\section{Introduction}
\label{sec:Introduction}

In the chiral limit where the light quark masses are set to zero, the
low--energy
effective Lagrangian of QCD in the presence of electromagnetic interactions
has, to
lowest order in $\alpha={e^2}/{4\pi}\simeq 1/137.03...$, an interaction term
without
derivatives of order
${\cal O} (p^0)$ in the chiral expansion~\cite{EGPR89}
\begin{equation}\label{eq:effem}
{\cal L}_{\mbox{\scriptsize\rm eff}}=\cdots +e^2 C\,\mbox{\rm tr}
\left(Q_{R}UQ_{L}U^{\dagger}\right)\,.
\end{equation} Here, $U$ is the matrix field which collects the octet of pseudoscalar
Goldstone fields and $Q_{R}=Q_{L}=\mbox{\rm diag}[2/3, -1/3, -1/3]$ the
right-- and
left--handed charges associated with the electromagnetic couplings of the light
quarks. Upon expanding $U$ in powers of the pseudoscalar fields there appear
quadratic terms like
\begin{equation}\label{eq:effemex}
{\cal L}_{\mbox{\scriptsize\rm eff}}=\cdots -2e^2 C\frac{1}{f_{\pi}^2} (\pi^+
\pi^- + K^+ K^-) + \cdots ,
\end{equation} showing that, in the presence of electromagnetic interactions, the
charged pion
and kaon fields become massive in agreement with the old  current algebra result
obtained by Dashen~\cite{Dashen69},
\begin{equation} (m_{K^+}^{2}-m_{K^0}^{2})\vert_{\mbox{\scriptsize\rm
EM}}=(m_{\pi^+}^{2}-m_{\pi^0}^{2})\vert_{\mbox{\scriptsize\rm EM}}=
\frac{2e^2 C}{f_{\pi}^2}+ \cdots.
\end{equation} In fact, the main contribution to the physical
$\pi^{+}-\pi^{0}$ mass difference {\em is} of electromagnetic origin
because quark
masses do not contribute significantly to the
$\pi^{+}-\pi^{0}$ mass splitting~\cite{GL82}.

The effective term in eq.~(\ref{eq:effem}) results from the integration of a
virtual photon in the presence of the strong interactions in the chiral
limit. It
is well known~\cite{Lowetal67,BdeR91,Bachir97,KU97} that the corresponding
coupling
constant
$C$ is given by an integral of a correlation function
$\Pi_{LR}(Q^2)$ which is the invariant amplitude of the two--point function
\begin{equation}\label{eq:lrtpf}
\Pi_{LR}^{\mu\nu}(q)=2i\int d^4 x\,e^{iq\cdot x}\langle 0\vert
\mbox{\rm T}\left(L^{\mu}(x)R^{\nu}(0)^{\dagger}
\right)\vert 0\rangle\,,
\end{equation} with currents
\begin{equation} L^{\mu}=\bar{d}(x)\gamma^{\mu}\frac{1}{2}(1-\gamma_{5})u(x)
\quad \mbox{\rm and} \quad R^{\mu}=\bar{d}(x)\gamma^{\mu}\frac{1}{2}(1+\gamma_{5})u(x)\,.
\end{equation} In the chiral limit
\begin{equation}\label{eq:lritpf}
\Pi_{LR}^{\mu\nu}(Q^2)=(q^{\mu}q^{\nu}-g^{\mu\nu}q^2)\Pi_{LR}(Q^2)\,,
\end{equation} and
\begin{equation}\label{eq:piem}
C=\frac{-1}{8\pi^2}\,\frac{3}{4}\,
\int_0^\infty dQ^2\,Q^2\Pi_{LR}(Q^2)\,,
\end{equation}
with $Q^2$ the euclidean momentum squared of the virtual photon.
   The function $\Pi_{LR}(Q^2)$ is, for all values of $Q^2$, an order
parameter of
spontaneous chiral symmetry breakdown (S$\chi$SB)  and thus, so is the
low--energy
constant $C$.  This means that the correlator in eq.~(\ref{eq:lrtpf}) has a
smooth
behaviour at short distances, so that the integral in eq.~(\ref{eq:piem})
converges
in the ultraviolet region~:
\begin{equation}\label{eq:wsrs}
\lim_{Q^2\rightarrow\infty} Q^4 \Pi_{LR}(Q^2)\rightarrow 0.
\end{equation} This behaviour also entails the two Weinberg sum rules~\cite{We67} in the
chiral limit. It has furthermore been shown~\cite{W83, CLT95} that
\begin{equation}\label{eq:witten} -Q^2\Pi_{LR}(Q^2)\ge 0 \qquad\mbox{\rm for}\;  \quad 0\le
Q^2\le\infty\,,
\end{equation} which in particular ensures the positivity of the integral in
eq.~(\ref{eq:piem}) and thus the stability of the QCD vacuum with respect
to  small
perturbations induced by electromagnetic interactions.

In this letter we shall first show that the same correlation function
$\Pi_{LR}(Q^2)$ governs the $\pi^{+}-\pi^{0}$ mass difference in the presence of
the electroweak interactions of the Standard Model. This observation and the
calculation which follows thereof are of course of phenomenological
interest {\it
per se}; but their real relevance lies in the fact that, as we shall show below,
they provide an excellent theoretical laboratory to study issues of {\it
long-- and
short--distance matching} in calculations of weak matrix elements of four--quark
operators. We also think that this example, which is under good analytical
control,
provides an excellent testground for numerical evaluations in
lattice--QCD and for
model calculations in general.

\section{The Electroweak $\pi^{+}-\pi^{0}$ Mass Difference}
\label{sec:Elweakmass}

In the presence of the electromagnetic and weak neutral currents of the
Electroweak
Model, the QCD lagrangian ${\cal L}_{\mbox{\rm{\tiny QCD}}}$ becomes
\begin{equation}
{\cal L}_{\mbox{\rm{\tiny QCD}}}\rightarrow {\cal L}_{\mbox{\rm{\tiny QCD}}}+
\bar{q}_{R}\gamma^{\mu}r_{\mu}q_{R} +\bar{q}_{L}\gamma^{\mu}l_{\mu}q_{L}\,,
\end{equation} where
\begin{eqnarray}\label{eq:external} l_{\mu} &= & eQ_{L}[A_{\mu}(x)-\tan\theta_{W}
Z_{\mu}(x)]+
\frac{e}{2\sin\theta_{W}\cos\theta_{W}}T_{3}Z_{\mu}(x)\,,\\ r_{\mu} &= &
eQ_{R}[A_{\mu}(x)-\tan\theta_{W} Z_{\mu}(x)]\,,
\end{eqnarray} and
\begin{equation} Q_{L}=Q_{R}=\frac{1}{3}\mbox{\rm diag} (2,-1,-1)\,,\qquad T_{3}=\mbox{\rm diag}(1,-1,-1)\,.
\end{equation} There are then two possible terms of order ${\cal O}(p^0)$ in the low--energy
effective lagrangian generated by the virtual
$Z$ integration~\footnote{The charged $W$--field only couples to left currents
and therefore cannot contribute to an ${\cal O}(p^0)$ self--energy.}:
\begin{equation}
+e^2 \tan^2 \theta_{W} C_{Z}\mbox{\rm tr} (Q_{R}UQ_{L}U^{\dagger})\quad\mbox{\rm and}\quad -e^2
\frac{1}{2\cos^2
\theta_{W}} C_{Z}\mbox{\rm tr} (Q_{R}UT_{3}U^{\dagger})\,,
\end{equation}
where
\begin{equation}\label{eq:CZ}
C_{Z}=\frac{1}{8\pi^2}\,\frac{3}{4}\,
\int_0^\infty dQ^2\,\frac{Q^2}{Q^2 +M_{Z}^2}\,Q^2\Pi_{LR}(Q^2)\,.
\end{equation}
These couplings give rise to physical mass terms
\begin{equation}
+e^2 \tan^2 \theta_{W} C_{Z}\mbox{\rm tr} (Q_{R}UQ_{L}U^{\dagger})=-2e^2
\frac{\sin^2 \theta_{W}}{\cos^2 \theta_{W}}
C_{Z}\frac{1}{f_{\pi}^2}\left(\pi^{+}\pi^{-}+ K^{+}K^{-}\right)+\cdots\,,
\end{equation}
and
\begin{equation}
-e^2 \frac{1}{2\cos^2 \theta_{W}} C_{Z}\mbox{\rm tr} (Q_{R}UT_{3}U^{\dagger})= 2e^2
\frac{1}{\cos^2 \theta_{W}} C_{Z}\frac{1}{f_{\pi}^2}
\left(\pi^{+}\pi^{-}+K^{+}K^{-}\right)+\cdots\,.
\end{equation}
Collecting together these  weak neutral current contributions with the
electromagnetic contribution in eq.~(\ref{eq:effemex}), we arrive at the quite
remarkable result that, to lowest order in $e^2$ and in the chiral limit,
\begin{equation}\label{eq:ewmass}
(m_{K^+}^2-m_{K^0}^2)\vert_{\mbox{\rm {\tiny SM}}}=
(m_{\pi^+}^2-m_{\pi^0}^2)\vert_{\mbox{\rm {\tiny SM}}}=
\frac{2e^2 C_{\mbox{\rm {\tiny SM}}}}{f_{\pi}^2}\,,
\end{equation} with
\begin{equation}\label{eq:CSM}
C_{\mbox{\rm {\tiny SM}}}=\frac{1}{8\pi^2}\frac{3}{4}\int_{0}^{\infty} dQ^2
\left(1-\frac{Q^2}{Q^2 +M_{Z}^2}\right) \left(-Q^2
\Pi_{LR}(Q^2)\right)\,.
\end{equation}
In the limit where $M_{Z}\rightarrow\infty$ we recover the usual expression of Low {\it
et al.}~\cite{Lowetal67}, while there is no induced mass difference in the limit
$M_{Z}\rightarrow 0$ where the $SU(2)\times U(1)$ gauge symmetry is unbroken. This can
readily be seen by rewriting the external fields in eq.~(\ref{eq:external}) in
terms of the usual
$W_{\mu}^{3}$ and $B_{\mu}$ gauge fields:
\begin{eqnarray}\label{eq:externalBW}
l_{\mu} &= &
\frac{e}{\cos\theta_{W}}\left[Q_{L}-\frac{1}{2}T_{3}\right]B_{\mu}(x) +
\frac{e}{2\sin\theta_{W}}T_{3}W_{\mu}^{3}(x)\,,\\ r_{\mu} &= &
\frac{e}{\cos\theta_{W}}Q_{R}B_{\mu}(x)\,.
\end{eqnarray} Neither the $W_{\mu}^{3}$ field, which has a pure left--handed coupling to
quarks, nor the
$B_{\mu}$ field, which has a left component proportional to the unit
matrix, can
generate an effective ${\cal O}(p^0)$ interaction of Goldstone fields. When the
$SU(2)\times U(1)$ gauge symmetry is broken down to $U(1)_{\mbox{\tiny\rm  EM}}$,
eqs.~(\ref{eq:ewmass}) and (\ref{eq:CSM}) show that the net effect of the
massive
physical
$Z$ is to reduce just a little bit the $\pi^{+}$ mass induced by
electromagnetism
alone.

