
self-consistent since, combining \eqref{eq:26} and \eqref{eq:35}, we get
BR$\bigl(\Sip\!\!\to p\gamma\bigr)\sim10^{\minus 3}$.

\section{Concluding Remarks}

The dominance of the mechanism considered here in the decay
$\Xim\!\!\to\!\Sim\gamma$ and $\Omm\!\!\to^{\!}\Xim\gamma$ is determined by
the quark structure of the operator $O_1$ in the effective weak Hamiltonian
$\Hw\,$. It is clear that this mechanism dominates in the amplitudes of weak
radiative decays only of negatively charged hyperons, since they consist of
down $s$ and $d$ quarks, and in the pole approximation the operator $O_1$
cannot generate a direct transition of the hyperons. In this case, large
differences are possible between the relative decay widths in the various
decays. Thus, we have found that
$$
{\rm BR}\bigl(\Omm\!\!\to^{\!}\Xim\gamma\bigr)
/
{\rm BR}\bigl(\Xim\!\!\to\!\Sim\gamma\bigr)
\sim 10^{\minus 1}\,.
$$
Experimental investigation of the decays $\Xim\!\!\to\!\Sim\gamma$ and
$\Omm\!\!\to^{\!}\Xim\gamma$ would be of great interest, since a sharp
excess of BR$\bigl(\Xim\!\!\to\!\Sim\gamma\bigr)$ compared to
\mbox{BR${\raisebox{-2pt}{${\genfrac{}{}{0pt}{3}
{\rm unitary}{{\rm limit}\;\;\;\;}}$}}\bigl(\Xim\!\!\to^{\!}\Sim\gamma\bigr)$}
would mean that there is some mysterious enhancement in the real part,
either due to non-dominant operators of the weak Hamiltonian, or because
of the $t$ quark contribution to the $s\!\to d\gamma$ transition. Our
unitary bound \eqref{eq:21} and the estimate \eqref{eq:27} of the relative
probability of the decay $\Xim\!\!\to\!\Sim\gamma$ indicate that this decay
can be observed in present-day intense hyperon beams, for instance at
the Serpukhov accelerator.

In conclusion we would like to thank L.\,G.\,Landsberg and
V.\,T.\,Smo\-lyankin,
who drew our attention to this problem, for fruitful discussions, and
M.\,B.\,Voloshin, L.\,B.\,Okun', and I.\,B.\,Khriplovich for critical
comments and discussions. One of us (Ya.\,K.) is grateful to
K.\,A.\,Ter-Martirosyan who pointed out the importance of the problem.

\section*{Appendix. Estimate of the $t$ quark Contribution\\
          $^{}\hspace{2.2cm}$to the $s\!\to\!d\gamma$ Amplitude}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}

The coefficient of the operator
$T\!=\!i\bar s\subs{\!R}\sigma_{\!\mu\nu}d\subs{L}F_{\!\mu\nu}$ has been
calculated in Ref.\cite{two} in the leading log approximation in the
four-quark model. In first order in~$\alpha_s$ it has the following form:
\begin{equation}\label{eq:A1}
t=-\coeff{\alpha_{^{\!}S}e^{}\Gf\sqrt2}{12\pi^3}\,
   \sin^{\!}\theta\subs{^{\!}C} \cos^{\!}\theta\subs{^{\!}C}^{\,}m_s
   \ln^{\!}\coeff{\subsup{m}{c}{2}}{m^2}\;,
\end{equation}
where $m_s\!\simeq\!150\mev$ is the mass of the current $s$ quark,
$m\!\simeq\!300\mev^{}$, and $m_c\sim1.5\gev$ is the mass of the $c$ quark.
It is easy to see that the contribution of one of the up quarks $(u,c,t)$
to the diagram of the Fig.\,\ref{fig:one}b is
\begin{equation}\label{eq:A2}
x_{u^{},^{}c^{},^{}t}\,\coeff{\alpha_{^{\!}S}e^{}\Gf\sqrt2}{12\pi^3}\,
m_s\ln^{\!}\coeff{\;\subsup{m}{u,c,t}{2}}{\subsup{M}{W}{2}}\;,
\end {equation}
where the coefficient $x_{u^{},^{}c,^{}t}$ is determined by the mixing of a
given quark with~$s$ and $d$ quarks. In the four-quark model
$$
x_u = \sin{\theta\subs{\!C}} \cos{\theta\subs{\!C}}\,,
\qquad
x_c =-\sin{\theta\subs{\!C}} \cos{\theta\subs{\!C}}\,,
$$
which leads to the expression \eqref{eq:A1}. In the six-quark model
using the explicit form of the Kobayashi-Maskawa matrix elements \cite{ten},
it is easy to show that
\begin{equation}\label{eq:A3}
\begin{split}
x_u &=  c_1s_1c_3\,,                            \\
x_c &= -s_1c_2(c_1c_2c_3-e^{i\delta}s_2s_3)\,,  \\
x_t &= -s_1s_2(c_1s_2c_3+e^{i\delta}c_2s_3)\,.
\end{split}
\end{equation}
Using \eqref{eq:A2} and \eqref{eq:A3} we get the following expression for
the coefficient $t\,$:
\begin{equation}\label{eq:A4}
t = -\coeff{\alpha_{^{\!}S}e^{}\Gf\sqrt2}{12\pi^3}\,m_sc_1s_1
     \Bigl[  c_3\ln^{\!}\coeff{\subsup{m}{c}{2}}{m^2}
            +\coeff{s_2}{c_1}
             \ln^{\!}\coeff{\subsup{m}{t}{2}}{\subsup{m}{c}{2}}
             \bigl(c_1s_2c_3+s_3c_2e^{i\delta}\bigr)
     \Bigr]\,,
\end{equation}
where we have replaced the mass of the $u$ quark by the dynamical
light-quark mass $m\!\simeq\!300\mev$ \cite{two}. Assuming $s_2,s_3\!\le\!0.5$
we can estimate the relative~\mbox{$t$ quark} contribution which is
$<\!0.31\ln(m_t/m_c)\,$. Even for $m_t\!\simeq\!100\gev$, which is very
unlikely, this does not exceed 1.2\,. Thus, for reasonable values of the 
$t$ quark mass, its contribution to the amplitude of the local
$s\!\to\!d\gamma$ transition does not change the amplitude by an order
of magnitude and thus cannot ensure the dominance of this transition
in weak radiative decays. 

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\end{document}