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%Date: Mon, 19 Jul 1993 18:16:22 -0400 (EDT)

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\begin{document}
\begin{flushright}
FSU--HEP--930719\\
July 1993
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\vglue 0.5cm
\begin{center}
{\large\bf PROBING THE $WW\gamma$ VERTEX IN RADIATIVE $b$-QUARK \\
DECAYS}
\footnote{To appear in the Proceedings of the {\it ``Workshop on $B$
Physics at Hadron Accelerators''}, Snowmass, CO, June~21 --~July~2, 1993.}
\vglue 0.6cm
{U.~BAUR\\
        \em Physics Department, Florida State University, \\
        \em Tallahassee, FL 32306, USA}
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\centerline{\bf Abstract}
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{\rightskip=3pc
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 \noindent
The recent CLEO results on on radiative $b$-quark decays are used to derive
constraints on anomalous $WW\gamma$ couplings. These constraints are
compared with expectations from $p\bar p\rightarrow e^\pm
p\llap/_T\gamma+X$ at the Tevatron. The usefulness of exclusive
radiative $B$ meson decay channels in probing the $WW\gamma$ vertex is
largely limited by present theoretical uncertainties in the calculation
of hadronic matrix elements.
\vglue 0.6cm}

One of the major goals of future experiments at the Tevatron is to probe
the structure of the $WW\gamma$ vertex in $W\gamma$ and $W^+W^-$
production. Such direct tests of three vector boson vertices through
tree level processes have to be contrasted with indirect tests which
involve one-loop processes. Whereas bounds derived from tree level
processes are essentially model independent, limits on anomalous
$WW\gamma$ couplings extracted from processes which are sensitive to
three vector boson couplings only at the one-loop level usually do
depend on specific assumptions \cite{LOOP}. The dependence on model
specific assumptions is most pronounced in quantities where anomalous
couplings lead to divergencies, {\it e.g.} the $S$, $T$ and $U$ parameters.

Some one-loop processes, such as $b\rightarrow s\gamma$, yield finite
answers due to the GIM mechanism. Recently, the CLEO Collaboration
reported \cite{CLEO} the observation of the decay $B\rightarrow
K^*\gamma$ with a branching fraction of $B(B\rightarrow K^*\gamma)=
(4.5\pm 1.5\pm 0.9)\cdot 10^{-5}$. In the following we analyze the
implications of this measurement on the anomalous $WW\gamma$ couplings,
$\Delta\kappa$ and $\lambda$, and compare the result with expectations
from future experiments at the Tevatron.

Our calculations are based on the results obtained in Ref.~\cite{ANOM}
for the inclusive radiative decay $b\rightarrow s\gamma$ for arbitrary
anomalous couplings $\Delta\kappa$ and $\lambda$. Apart from
non-standard
contributions to the $WW\gamma$ vertex we assume the Standard Model to
be valid. QCD corrections are
incorporated following Ref.~\cite{ROX}. To estimate the branching
fraction of the exclusive decay mode $B\rightarrow K^*\gamma$ we use the
approach of Ref.~\cite{ALI}. In this model, $B(B\rightarrow K^*\gamma)$
is estimated by integrating the invariant mass distribution of the
hadrons recoiling against the photon from the $m_K+m_\pi$ threshold up
to ${\cal O}(1$~GeV), assuming that this range is completely saturated
by the $K^*$ resonance. The upper integration limit, however, is only
loosely defined. Together with uncertainties in the $B$ meson wave
function, this results in rather large uncertainties in the estimated
$B\rightarrow K^*\gamma$ branching ratio. For the present lower
experimental limit on the top quark mass \cite{TOP}, $m_{\rm
top}=108$~GeV, we find
$B(B\rightarrow K^*\gamma)=(2-9)\cdot 10^{-5}$. For $m_{\rm
top}=200$~GeV, we obtain $B(B\rightarrow K^*\gamma)=(3-12)\cdot 10^{-5}$.
These ranges are consistent with the branching ratios obtained in other
models \cite{OTHER}.

The resulting constraints on $\Delta\kappa$ and $\lambda$ depend
explicitly on $m_{\rm top}$, and are shown in Fig.~1.
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voffset=-15 hscale=40 vscale=40 angle=90}
Figure~1: Allowed regions in the $\Delta\kappa -\lambda$ plane for
$m_{\rm top}=108$~GeV and $m_{\rm top}=200$~GeV. The
region allowed by present $B\rightarrow K^*\gamma$ data is indicated by
the shaded bands. The short-dashed lines outline the limits
from $B\rightarrow K^*\gamma$ expected from CDF with an integrated
luminosity of 100~pb$^{-1}$. The long-dashed lines show the bounds from
the CLEO upper limit on the branching ratio for the inclusive decay
$b\rightarrow s\gamma$. The hatched area,
finally, displays the allowed region in the $\Delta\kappa -\lambda$ plane which
is expected to result from $p\bar p\rightarrow e^\pm p\llap/_T\gamma+X$
at the Tevatron with $\int\!{\cal L}dt=100$~pb$^{-1}$.
\end{figure}
%
In order to obtain $1\sigma$ limits from $B\rightarrow K^*\gamma$, we
have added the statistical and systematic errors in the branching ratio
linearly. Despite the large uncertainties in the calculation of the
$B\rightarrow K^*\gamma$ decay rate, the CLEO measurement excludes large
regions of the $\Delta\kappa - \lambda$ plane. At the $1\sigma$ level,
only two rather narrow bands remain allowed. The width of these bands
depends quite strongly on $m_{\rm top}$. The region between the two
bands is not excluded with a very high significance; it still allowed at
the $2\sigma$ level.