It is possible to make a phenomenological evaluation of the integral in
eq.~(\ref{eq:CSM})  using the fact that the function $\Pi_{LR}(Q^2)$ obeys an
unsubtracted dispersion relation
\begin{equation}\label{eq:disp}
\Pi_{LR}(Q^2)=\int_{0}^{\infty}dt\frac{1}{t+Q^2}\left[\frac{1}{\pi}\mbox{\rm Im}
\Pi_{V}(t)-\frac{1}{\pi}\mbox{\rm Im}\Pi_{A}(t)\right]\,,
\end{equation}
with $\frac{1}{\pi}\mbox{\rm Im}
\Pi_{V}(t)$ and $\frac{1}{\pi}\mbox{\rm Im}
\Pi_{A}(t)$  the physical vector and axial--vector spectral functions which are
measured in hadronic $e^{+} e^{-}$ annihilations and in hadronic
$\tau$ decays.
The physical spectral functions, however, are not quite the ``chiral limit''
spectral functions which should be inserted in eq.~(\ref{eq:CSM}). It is
possible to do the appropriate corrections, as discussed e.g. in
ref.~\cite{DGHS98}, but for the purposes of this letter we wish to follow a more
theoretical procedure. We propose to evaluate the integral in eq.~(\ref{eq:CSM})
within the framework of QCD in the limit of a large number of colours
$N_c$~\cite{'tH74,RV77,W79}, which we shall denote by QCD$(\infty)\;$ for short.

The spectral function associated with
$\Pi_{LR}(Q^2)$ in QCD$(\infty)\;$ consists of the difference of an infinite number of
narrow
vector states and an infinite number of narrow axial--vector states,
together with
the Goldstone pion pole:
\begin{equation}
\frac{1}{\pi}\mbox{\rm Im}\Pi_{LR}(t) =\sum_{V}f_{V}^2 M_{V}^2\delta(t-M_{V}^2)
-\sum_{A}f_{A}^2 M_{A}^2\delta(t-M_{A}^2)-f_{\pi}^2\delta(t)\,.
\end{equation} Since $\Pi_{LR}(Q^2)$ obeys the dispersion relation~(\ref{eq:disp}), we find
that
\begin{equation}\label{eq:LRN1}
-Q^2\Pi_{LR}(Q^2)=f_{\pi}^2+\sum_{A}f_{A}^2
M_{A}^2\frac{Q^2}{M_{A}^2+Q^2} -\sum_{V}f_{V}^2
M_{V}^2\frac{Q^2}{M_{V}^2+Q^2}\,.
\end{equation}
Furthermore, the two Weinberg sum rules that follow from eq.~(\ref{eq:wsrs})
constrain the couplings and masses of the narrow states as follows:
\begin{equation}\label{eq:weinbergsrs}
\sum_{V}f_{V}^2 M_{V}^2-\sum_{A}f_{A}^2 M_{A}^2=f_{\pi}^2
\quad
\mbox{\rm and}\quad
\sum_{V}f_{V}^2 M_{V}^4-\sum_{A}f_{A}^2 M_{A}^4=0\,,
\end{equation}
ensuring the convergence of the integral in eq.~(\ref{eq:CSM}) in QCD$(\infty)\;$, with the
result
\begin{equation}\label{eq:CSMexact}
C_{\mbox{\rm {\tiny SM}}}=\frac{-3}{32\pi^2}\left[\sum_{A}\frac{M_{Z}^2}{M_{Z}^2-M_{A}^2}f_{A}^2
M_{A}^4
\log\frac{M_{Z}^2}{M_{A}^2}-\sum_{V}\frac{M_{Z}^2}{M_{Z}^2-M_{V}^2}f_{V}^2
M_{V}^4
\log\frac{M_{Z}^2}{M_{V}^2}\right]\,.
\end{equation}

The original evaluation by Low {\it et al.}~\cite{Lowetal67} of the integral in
eq.~(\ref{eq:piem})  was made by considering only the phenomenological
contributions from the pion pole and the lowest vector and axial--vector
states in
the narrow width approximation. This approximation, which reproduces very
well the
experimental determination of the physical
$\pi^{+}-\pi^{0}$ mass difference, can nowadays be viewed as the
approximation to
QCD$(\infty)\;$ which consists in restricting the part of the hadronic spectrum which is
responsible for S$\chi$SB in the channels with the $J^{P}$ quantum numbers
$1^{-}$ and $1^{+}$ to their lowest energy states. The rest of the infinite
number
of narrow states are then treated as dual to their corresponding
QCD$(\infty)\;$ perturbative continuum, and therefore do not contribute to order
parameters of
S$\chi$SB. An explicit formulation in terms of an effective Lagrangian of this
lowest meson dominance (LMD) approximation to QCD$(\infty)\;$, with inclusion of the
$0^-$ and
$0^{+}$ channels as well,  has been recently
discussed in ref.~\cite{PPdeR98}. In the LMD approximation to
QCD$(\infty)\;$, the integral in eq.~(\ref{eq:CSM}) gives the result
\begin{equation}\label{eq:CSMLMD} C_{\mbox{\rm {\tiny SM}}}\simeq\frac{3}{4}\frac{f_{\pi}^2}{8\pi^2}
M_{V}^2\left[
\frac{M_{A}^2}{M_{A}^2-M_{V}^2}\log\frac{M_{A}^2}{M_{V}^2}-
\frac{M_{A}^2}{M_{Z}^2}\left(\log\frac{M_{Z}^2}{M_{V}^2}-
\frac{M_{A}^2}{M_{A}^2-M_{V}^2}\log\frac{M_{A}^2}{M_{V}^2}
\right)\right]\,,
\end{equation}
where we have expanded in powers of $1/M_{Z}^2$ and retained only the leading
term. The first term in the r.h.s. of eq.~(\ref{eq:CSMLMD}) is the contribution
from the virtual photon integration. The
$Z$--induced contribution consists of a ``large''
$\log M_{Z}^2$ term and a ``constant'' term. Overall, it represents a
correction of
0.097\% \footnote{For the numerical evaluation, we have taken $M_V$ and
$M_A$ as the $\rho(770)$ and the $a_1(1260)$ masses, respectively. Our
normalization convention corresponds to $f_\pi = 92.4$ MeV.}
to the lowest order
electromagnetic contribution. The effect, as expected, is very small but
nevertheless larger than the present experimental accuracy in the
determination of
the $\pi^{+}-\pi^{0}$ mass difference. Although this observation may bear some
interest by itself, we shall not pursue it any further here. We rather
dedicate the rest of this note to several theoretically interesting aspects
of this
calculation.

