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\begin{flushright}
OHSTPY-HEP-T-93-008\\
USM-TH-62\\
July 1993
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\begin{center}
{\bf PCAC relation for $D^*\to D$ axial form factors}\footnote{To appear
in the Proceedings of the {\it Workshop on B Physics at Hadron
Accelerators}, June 1993, Snowmass, Colorado.}
\end{center}

\begin{center}
Claudio O. Dib\\
Universidad Federico Santa Mar\'\i a, Valpara\'\i so, Chile\\
and\\
Josep Taron\\
The Ohio State University, Columbus, OH, U.S.A.
\end{center}

The spin-flavour symmetry among hadrons containing one heavy quark (b, c) and
the chiral symmetry $SU(3)_L \times SU(3)_R$ associated with the light quarks
(u, d, s), spontaneously broken down to $SU(3)_V$, can be invoked
simultaneously to provide relations among matrix elements involving such heavy
hadrons and soft Goldstone bosons ($\pi$, $K$, $\eta$).
This treatment has recently been formulated under the form of an effective
theory which incorporates the heavy quark and chiral symmetries at the same
time \cite{W,D,G,C}. The effective lagrangian consists of an infinite number of
terms, with an increasing number of derivatives, each of which is an expansion
in
inverse powers of the heavy quark masses. The coefficients of the different
terms can not be fixed by symmetry arguments alone, and will be fitted from
experiment.

To lowest order in both chiral and $1/m_Q$ expansions, the interaction
lagrangian contains
a term of the form (see \cite{W1}):
\begin{equation}
\label{E1}
 -{2g\over f_\pi} \left( \partial^\nu M_{ba} D^\dagger_a D^*_{b\;\nu} +
{\rm h.c.}\right),
\end{equation}
%
where $M$ is the $3\times 3$ Goldstone boson matrix, and $D_a$
and $D^*_{a\;\nu}$ (a=
u,d,s) stand for the $SU(3)_V$ triplet pseudoscalar and vector fields,
respectively. The
constant $g$ is nothing but the effective coupling of $D^* D \pi$, which is a
dimensionless quantity of order unity and can be fitted from
$\Gamma(D^*\to D\pi)$.

The object of this note is to clarify a discussion brought up by E. Levin
about relations
of coupling constants in the effective lagrangian.
The relation derived below is the analog of the Goldberger-Treiman relation
for nucleons
\cite{GT}, for the case of the form factors of the matrix element
\begin{equation}
\label{E2}
\langle D(p')|A^\mu|D^*(p,\epsilon)\rangle = g_1(q^2)\epsilon^\mu +
g_2(q^2)(\epsilon\cdot
q) (p+p')^\mu + g_3(q^2)(\epsilon\cdot q) q^\mu ,
\end{equation}
%
where $q = p - p'$, and  $A^\mu$ is the axial current of the chiral symmetry,
which is
conserved in the chiral limit ({\it i.e.} $m_\pi\to 0$). Thus
$$
\langle D(p')|\partial_\mu A^\mu|D^*(p,\epsilon)\rangle = (\epsilon\cdot q)\;
[~ g_1(q^2) + g_2(q^2) ( m_{D^*}^2 - m_D^2)  + g_3(q^2) q^2~]
$$
\begin{equation}
\label{E3}
 = {\cal O}(m_\pi^2) \to 0 .
\end{equation}
%
Taking $q^2$ close to zero, the form factor $g_3(q^2)$ is dominated by the
pion pole at
$q^2=0$. Then, for $q^2\to 0$ it follows that:
\begin{equation}
\label{E4}
 g_1(q^2) +
g_2(q^2) ( m_{D^*}^2 - m_D^2)  + {\rm Res}~g_3(q^2) \Big|_{q^2 = 0} = 0 .
\end{equation}
%
The matrix element (\ref{E2}) is dominated at $q^2\to 0$ by the diagram:
\vspace{1.5in}

\noindent where

$$ \langle 0|A^\mu|\pi(q)\rangle = i f_\pi q^\mu =
{\;\;\;\;\;\;\;\;\;\;\;\qquad\qquad\qquad}
$$
\vskip 0.5in
\noindent and the $D^* D \pi$ vertex is
$g_{\scriptscriptstyle D^*D\pi} ~(\epsilon\cdot q)$;
%, where
$g_{\scriptscriptstyle D^*D\pi}$ corresponds to $g$ in
the effective lagrangian (\ref{E1}).
One then finds
\begin{equation}
\label{E5}
{\rm Res}~g_3(q^2)\Big|_{q^2=0} = -~f_\pi ~~ g_{\scriptscriptstyle D^*D\pi} .
\end{equation}
%
In the infinite c-quark mass limit, the pseudoscalar meson $D$ and the
vector meson $D^*$
become degenerate in mass, and the relation (\ref{E4}) reads:
\begin{equation}
\label{E6}
g_1(0) = f_\pi ~~ g_{\scriptscriptstyle D^*D\pi} .
\end{equation}
%
It is easy to verify that the leading terms of the lagrangian in Ref.
\cite{W1} satisfy
this relation for $q^\mu \to 0$. Indeed, the hadronized $A^\mu$ current reads
\begin{equation}
\label{E7}
A^A_\mu = - 2 g \left( D^\dagger_a D^*_b ~+~ {\rm h.c.} \right)
T^A_{ba} ~+~ {\cal
O}(q_\mu),
\end{equation}
%
which verifies Eq.~(\ref{E6}).

Notice that relation (\ref{E4}) is valid for finite mass values of the
heavy mesons, and
the mass difference $m_{D^*} - m_D$ does not originate from chiral
symmetry breaking but
from hyperfine splitting, which is of order ${\cal O}(1/m_Q)$.

\bigskip
We benefited from helpful conversations with J. Amundson and E. Levin.
This work was supported in part by Fondecyt, Chile, grant No. 92-0806, and
in part by U.S.A. grant DE-FG02-91-ER40690.

\begin{thebibliography}{99}
\bibitem{W}
M.B. Wise, Phys. Rev. \underline{D45}, 2188 (1992).

\bibitem{D}
G. Burdman and J. Donoghue, Phys. Lett. \underline{B280}, 287 (1992).

\bibitem{G}
J.L. Goity, Phys. Rev. \underline{D46}, 3929 (1992).

\bibitem{C}
T.M. Yan {\it et al.}, Phys. Rev. \underline{D46}, 1148 (1992).

\bibitem{W1}
M.B. Wise, Caltech preprint CALT-68-1860, 1993, unpublished.

\bibitem{GT}
M.L. Goldberger and S.D. Treiman, Phys. Rev. \underline{110}, 1178 (1958).

\end{thebibliography}
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