The CLEO collaboration recently also presented a new upper 95\% CL limit on
the branching ratio of the inclusive decay $b\rightarrow s\gamma$
\cite{APS} of $B(b\rightarrow s\gamma)<5.4\cdot 10^{-4}$, derived from
the inclusive photon energy spectrum in $B$ decays. The
$b\rightarrow s\gamma$ decay rate is much more accurately predicted
theoretically than
that of the exclusive channel $B\rightarrow K^*\gamma$. The region in
the $\Delta\kappa - \lambda$ plane which is consistent with the CLEO
limit on $b\rightarrow s\gamma$ is the one between the two long-dashed
lines in Fig.~1. The bounds obtained from inclusive radiative $b$ decays
reduce the region allowed by $B\rightarrow K^*\gamma$ somewhat. Similar
results have also been obtained in Ref.~\cite{TOM}.

The current measurement of the $B\rightarrow K^*\gamma$ branching
fraction is based on 13~signal events~\cite{CLEO}. A much larger event
sample is possible in the near future from CLEO and, with a special
photon trigger~\cite{CDF}, also from CDF. If this trigger were
implemented, up to 100 $B\rightarrow K^*\gamma$ events are expected in
the 1993-94 run. Assuming that the central value of the branching ratio
does not change, and systematic errors coincide with those of
Ref.~\cite{CLEO}, the anticipated improvement is shown in Fig.~1 by the
short-dashed lines. Since theoretical uncertainties dominate, the
resulting bounds are only slightly better than those obtained with the
present data. A substantial improvement in the calculation of
$B(B\rightarrow K^*\gamma)$, however, may result from a lattice
computation of the hadronic matrix element in the near future. The CDF
photon trigger may
also allow the observation of radiative $B_s$ decays in the channel
$B_s\rightarrow\phi\gamma$ \cite{CDF}. The number of events expected is similar
to the rate foreseen for $B\rightarrow K^*\gamma$. So far, no
theoretical calculation of the $B_s\rightarrow\phi\gamma$ branching ratio
has been performed.

To contrast the bounds on $\Delta\kappa$ and $\lambda$ from radiative
$B$ decays with those from diboson
production, we have also included the $1\sigma$ limits expected from
$p\bar p\rightarrow W^\pm\gamma+X\rightarrow e^\pm p\llap/_T\gamma+X$
with 100~pb$^{-1}$ \cite{BHO} in Fig.~1. $W\gamma$ production is
expected to yield much stronger bounds for $\lambda$ while $B\rightarrow
K^*\gamma$ tends to give better limits for $\Delta\kappa$. The two processes
thus complement each other.

In conclusion, we have shown that present CLEO data on radiative $b$ decays
yield valuable information on anomalous $WW\gamma$ couplings. Future
improvements of limits extracted from exclusive $B$ (and $B_s$) decays
depend mostly on the ability to obtain more accurate estimates of the
hadronic $B$ decay matrix elements. Combined with limits expected from $p\bar
p\rightarrow W\gamma$, $\Delta\kappa$ and $\lambda$ can be highly
constrained in the near future.

\onehead{}{ACKNOWLEDGEMENTS}

I would like to thank A.~Kronfeld, T.~LeCompte, R.~Springer and
J.~Thaler for useful
discussions. This research was supported by the U.S.~Department of
Energy under Contract No. DE-FG05-87ER40319.

\begin{thebibliography}{}{99}

\bibitem{LOOP} K.~Hagiwara {\it et al.}, MAD/PH/737 preprint
(March~1993); C.~P.~Burgess {\it et al.}, McGill-93/14 preprint (June~1993).

\bibitem{CLEO} R.~Ammar {\it et al.} (CLEO Collaboration), CLNS-93-1212
preprint (May~1993), to appear in {\it Phys. Rev. Lett.}

\bibitem{ANOM} S.~Chia, {\it Phys. Lett.} {\bf 240B} 465 (1990);
K.~Numata, {\it Z.~Phys.} {\bf C52} 691 (1991); K.~Peterson, {\it Phys.
Lett.} {\bf 282B} 207 (1992).

\bibitem{ROX} B.~Grinstein, R.~Springer and M.~B.~Wise, {\it Phys. Lett.
} {\bf 202B} 138 (1988).

\bibitem{ALI} A.~Ali and C.~Greub, {\it Phys. Lett.} {\bf 259B} 182
(1991).

\bibitem{TOP} N.~Shaw, talk given at the {\it Workshop on Physics at
Current Accelerators and the Supercollider}, Argonne Nat. Lab. June~2
--~5, 1993.

\bibitem{OTHER} see {\it e.g.} A.~Ali, DESY 92-058 preprint (April~1992)
and references therein.

\bibitem{APS} E.~Thorndike, talk given at the {\it 1993 APS Meeting},
Washington, D.C., April, 1993.

\bibitem{TOM} T.~Rizzo, ANL-HEP-PR-93-19 preprint (April~1993).

\bibitem{CDF} T.~LeCompte and J.~Mueller, private communication.

\bibitem{BHO} U.~Baur, T.~Han, and J.~Ohnemus, FSU-HEP-930519 preprint
(May~1993).

\end{thebibliography}

\end{document}