\section {$\Pi_{LR}(Q^2)$ and the Operator Product Expansion}
\label{sec:OPE}

As recently discussed in ref.~\cite{KdeR97}, besides
the two Weinberg sum rules in eqs.~(\ref{eq:weinbergsrs}), there are further
constraints between masses and couplings of resonances on the one hand and the
local order parameters which govern the operator product expansion (OPE) of the
$\Pi_{LR}(Q^2)$ function on the other. In particular, it has been
shown~\cite{KdeR97} that in the large--$N_c$ limit, the leading $d=6$ order
parameter is given by the expression
\begin{eqnarray}\label{eq:phi3}
\lim_{Q^2\rightarrow\infty}Q^6\Pi_{LR}(Q^2) & = &
-4\pi^2\left(\frac{\alpha_s}{\pi}+{\cal O}(\alpha_s^2)\right)
\langle\bar{\psi}\psi\rangle^2 \nonumber\\ & = & \sum_{V}
f_{V}^2 M_{V}^6-\sum_{A} f_{A}^2 M_{A}^6\,,
\end{eqnarray} where $\langle\bar{\psi}\psi\rangle$ is the usual single flavour quark
bilinear condensate ($\psi=u,d,$ or $s$) in the chiral limit, and the Wilson
coefficient is the result of a lowest order calculation~\cite{SVZ79} in
powers of
$\alpha_s$. We shall show that it is precisely this
$d=6$ term which controls the $\log M_{Z}^2$ contribution to the
$Z$--induced part of the integral $C_{Z}$ in eq.~(\ref{eq:CZ}). Indeed,
following a Wilsonian approach, we can
split this integral into a low--energy region $0\le Q^2 \le
\mu^2 \ll M_Z^2$ where the $Z$ field is integrated out, {\it i.e.} where we approximate
${1}/({Q^2 +M_{Z}^2})$ by
${1}/{M_{Z}^2}$, and a high--energy region $\mu^2\le Q^2\le\infty$ where the
$Z$--propagator is fully kept but the
$\Pi_{LR}(Q^2)$ function is approximated by the leading $d=6$ term in
eq~(\ref{eq:phi3}). These approximations are equivalent to neglecting
higher order
corrections in $1/M_Z^2$ and in $1/\mu^2$. We then obtain
\begin{eqnarray}\label{eq:OPEsplit} C_{Z} & = &
\frac{3}{32\pi^2}\frac{1}{M_{Z}^2}\left[\sum_{A}f_{A}^2 M_{A}^6
\log\frac{M_{A}^2}{\mu^2}-\sum_{V}f_{V}^2 M_{V}^6
\log\frac{M_{V}^2}{\mu^2}\right] \nonumber \\
 &  & \frac{-3}{32\pi^2}\frac{1}{M_{Z}^2}\left[\sum_{A}f_{A}^2 M_{A}^6-
\sum_{V}f_{V}^2 M_{V}^6\right]\log\frac{M_{Z}^2}{\mu^2}\,,
\end{eqnarray} where the first line is the result from the low--energy region and the
second
line the one from the high--energy region. We observe that the separation scale
$\mu^2$ cancels in the sum of the two terms; in other words, there is an exact
matching between the long--distance contribution and the short--distance
contribution. This cancellation also occurs in the LMD approximation to QCD$(\infty)\;$ and
the result coincides then with the one given in eq.~(\ref{eq:CSMLMD}). The
reason
why we can exhibit this exact matching is of course due to the fact that the
function $\Pi_{LR}(Q^2)$ is an order parameter for all values of $Q^2$ and
therefore has duality properties under the transformation $Q^2\rightleftharpoons
{1}/{Q^2}$ which, in the large--$N_c$ limit, we have been able to work out
explicitly.

It is also interesting to look at eq.~(\ref{eq:OPEsplit}) from the point of
view of
effective field theory. The full integral in eq.~(\ref{eq:CZ}) can be split as
follows
\begin{equation}\label{eq:effft} C_{Z}=\lim_{\mu\rightarrow\infty}\left\{C_{Z}(M_{Z}\rightarrow\infty,\mu)+
\delta C_{Z}(M_{Z},\mu)\right\}\, ,
\end{equation} where $\delta C_{Z}(M_{Z},\mu) \equiv C_{Z}-C_{Z}(M_{Z}\rightarrow\infty,\mu)$.
Each of
the two terms in the sum is now UV--divergent and this is why an
ultraviolet cutoff
$\mu$ is needed. The first term in eq.~(\ref{eq:effft}) corresponds to the
result
from the effective low--energy theory, where the $Z$ field has been
integrated out
and it is precisely given by the first line in the r.h.s. of
eq.~(\ref{eq:OPEsplit}). The second term $\delta C_{Z}(M_{Z},\mu)$
corresponds  to
the matching condition in the effective field theory language and the result is
precisely the one given in the second line of eq.~(\ref{eq:OPEsplit}).
Formally one
can set $\mu=M_{Z}$ (the effective field theory scale in this case) in these
expressions since they are actually
$\mu$ independent. Then $\delta C_{Z}(M_{Z},M_{Z}) =0$ and
$C_{Z}=C_{Z}(M_{Z}\rightarrow\infty,M_{Z})$.

We are now in a position to discuss how the above calculation  proceeds in the
framework which is usually adopted to tackle weak matrix element
calculations.~\footnote{See e.g. the lectures of A.~Buras in ref.~\cite{Buras98}
and references therein.}

\section{Four--Quark Operators}
\label{sec:fourqo}

The relevant term in the Lagrangian of the Standard Model which is
responsible for the $Z$--induced contribution to the
$\pi^{+}-\pi^{0}$ mass difference is the neutral current interaction
term
\begin{equation}
{\cal L}_{\mbox{\rm {\tiny NC}}}=\frac{e}{2\sin\theta_{W}\cos\theta_{W}}
\left[\bar{q}_{L}\gamma^{\mu}T_{3}q_{L}-2\sin^{2}\theta_{W}
\bar{q}_{L}\gamma^{\mu}Q_{L}q_{L}-2\sin^{2}\theta_{W}
\bar{q}_{R}\gamma^{\mu}Q_{R}q_{R}\right]Z_{\mu}\,.
\end{equation} When looking for the induced effective Lagrangian of order
${\cal O}(p^0)$ which contributes to Goldstone boson masses, it is
sufficient to consider left--right operators. In the absence of the
strong interactions, the effective four--quark Hamiltonian which
emerges after integrating out the $Z$ field  is
\begin{eqnarray}\label{eq:eff}
 - {\cal H}_{\mbox{\rm{\tiny eff}}} & = & \frac{-1}{M_{Z}^2}\frac{e^2}{\cos^2 \theta_{W}}\left[
\sin^2 \theta_{W} Q_{LR}
-\frac{1}{2}\left(\bar{q}_{L}\gamma_{\mu}T_{3}q_{L}\right)
\left(\bar{q}_{R}\gamma^{\mu}Q_{R}q_{R}\right)\right] \nonumber\\
 & = &
\frac{e^2}{M_{Z}^2} Q_{LR}-
\frac{e^2}{M_{Z}^2}\frac{1}{\cos^2
\theta_{W}}\frac{1}{6}\left(\sum_{q}\bar{q}\gamma_{\mu}q\right)
\left(\bar{q}_{R}\gamma^{\mu}Q_{R}q_{R}\right)\,,
\end{eqnarray} where
\begin{equation}
 Q_{LR}\equiv\left(\bar{q}_{L}\gamma_{\mu}Q_{L}q_{L}\right)
\left(\bar{q}_{R}\gamma^{\mu}Q_{R}q_{R}\right)\,,
\end{equation} and summation over quark colour indices within brackets is
understood. In fact, to ${\cal O}(p^0)$,
only the first term proportional to the four--quark operator $Q_{LR}$
can contribute. In the presence of the strong interactions, the
evolution of $Q_{LR}$ from the scale
$M_{Z}^2$ down to a scale
$\mu^2$ can be calculated in the usual way, provided this $\mu^2$ is
still large enough for a perturbative QCD (pQCD) evaluation to be valid.
In the leading logarithmic approximation in
pQCD, and to leading non--trivial order in the $1/N_c$ expansion, the relevant
mixing in this evolution which we need to retain is simply given by
\begin{equation}\label{eq:mixing}
Q_{LR}(M_{Z}^2) =
Q_{LR}(\mu^2)-3\frac{\alpha_{\mbox{\rm {\scriptsize s}}}}{\pi}\frac{1}{2}\log\frac{M_{Z}^2}{\mu^2}
D_{RL}(\mu^2)+\cdots\,,
\end{equation} where $D_{RL}$ denotes the four--quark density--density operator
\begin{equation} D_{RL} \equiv
\sum_{q,q'} e_{q}e_{q'}(\bar{q'}_{L}q_{R})(\bar{q}_{R}q'_{L})\,,
\end{equation} with $e_{q}$ and $e_{q'}$ the quark charges in units of the electric
charge. This can be seen as follows: in the $\overline{MS}$ renormalization
scheme,
the full evolution of the   Wilson coefficients $c_{Q}$ and $c_{D}$ of the
operators
$Q_{LR}$ and
$D_{LR}$ at the one loop level is governed by the equations
(subleading contributions in the $1/N_c$ expansion have been neglected)
\begin{equation}
\mu^2\frac{d}{d\mu^2}\left( \begin{array}{c} c_{Q} \\ c_{D} \end{array}\right)
=\frac{1}{4}\frac{\alpha_{\mbox{\rm {\scriptsize s}}} N_c}{\pi}\left(\begin{array}{cc} \cdots & \cdots \\
\frac{6}{N_c} & -3+\cdots \end{array}\right)\left( \begin{array}{c}
c_{Q} \\ c_{D} \end{array}\right)\,,
\end{equation}
with boundary conditions: $c_{Q}(M_Z)=1$ and
$c_{D}(M_Z)=\frac{3}{2}\frac{\alpha_{\mbox{\rm {\scriptsize s}}}}{\pi}$. The result in eq.~(\ref{eq:mixing})
follows when taking $c_{D}(M_Z)=0$, which is appropriate when keeping
the one-loop leading log only, and from the off--diagonal term in the
(transposed) anomalous dimension matrix.

We are then confronted with a typical problem of bosonization of four--quark
operators.
The bosonization of $D_{RL}$ is only needed to leading order in the
$1/N_c$ expansion. To that order and to order ${\cal O}(p^0)$ in the
chiral expansion it can be readily obtained from the bosonization of
the factorized density currents, with the result~\footnote{See e.g. the
lectures in
ref.~\cite{deR95} and references therein.}
\begin{equation}
D_{RL} \equiv \sum_{q,q'}
e_{q}e_{q'}(\bar{q'}_{L}q_{R})(\bar{q}_{R}q'_{L})\rightarrow
2B\frac{f_{\pi}^2}{4}\times 2B\frac{f_{\pi}^2}{4}
\mbox{\rm tr} \left(UQ_{L}U^{\dagger}Q_{R}\right)\,,
\end{equation}
where $B$ is the low energy constant which describes the bilinear quark
condensate in the chiral limit,
\begin{equation}
B=-\frac{\langle{\bar\psi}\psi\rangle}{f_{\pi}^{2}}\,.
\end{equation}
We find that the overall contribution of the term proportional to the
$D_{RL}(\mu^2)$ four--quark operator, which we denote $C_{Z}\vert_{D_{RL}}$, is
given by the expression
\begin{equation}\label{eq:4sd}
C_{Z}\vert_{D_{RL}} =\frac{-1}{M_{Z}^2}
3\frac{\alpha_{\mbox{\rm {\scriptsize s}}}}{\pi}\frac{1}{4}\langle\bar{\psi}\psi\rangle^2
\frac{1}{2}\log\frac{M_{Z}^2}{\mu^2}\,,
\end{equation}
and it is exactly the same result as the
one coming from the $d=6$ term of the OPE in the previous calculation,
the second line in eq.~(\ref{eq:OPEsplit}), once the duality constraint in
eq.~(\ref{eq:phi3}) is taken into account, {\it  i.e.} precisely the same
contribution as required
by the matching condition in the effective field theory analysis
above. Equation (\ref{eq:4sd}) does not change in an $\overline{MS}$
renormalization scheme when restricted to the one-loop leading log.

The problem is then reduced to the bosonization of the operator
$Q_{LR}(\mu^2)$. We are confronted here with a typical calculation of a
hadronic matrix element of a four--quark operator, in this case the matrix
element
$\langle\pi^{+}\vert Q_{LR}(\mu^2)\vert\pi^{+}\rangle$. The factorized
component
of the operator $Q_{LR}$, which is leading in $1/N_c$, cannot contribute to the
${\cal O}(p^0)$ term of the low--energy effective lagrangian. The contribution
we want
from this matrix element is therefore the next--to--leading one in
the $1/N_c$ expansion and it requires the evaluation of
non--factorizable four--quark matrix elements which is {\it a priori} a highly
non--trivial task. Yet, we know that there is a straightforward integral
representation of the observable we want to compute in terms of a two--point
function, and we have succeeded in doing the calculation that way within the
$1/N_c$--expansion framework. This seems to indicate that there should
be a way to do
the calculation of the matrix element $\langle\pi^{+}\vert
Q_{LR}(\mu^2)\vert\pi^{+}\rangle$ as well.
Indeed, we shall next show
that, for this particular matrix element, our present knowledge of analytic
results in non--perturbative QCD allows us to do the calculation exactly to
next--to--leading order in the $1/N_c$ expansion. The deep reason behind it is
that we know from the discussions in the previous sections that only two--point
functions can appear in the dynamics of this matrix element, and therefore it
is enough to have a formulation of the low--energy effective
lagrangian of QCD$(\infty)\;$ compatible with the OPE constraints at short distances
for
two--point functions. These are precisely the constraints  which have
been recently discussed in ref.~\cite{KdeR97}.

The calculation proceeds along much the same lines as first
suggested in papers by Bardeen, Buras and G{\'e}rard~\cite{BBG87,Bardeen89}
sometime ago~\footnote{See also  refs.~\cite{BGK91,FG95,Ko98} and references
therein for more recent work.}, except that we shall go beyond loops
generated by Goldstone particle interactions alone in order to achieve a
correct matching with the logarithmic scale dependence of the
short--distance contribution in eq.~(\ref{eq:4sd}). We have evaluated the matrix
element
$\langle\pi^{+}\vert Q_{LR}(\mu^2)\vert\pi^{+}\rangle$
within the framework of an effective Lagrangian which is a
straightforward generalization to an arbitrary number of massive $J^P=1^-$
and $J^P=1^+$ mesonic states of the effective Lagrangian corresponding to
the LMD
approximation to QCD$(\infty)\;$ recently discussed in ref.~\cite{PPdeR98}. Keeping only
the terms that are relevant for the evaluation of the ${\cal O}(p^0)$
contributions, we
have
\begin{eqnarray}
{\cal L}_{{\mbox{\rm {\scriptsize eff}}}} &=& \frac{f_\pi^2}{4}
\,\mbox{\rm tr} \,D_{\mu}U^+D^{\mu}U + L_{10}\,\mbox{\rm tr}\, U^+F_R^{\mu\nu}UF_{L,\mu\nu}
\nonumber\\
&&\ -\frac{1}{4}\sum_V \mbox{\rm tr}\left(
V_{\mu\nu}V^{\mu\nu}-2M_V^2V_{\mu}V^{\mu}\right)
-\frac{1}{4}\sum_A \mbox{\rm tr}\left( A_{\mu\nu}A^{\mu\nu}-2M_A^2A_{\mu}A^{\mu}\right)
\nonumber\\
&&\ -\frac{1}{2\sqrt{2}}\sum_V f_V\mbox{\rm tr} V_{\mu\nu}f_+^{\mu\nu}
-\frac{1}{2\sqrt{2}}\sum_A f_A\mbox{\rm tr} A_{\mu\nu}f_-^{\mu\nu}
+\cdots\,,
\end{eqnarray}
with the same notation as in ref.~\cite{PPdeR98},
$L_{10}$ being one of the ${\cal O}(p^4)$ constants of the Gasser-
Leutwyler Lagrangian~\cite{GL85}.

The bosonized expressions of the currents
$\left(\bar{q}_{L}\gamma^{\mu}Q_{L}q_{L}\right)$ and
$\left(\bar{q}_{R}\gamma^{\mu}Q_{R}q_{R}\right)$ are obtained from the
resulting effective action
$\Gamma_{{\mbox{\rm {\scriptsize eff}}}}$ at tree level,
\begin{equation}
\left(\bar{q}_{L}\gamma^{\mu}Q_{L}q_{L}\right) \rightarrow
(Q_{L})_{ij}\frac{\delta\Gamma_{{\mbox{\rm {\scriptsize eff}}}}}
{\delta l_{\mu}^{ij}(x)}\quad\mbox{\rm and}\quad
\left(\bar{q}_{R}\gamma^{\mu}Q_{R}q_{R}\right) \rightarrow
(Q_{R})_{ij}\frac{\delta\Gamma_{{\mbox{\rm {\scriptsize eff}}}}}
{\delta r_{\mu}^{ij}(x)}\,,
\end{equation}
where $i,j=u,d,s$ are flavour indices.
The contributions from a loop of virtual pions, vector and axial--vector
resonances
are computed by applying a finite ultraviolet cut-off $\mu^2$
to the integral over the
corresponding (euclidean) virtual loop momentum.
One technical point, as already
discussed in ref~\cite{PdeR91} (see eqs.~(7.24) to
(7.26) of this reference), requires however attention. It is the fact that
to next--to--leading order  in
the $1/N_c$ expansion, and besides the loops generated by the bosonized factorized
currents that we have just discussed, there
are also contact terms generated by the second variation of the effective
action. In our case these terms are:
\begin{eqnarray}
Q_{LR}^{{\mbox{\rm {\scriptsize contact}}}} & =
& (Q_{L})_{ij}(Q_{R})_{kl}\frac{\delta^2\Gamma_{{\mbox{\rm {\scriptsize
eff}}}}}{\delta l_{\mu}^{ij}(x)\delta r^{\mu, kl}(x)} \nonumber
\\
 & = & \left\{-2f_{\pi}^2\,\mbox{\rm tr}\left( Q_{R}UQ_{L}U^{\dagger}\right)\delta(0)-
6L_{10}\,\mbox{\rm tr}\left(Q_{R}UQ_{L}U^{\dagger}\right)\Box\delta
(0)+\cdots\right\}\,,
\end{eqnarray}
where the $\delta$--function contributions are to be interpreted within the
same cut-off regularization~:
\begin{equation}
\delta(0)\rightarrow \pi\int_{0}^{\mu^2} dQ^2 Q^2  = \frac{\pi}{2}\mu^4
\qquad\mbox{\rm and}\qquad
\Box\delta(0) \rightarrow \pi\int_{0}^{\mu^2} dQ^2 Q^4  = \frac{\pi}{3}\mu^6\,.
\end{equation}
Adding the two types of contributions, the final result reads
\begin{eqnarray}\label{eq:QLRresult}
\langle\pi^{+}\vert
Q_{LR}(\mu^2)\vert\pi^{+}\rangle\ &  = & \ \frac{+1}{16\pi^2}
\left\{ \frac{3}{2}\mu^4 + 4\frac{L_{10}}{f_{\pi}^2}\mu^6 + \right. \nonumber \\
 &  & \left. +\frac{3}{f_{\pi}^2}\int_{0}^{\mu^2} dQ^2 Q^6
\bigg[ \sum_{V}\frac{f_{V}^2}{Q^2+M_V^2} -
       \sum_{A}\frac{f_{A}^2}{Q^2+M_A^2}\bigg]\right\}\,.
\end{eqnarray}
The contribution of the Goldstone bosons alone corresponds to
the two terms in the first line of the r.h.s. of eq.~(\ref{eq:QLRresult}). They
display a typical polynomial
dependence with respect to the cut-off $\mu$, which can hardly provide a
reasonably good matching with the logarithmic scale dependence coming from
the short--distance contributions in
$C_{Z}\vert_{D_{RL}}$. In fact, in an $\overline{MS}$ regularization scheme, as
commonly chosen for the evaluation of the short--distance Wilson coefficients,
these power divergences will automatically disappear.  Simply adding higher
resonances does not by itself solve the problem of matching the long-- and the
short--distances either; however, when the information coming from the
short--distance properties of the correlator in eq.~(\ref{eq:lrtpf}), {\it i.e.}
the two Weinberg sum rules (\ref{eq:weinbergsrs}) and the sum
rule~\cite{GL84,KdeR97}
\begin{equation}\label{eq:L10}
-4L_{10}=\sum_{V} f_{V}^2-\sum_{A}f_{A}^2\,,
\end{equation}
is taken into account, the result of eq.~(\ref{eq:QLRresult}) can indeed be
recast
into a form which reproduces the first line of eq.~(\ref{eq:OPEsplit}). Notice
that in an $\overline{MS}$ regularization scheme, the integral in eq.
(\ref{eq:QLRresult}) should be understood in $n=4-\epsilon$ dimensions and therefore
multiplied by
$\mu^{4-n}$ with $\mu$ the
$\overline{MS}$ regularization scale. The result, when combined with
eq.~(\ref{eq:4sd}) finally yields eq.~(\ref{eq:OPEsplit}) once again. We
insist on
the fact that, regardless of the regularization one chooses,
the calculation we
have done of the matrix element $\langle\pi^{+}\vert
Q_{LR}(\mu^2)\vert\pi^{+}\rangle$ is an exact calculation to next--to--leading
order in the $1/N_c$ expansion and to ${\cal O}(p^0)$ in the chiral expansion.

\section{Comments and Outlook}
\label{sec:conlook}

The main purpose of the above analysis has been to show that the
calculation of the
electroweak contribution to the $\pi^{+}-\pi^{0}$ mass difference in the chiral
limit is an interesting theoretical
laboratory for testing issues connected with
the matching of long and short distances in the evaluation of hadronic weak
matrix elements in general.

On the one hand, upon identifying the relevant low--energy constant in
terms of the appropriate QCD correlator, we have been able to obtain, in the
large--$N_c$ limit, an exact result in terms of resonance parameters
with masses and couplings constrained by the short--distance
properties of this correlator.

On the other hand, due to the presence of the large scale set by $M_Z$, we
were able to proceed as done usually in the study of weak non--leptonic
processes.
In this respect, we encounter a typical situation: the effective
Hamiltonian at tree level is given by a current--current four quark
operator $Q_{LR}$, which, upon taking into account short--distance pQCD
corrections, mixes with the density--density operator $D_{RL}$. The matrix
element of the latter to ${\cal O}(p^0)$,
which is only needed at the leading order in the large--$N_c$
limit, is easily computed. In the same limit, $\langle\pi^{+}\vert
Q_{LR}(\mu^2)\vert\pi^{+}\rangle$ vanishes, and next--to--leading,
non--factorizable, contributions have to be included.
A suitable matching of the long--distance contribution with the logarithmic
scale dependence coming from the short--distance contribution cannot be
achieved, in this case, by evaluating the relevant hadronic matrix elements
in terms of an effective theory incorporating only the Goldstone bosons.
When the
contribution of vector and axial--vector states satisfying the appropriate
short--distance constraints are included, we find an exact matching.

We think that the observations which follow from this example point towards
promising perspectives for a systematic determination of hadronic weak  matrix
elements within the framework of QCD$(\infty)\;$. The required steps are the
following: first
one needs an identification of the low--energy constants of the $\Delta S=1$ and
$\Delta S=2$ effective chiral Lagrangian in terms of integrals of QCD Green's
functions, {\it i.e.} the equivalent of our eq.~(\ref{eq:CSM}) above. Next one
should
proceed to the study of the constraints which the OPE imposes on these Green's
functions in the large--$N_c$ limit, {\it i.e.} the equivalent of the Weinberg
sum rules
in eqs.~(\ref{eq:weinbergsrs}) and the sum rules in eqs.~(\ref{eq:phi3}) and
(\ref{eq:L10}). The final step is the construction of an effective
Lagrangian, along
the lines shown in ref.~\cite{PPdeR98}, which incorporates these
constraints. This
program is at present under study.

In the meantime, it would be interesting to see
a calculation done using numerical simulations of lattice QCD of the matrix
element
$\langle\pi^{+}\vert Q_{LR}(\mu^2)\vert\pi^{+}\rangle$, so as to be able to
appreciate the performance of the lattice approach in a simple case which we
understand well analytically.

\vspace*{7mm} {\large{\bf Acknowledgments}}
\vspace*{3 mm}

\noindent
We wish to thank J.~Bijnens, M.~Perrottet, A.~Pich and J.~Prades for
discussions on topics related to the work reported here.  This work has been
supported in part by TMR, EC-Contract No. ERBFMRX-CT980169 (EURODA$\phi$NE). The
work of S.~Peris has also been partially supported by the research project
CICYT-AEN95-0882.


\section{Introduction}
In classical mechanics, a particle can be said to follow a definite
trajectory, and so it is clear what is meant by the time at which a
particle arrives in a given place.  If one considers an ensemble of
particles, it is then easy to say what is meant by a distribution of arrival
times.  In standard quantum theory, on the other hand, particles are not
said to follow trajectories, and so the meaning of arrival time in quantum
theory has been rather controversial.  Some elements of this controversy
include statements such as ``the time-of-arrival cannot be precisely defined
and measured in quantum mechanics'' (quoted from \cite{aha}; see also
\cite{Allcock, Mielnik, HZ}); statements that time-of-arrival {\em must} be
definable, for example ``Since the distribution of arrival
times at a given spatial point is, in principle, a measurable quantity
that can be determined via a time-of-flight experiment, it is reasonable
to ask for an apparatus-independent theoretical prediction''
 \cite{Del1}; specific proposals for defining a time-of-arrival
distribution (for example, in \cite{ MBM, GRT, DM}), and suggestions
(for example, in \cite{GRT} and \cite{Del2})
that experiments could determine which if any of these proposals is correct.

In the causal theory of Bohm \cite{Bohm, BH}, particles {\em do} follow
definite trajectories, and so the definition of arrival-time distributions
is again unambiguous. Leavens, most recently in \cite{Leavens}, has
studied the
arrival-time distribution of a free particle in Bohmian theory, and found
results which differ from the proposal made in \cite{GRT}.  Deotto and
Ghirardi
\cite{DGh}, and also Holland \cite{Holland} have proposed what I shall
call Bohm-like theories: theories in which particles follow trajectories
which differ from the trajectories of Bohmian theory, but which
nevertheless reproduce all of the observational results of standard
quantum theory, in the same way that Bohmian theory does. In this paper I will
study a simple example of a Bohm-like theory, and demonstrate that in
certain cases this theory will produce arrival-time distributions
which are different that those produced by (standard) Bohmian theory.
I will also use
this same demonstration to discuss, quite apart from any consideration
of Bohmian theory, an ambiguity in the meaning of arrival-time distributions
for three-dimensional problems within standard quantum theory.

I begin by using terminology which is appropriate is classical mechanics.
Consider a particle with position coordinates $(x,y,z)$, which at the
initial time $t=0$ has $x<0$. Let $S_+$ denote the region of space in which
$x\geq 0$, $S_-$ the region in which $x<0$, and $S_0$ the $x=0$ plane. Let
$T$ be the first time after $t=0$ at which the particle arrives at $S_0$
(equivalently, crosses from $S_-$ to $S_+$); by definition, $T>0$. I will
mostly be concerned with the integrated arrival-time distribution,
which I denote as $P(t)$; that is, $P(t)$ is the probability that $T\leq t$.

The question to be discussed is whether this distribution $P$ can be
precisely and unambiguously defined in quantum theory.  If it can be,
it could be expected to satisfy, at least, the following properties:

\begin{description}
\item[i)] $P(t)$ is monotonically increasing: $P(t) \geq P(t')$ for
  $t\geq t'\geq 0$.
\item[ii)] Define $A(t)=dP(t)/dt$; from i), we have $A(t)\geq 0$.
  Then $A(t)dt$ represents the probability that $T=t$.
\item[iii)] Let $\overline{T}$ be the average value of $T$
  (averaged over those cases in which the particle does eventually
  arrive at $S_0$). From ii), this is given by
  \begin{equation}
  \overline{T}=\frac{\int_{0}^{\infty}t\,A(t)\,dt}{\int_{0}^{\infty}A(t)\,dt}
  \end{equation}
  This can be re-written, with
  $P_{\infty }:=\lim_{t\rightarrow \infty}P(t )$, as
  \begin{equation}
    \overline{T}=\frac{\int_{0}^{\infty}dt\,[P_{\infty }-P(t)]}{P_{\infty }}
  \end{equation}
\item[iv)] Now let $Q(t)$ be the probability that the particle would be
  found, at time $t$, in $S_+$; that is,
  \begin{equation}
     Q(t)=\int \int \int|\Psi(x,y,z,t)|^{2}\,\theta (x)\,dx\,dy\,dz.
  \label{ez}\end{equation}
  The initial condition we are assuming means that $Q(t=0)=0$. We expect that
  \begin{equation}P(t)\geq Q(t).\label{i1}\end{equation}
  If the particle could surely be found in $S_+$ at time $t$
  if it had arrived there at time $t'$ (with $0\leq t'\leq t$), then the
  relation (\ref{i1}) could be replaced by an equality.
\item[v)] From i) and iv), we have
  \begin{equation}P(t)\geq \max_{t\geq t'\geq 0}Q(t').\end{equation}
\end{description}

In the next section I will present an example in which a Bohm-like theory
produces a different result for $P(t)$ than does the usual Bohmian theory.
The implications of this example are discussed in the final section.

\section{Example}
Consider a free ($V=0$) particle which, at the initial time $t=0$, is
described by
the following minimum-uncertainty wave packet centered at the point
$x=-x_1$, $y=0$, $z=0$:
\begin{eqnarray}
\Psi(x,y,z,t=0)&=&\left[\frac{1}{\pi^{3}a^{2}b^{2}c^{2}}\right] ^{\frac{1}{4}}
   \exp[ikx]\exp\left[-\frac{(x+x_{1})^{2}}{2a^{2}}\right]\nonumber\\
   & &\mbox{}\times \exp\left[-\frac{y^{2}}{2b^{2}}\right]
   \exp\left[-\frac{z^{2}}{2c^{2}}\right],\label{e1}\end{eqnarray}
where $a$, $b$, $c$, and $k$ are positive constants, and where I
have set both the mass of the particle and the value of $\hbar$ to one.
We want the particle to start out with $x< 0$; this corresponds to
$x_{1}> 0$. Strictly speaking, this wave packet does not
satisfy the condition $Q(t=0)=0$, because its tail extends to positive
values of $x$. However, $Q(0)$ can be made arbitrarily
small by taking $(x_{1}/a)$ large (see Eq.~\ref{ew} below).

Define $\alpha =(a^{2}+it)^{\frac{1}{2}}$;
$\beta =(b^{2}+it)^{\frac{1}{2}}$; $\gamma =(c^{2}+it)^{\frac{1}{2}}$.
Then
\begin{eqnarray}
  \Psi(x,y,z,t)&=&\left[\frac{a^{2}b^{2}c^{2}}{\pi ^{3}}\right]^{\frac{1}{4}}
   \frac{\exp[i(kx-k^{2}t/2)]}{\alpha \beta \gamma}
   \exp\left[-\frac{(x+x_{1}-kt)^{2}}{2\alpha ^{2}}\right]\nonumber\\
   & &\mbox{}\times \exp\left[-\frac{y^{2}}{2\beta ^{2}}\right]
   \exp\left[-\frac{z^{2}}{2\gamma ^{2}}\right].\label{e2}\end{eqnarray}
At time $t$, the center of the wave packet is at $x=-x_{1}+kt$ (with
$x_{1}>0$ and $k>0$), $y=z=0$.
It is straightforward to calculate
\begin{equation}
  Q(t)=\mbox{\small $\frac{1}{2}$}{\rm erfc}(\eta), \label{e3}
  \end{equation}
where $\eta:=a(x_{1}-kt)/|\alpha |^2$, and erfc is the complementary
error function: erfc$(\eta )=2\pi ^{-\frac{1}{2}}\int_{\eta}^{\infty}
\exp (-\xi ^{2})\,d\xi $.  It follows from Eq.~\ref{e3} that
\begin{equation}
  Q(0)=\mbox{\small $\frac{1}{2}$}{\rm erfc}(x_{1}/a),
  \label{ew}\end{equation}
and that
\begin{equation}
  \lim_{t\rightarrow \infty}Q(t)=\mbox{\small $\frac{1}{2}$}
  {\rm erfc}(-ak).\label{ey}\end{equation}
We see from (\ref{ey}) that $Q$ does not approach one even as $t\rightarrow
\infty$; this is because the wave packet spreads out as its center moves
toward large $x$, so that a finite fraction of the tail of the packet
remains in $S_-$.

In Bohmian theory \cite{Bohm, BH} the particle is considered to have a
definite position, which will be denoted as ${\bf r}=(X,Y,Z)$. Let
${\bf v}_b$ denote the Bohmian velocity of the particle (that is,
${\bf v}_{b}=d{\bf r}/dt)$; then if we write $\Psi =R\exp(iS)$, in
Bohmian theory ${\bf v}_b$ is given by
\begin{equation}
  {\bf v}_b = {\bf \nabla} S \label{e4}\end{equation}
It then follows that
\begin{equation}
  {\bf \nabla}\cdot (|\Psi |^{2}{\bf v}_{b})=-\frac{\partial |\Psi |^{2}}
   {\partial t} \label{e5}\end{equation}
In fact, the product $(|\Psi |^{2}{\bf v}_{b})$ is just the usual quantum
probability current, and Eq.~\ref{e5} is just the
equation of conservation of probability in standard quantum theory.
One associates with a given wave function $\Psi$ an ensemble of particles,
whose distribution agrees with the quantum probability density
$|\Psi|^2$; Eq.~\ref{e5} assures that this agreement, if it exists at
the initial time, persists for all time, and this in turn means that
Bohmian theory will reproduce all of the experimental predictions of
standard quantum theory \cite{ex}.  Thus Bohmian theory is not in
conflict with, but rather is a completion of, standard quantum theory.

Since each particle in the Bohmian ensemble follows a definite trajectory,
the interpretation of arrival-time distributions is unambiguous.
The quantity $P(t)$ defined above is simply the fraction of particles in
the ensemble which have $X\geq 0$ for any time $t'$ with $0\leq t' \leq t$,
and of course $Q(t)$ is the fraction which have $X\geq 0$ at time $t$.
For the wave function in Eq.~\ref{e2}, the components of the Bohmian
velocity turn out to be
\begin{equation}
 v_{bx}(X,Y,Z,t)=[k+(X+x_{1})t]/|\alpha |^{4},\label{e6}\end{equation}
\begin{equation}
 v_{by}(X,Y,Z,t)=Yt/|\beta |^{4},\label{e7}\end{equation}
\begin{equation}
 v_{bz}(X,Y,Z,t)=Zt/|\gamma |^{4},\label{e8}\end{equation}
Since $x_{1}>0$, we see from Eq.~\ref{e6} that, for all $t\geq 0$,
\begin{equation}
 v_{bx}(X=0,Y,Z,t)>0\label{e9}\end{equation}
This means that if the particle does enter $S_+$, it can
never leave.  Since that was the condition which gives equality in
the relation (\ref{i1}), we see that in this example, Bohmian theory gives
\begin{equation}
  P(t)=Q(t), \label{e10}\end{equation}
where $Q(t)$ is given by Eq.~\ref{e3}.

Because of the factorized form of $\Psi$ in this example,
the x-component of the Bohmian motion is the same as in the one-dimensional
example of a minimum-uncertainty packet studied in \cite{Leavens}. In fact
we can, without having to solve for the Bohmian trajectories in detail,
recover one of the main results of \cite{Leavens}, namely that a finite
fraction of the Bohmian ensemble never makes it to the region $S_+$.
That fraction is just $(1-P_{\infty})$; by
Eq.~\ref{e10} this equals  $(1-\lim_{t\rightarrow \infty}Q(t))$,
which we saw in Eq.~\ref{ey} is not zero.

It is Eq.~\ref{e5} which insures, for Bohmian theory,
that an ensemble of particles with
initial distribution given by $|\Psi |^2$ reproduces the
experimental predictions of standard quantum theory.  One can formulate
an alternative theory, which I will refer to as a Bohm-like theory,
in which a particle again has a definite position, but in which the
velocity (call it ${\bf v}_{bl})$ may differ from the Bohmian velocity
${\bf v}_b$ (given in Eq.~\ref{e4}).  Let $\delta {\bf v}$ denote
the difference between ${\bf v}_{bl}$ and ${\bf v}_b$:
\begin{equation}
   {\bf v}_{bl}= {\bf v}_{b}+ \delta {\bf v}\label{a1}\end{equation}
Then in order for this Bohm-like theory to agree with standard quantum
theory in the same sense that Bohmian theory does, one must require that
${\bf v}_{bl}$ also satisfy Eq.~\ref{e5}; that is, one must require
\begin{equation}
  {\bf \nabla}\cdot(|\Psi |^{2}\delta {\bf v})=0.\label{a2}\end{equation}
Deotto and Ghirardi \cite{DGh} have shown that it is possible to choose
${\bf v}_{bl}$ in such a way as to satisfy several requirements that
one may reasonably expect, in particular what they call ``genuine''
Galilean covariance. I will consider a simplified form of the theory
suggested in \cite{DGh}; I will take
\begin{equation}
   \delta {\bf v}=\lambda ({\bf \nabla}|\Psi |^{2})\times  {\bf v}_{b},
   \label{a3}\end{equation}
where $\lambda $ is a constant and  ${\bf v}_{b}$ is still given by
Eq.~\ref{e4};  ${\bf v}_{bl}$ is then given by Eq.~\ref{a1}.
This  ${\bf v}_{bl}$ will certainly not satisfy all of the requirements
imposed in \cite{DGh}; I will argue in the next section that this makes
this simple example of a  Bohm-like theory  implausible, but not
demonstrably incorrect.  For now, I will proceed to discuss the consequences
of the choice (\ref{a3}). This choice does at least satisfy Eq.~\ref{a2};
to see that, note that ${\bf v}_{b}={\bf \nabla}S$ and that
${\bf \nabla}\cdot (|\Psi |^{2}{\bf \nabla}|\Psi|^{2}\times {\bf \nabla}S)$
vanishes identically for any $|\Psi|^2$ and any $S$.

It is possible to discuss the distribution $P(t)$ that this Bohm-like
theory will imply for the example given by Eq.~\ref{e2} without having to find
the trajectories explicitly.  If it were the case that the x-component
of ${\bf v}_{bl}$ were positive everywhere on the plane $S_0$ for all
times $t\geq 0$, we could conclude that $P(t)=Q(t)$, just as we did
in Eq.~\ref{e10} for the standard Bohmian theory. As we shall see below,
if this condition on the x-component of ${\bf v}_{bl}$ is {\em not}
satisfied, then this Bohm-like theory will necessarily imply a
{\em different} distribution $P(t)$ than does standard Bohmian theory. From
Eqs.~\ref{e2}, \ref{e7}, \ref{e8}, and \ref{a3}, the x-component of
$\delta {\bf v}$ at $X=0$ is
\begin{equation}
  \delta v_{x}(0,Y,Z,t)=2\lambda|\Psi(0,Y,Z.t)|^{2}(c^{2}-b^{2})YZt
  /(|\beta|^{4}|\gamma|^{4}),\label{a4}\end{equation}
while from Eq.~\ref{e6},
\begin{equation}
  v_{bx}(0,Y,Z,t)=(k+x_{1}t)/|\alpha|^{4}.\label{a5}\end{equation}
Let me now take $\lambda >0$ and $(c^{2}-b^{2})<0$.  Then in the two
quadrants of the plane $S_0$ with the product $YZ$ negative, $\delta  v_{x}$
will be positive, and since $v_{bx}$ is positive everywhere on $S_0$,
we will have $v_{blx}\;(=v_{bx}+\delta v_{x})>0$. On the other hand, in the
quadrants with $YZ$ positive, $\delta v_{x}$ is negative, and so
$v_{blx}$ will be positive if and only if
$|\delta v_{x}|\leq v_{bx}$.
For a fixed value of $t$, the maximum value of $|\delta v_{x}(0,X,Y,t)|$ occurs
at the points $Y=\pm |\beta|^{2}/(\sqrt{2}b)$,
$Z=\pm |\gamma|^{2}/(\sqrt{2}c)$. This maximum value is
\begin{equation}
|\delta v_{x}|_{{\rm max}}=\lambda\frac{a(b^{2}-c^{2})t}{\pi^{\frac{3}{2}}
  e|\alpha|^{2}|\beta|^{4}|\gamma|^{4}}\exp\left[\frac{-a^{2}(x_{1}-kt)^{2}}
  {|\alpha|^{4}}\right].\label{a7}\end{equation}
From Eq.~\ref{a7}, $|\delta v_{x}|_{{\rm max}}$ is zero at $t=0$ and is
proportional to $t^{-4}$ as $t\rightarrow \infty$, while from Eq.~\ref{a5},
$v_{bx}$ is non-zero at $t=0$ and is proportional to $t^{-1}$ as
$t\rightarrow \infty$. Therefore it is possible to have a value of
$\lambda$ sufficiently small so that $|\delta v_{x}|_{{\rm max}}<v_{bx}$
for all times $t\geq0$. In that case, $v_{blx}$ would be positive
everywhere on $S_0$ for all $t\geq0$, and so the values of
$P(t)$ in this Bohm-like theory and in the standard Bohmian theory
would agree.

Now let me take $\lambda$ to be sufficiently large so that
{$|\delta v_{x}|_{{\rm max}}>v_{bx}$ for some time $t>0$.
This means that, for some values of $Y$, $Z$, and $t$,
\mbox{$v_{blx}(0,Y,Z,t)<0$},
which implies that some members of the Bohmian
ensemble are returning from $S_+$ to $S_-$.
Let $t_r$ be within an interval of time in which this return
is occurring. At any time $t$, the fraction of the ensemble in $S_+$
equals $Q(t)$, but at $t_r$ there is an additional fraction of
``returned'' members, which are in $S_-$ at $t_r$ but were in $S_+$
at some time prior to $t_r$.  This means that $P(t_{r})$ (which is the
total fraction of ensemble members that were in $S_+$
at {\em any} time $t'\leq t$) must be greater than $Q(t_{r})$.

To be certain of this conclusion, we must show that, of the ensemble
members which returned from $S_+$ prior to $t_r$, at least
a finite fraction still are in $S_-$ at $t_r$.  Let $D$ be an open, bounded
region of the  plane $S_0$, such that at every point of $D$ and for an
interval of time around $t_r$, $v_{blx}$ is negative; such a region
must exist, if $\lambda$ is sufficiently large.  For sufficiently small
$\epsilon $, it must be possible to find a subset $D_{\epsilon }\subset D$
such that the distance between any point in $D_{\epsilon}$ and any point
on $S_0$
not in $D$ is at least $\epsilon$. Now it can be shown that, for Y and Z
bounded, the magnitude of the component of ${\bf v}_{bl}$ parallel to
$S_0$ is bounded, independently of $X$ and $t$; call such a
bound $|v_{\parallel}|_{{\rm max}}$.  Thus any member of the ensemble
which returns to  $S_-$ through $D_{\epsilon}$ must spent at least an
amount of time $\tau =\epsilon/|v_{\parallel}|_{{\rm max}}$ in $S_-$
(because it takes at least time $\tau$ for it to clear the region $D$).
Thus all members of the ensemble which return to $S_-$ through
$D_{\epsilon}$ in the time interval $[t_{r}-\tau ,\;  t_{r}]$
will still be in $S_-$ at time $t_r$.

We therefore see that, with the wave function as given in Eq.~\ref{e2},
the Bohm-like theory defined by Eq.~\ref{a3} with a sufficiently-large
value of $\lambda $ will imply that $P(t)>Q(t)$, for some values of $t$.
Since with this wave function the standard Bohmian theory gives
$P(t)=Q(t)$ for all $t$, we conclude that these two theories can
give different arrival-time distributions $P(t)$.

\section{Discussion}
The choice for $\delta {\bf v}$ made in Eq.~\ref{a3} does not respect many
of the conditions set out by Deotto and Ghirardi \cite{DGh}. For example,
the cross product of two vectors is a pseudo vector, although a velocity
must of course be a true vector.  To take this choice seriously, one would
have to say that Eq.~\ref{a3} is only valid in a particular coordinate
system; if you want to know  $\delta {\bf v}$ in some other coordinate
system, use Eq.~\ref{a3} to calculate it in the particular system, and
then transform.  Deotto and Ghirardi require that there not be any
preferred coordinate system; while this requirement is certainly quite
reasonable, it is not, strictly speaking, necessary.  As long as the
Bohm-like theory reproduces the observational consequences of standard
quantum theory, the preferred coordinate system remains hidden; its
existence can be neither confirmed nor refuted by any experimental result.

Still, Deotto and Ghirardi, and in a different way Holland \cite{Holland},
have shown that is is possible to formulate a Bohm-like theory which is
considerably more plausible than the one defined by Eq.~\ref{a3}.
It is certainly an
important question for the program of studying Bohmian theories, to
judge which of the possible alternatives for the particle velocity
is the most plausible.  However, plausibility is not an issue we must
be concerned with here. One could certainly criticize the calculations
presented here, because of the implausibility of the choice
(\ref{a3}) or for that matter because the condition $Q(0)=0$ is not strictly
satisfied.  The calculation presented here does have the virtue of
simplicity, and it is hard to believe that the result obtained
(that $P(t)$ differs from that implied by the standard Bohmian theory)
is an artifact either of the transformation properties of Eq.~\ref{a3}
or of the (arbitrarily small) tail of the initial wave function.
Rather, this result gives one confidence to conjecture that for
{\em any} Bohm-like theory (with non-trivial  $\delta {\bf v}$)
there exists an example of a wave function with $Q(0)=0$ exactly,
for which that theory and standard Bohmian theory yield different $P(t)$.

The example presented here does not imply any additional ambiguity
within the Bohmian program, beyond that already recognized in \cite{DGh}
and \cite{Holland}.  It is obvious that, when theories make different
choices for $\delta {\bf v}$, there will be some quantities for which
those theories will imply different results. What this example does show
is that such theories will differ on a quantity, namely the distribution
of arrival times, that one might have hoped would be definable strictly in
terms of the wave function, and so would be independent of any particular
completion of standard quantum theory.

To discuss this matter further, let us consider to what extent the
demonstration by
Leavens \cite{Leavens}, that results from Bohmian theory
disagree with those from the theory of Grot, Rovelli, and Tate \cite{GRT},
should be counted as evidence against the latter theory.   If
one is committed to believing in the truth of Bohmian theory, one will
consider as incorrect anything which disagrees with it.
But even if one has no such commitment, one might be at least suspicious
of any result expressed solely in terms of the wave function
which disagrees with the Bohmian result, not because the
Bohmian theory is {\em necessarily} correct, but just because it
{\em might} be. Certainly there are quantities which can be precisely
defined and calculated within Bohmian theory, to which standard quantum
theory assigns no meaning.  However, to the extent that
Bohmian theory can be considered a completion of standard quantum theory,
one might expect that, for any quantity that {\em can} be calculated
within standard quantum theory, the Bohmian calculation would agree.
Similarly, one might expect any quantity calculable from the
wave function to agree with {\em all} completions (including any
that might be judged implausible). As we have seen, for the distribution
of arrival times this is not possible.

Of course, if one asks about the results of a particular experiment designed
to measure times of arrival, quantum theory should be able to
give an unambiguous answer, and Bohmian theory as well as any Bohm-like
theory should agree with that answer.  The issue we are considering is
whether that answer can be stated, within standard quantum theory, in a
way which is independent of the particular way in which the arrival times
are to be measured.  In standard quantum theory, no result is meaningful
unless it is measured; the quantity $Q(t)$ defined in Eq.~\ref{ez} must be
interpreted as the probability that the particle be found in $S_+$
at time $t$, rather than the probability that it {\em is}
there.  Nevertheless, we do not have to consider the particular way in
which the particle's position is measured; in terms of Bohm-like theories,
we can say that they all must agree on the quantity $Q(t)$.  One might
have thought that $P(t)$ would enjoy the same status; after all, $P(t)$ is,
roughly speaking, like the disjunction of $Q(t')$ for $0\leq t' \leq t$.
Unfortunately, a determination of position at one time will disturb
the determination at any other time, and different Bohm-like
theories, while constrained to have identical ensembles of positions
at any one time, differ precisely because they have different trajectories.

The ambiguity in the time-of-arrival distribution
revealed by the example discussed
here would not be present in a one-dimensional example.  The
analogue of Eq.~\ref{a2} for one dimension, namely $\partial(|\Psi|^{2}
\delta v)/\partial x=0$, together with suitable boundary conditions,
would require $\delta v=0$. One would expect that general arguments for
or against the definability of the arrival-time distribution, such as
those quoted at the beginning of this paper, would be equally cogent
in one and in three dimensions. It does seem, however, that there
is an ambiguity in three dimensions which is not apparent if one only
considers one-dimensional examples.  Perhaps one dimension is
misleadingly simple; one may be tempted by the isomorphism between
configuration space and temporal space to ignore the special role
played by time in non-relativistic quantum mechanics.

The discussion above has been within the context of Bohmian and
Bohm-like theories, but many of the same points can be put entirely
within the context of standard quantum theory, by considering the
quantum probability current (to be denoted ${\bf J}$).
As Squires has pointed out \cite{Squires}, the freedom to chose
alternative expressions for the velocity in Bohm-like theories is
a direct reflection of the under-determination of the quantum probability
current in more than one spatial dimension. Mielnik \cite{Mielnik}
suggested that a reasonable first guess for a time-of-arrival
density would be the component of ${\bf J}$ normal to the arrival
surface; in our case this would mean identifying $J_{x}(x=0,y,z,t)$
as the probability for arriving at the point $(0,y,z)$ at time $t$.
Mielnik then went on to show that this could not be correct in general,
since there must exist examples in which this component becomes negative.
It is sometimes suggested (for example, in \cite{DM}) that $J_{x}$
does indeed give the correct arrival time density, in those cases in
which it is always positive.

Let ${\bf J}_c$ denote the customary form for the
quantum probability current  (which is just the product
of $|\Psi|^2$ with ${\bf v}_b$ which is given in Eq.~\ref{e4}); then
without now identifying  ${\bf v}_b$ as the velocity of anything, we
can see from Eq.~\ref{e9} that for the wave function given in Eq.~\ref{e2},
$J_{cx}$ is indeed positive everywhere on the  plane $S_0$.  So, if we
follow the above suggestion, we would say that $J_{cx}(x=0,y,z,t)$
does indeed give the arrival-time density for this wave function.

Now define ${\bf J}_{l}$ to be $(|\Psi|^{2}{\bf v}_{bl})$,
where ${\bf v}_{bl}$
is given by Eqs.~\ref{a1} and \ref{a3} (and also need not be identified
as the velocity of anything). Then from Eqs.~\ref{e5} and \ref{a2} it follows
that
\begin{equation}
  {\bf \nabla}\cdot {\bf J}_{l}=-\frac{\partial |\Psi|^{2}}
  {\partial t}, \end{equation}
which means that we can, if we wish, violate custom and call
${\bf J}_{l}$ (instead of ${\bf J}_{c}$) the quantum probability current.
So we might as well say that $J_{lx}$ gives the arrival-time density,
in those cases in which $J_{lx}$ is always positive.

The calculations of the previous section show that, for the wave function
given in Eq.~\ref{e2}, if $\lambda$ happens to be small enough,
then $J_{lx}$ is positive everywhere on $S_0$. So, for small
enough $\lambda$, the two possibilities for the probability current
(${\bf J}_c$ and ${\bf J}_{l}$) give us two possibilities for the
arrival-time density ($J_{cx}$ and $J_{lx})$ which disagree
with each other \cite{int}.  One certainly can make an arbitrary
choice between  ${\bf J}_c$ and ${\bf J}_{l}$, that is, one can pick
either one of them and choose to define that one to be
the probability current. That choice, however, does not have any experimental
implications---no experimental result can depend upon which definition
one happens to make---so it would not make sense to expect the choice
to be either confirmed or refuted by any experiment.  More generally,
one certainly can (and sometimes one does \cite{GRT}) identify some
quantity which can be calculated purely in terms of the wave function,
and choose to call that quantity a time-of-arrival distribution.
One can then discuss the question of whether this quantity does
have \cite{GRT} or does not have \cite{ORU} properties that one might
intuitively expect such a distribution to have.  However, one should not
then expect that any experiment will confirm the felicity of that choice.

\section*{Acknowledgement}
I  acknowledge the hospitality of the
Lawrence Berkeley National Laboratory.

\vspace{2cm}


\section{Introduction}
\label{intro}
The hard X-ray transient GRO\,J0422+32\ was discovered by \GRO/BATSE on
1992 Aug 5 \cite{P92} at a flux of 0.2 Crab.  It rapidly rose to a
level of 3 Crab, before beginning a gradual decline over subsequent
months.  The X-ray source was identified with a 13th magnitude optical
nova \cite{Castro93}, which showed a similar decline.  The later
stages of the outburst were characterised by an unprecedented series
of mini-outbursts.  Observations in the early stages of its discovery
suggested similarities to other black hole X-ray transient (BHXRT)
binaries; subsequent quiescent studies (Casares et al.\ 1995,
Filippenko et al.\ 1995, Orosz \& Bailyn 1995) have confirmed its
binary nature and suggested a compact object mass of
2.5--5.0\,M$_{\odot}$, larger than any known neutron star and probably
a black hole.  A more recent study has suggested that the mass may
actually be much greater than this \cite{B97}.  In some ways,
GRO\,J0422+32\ is the most extreme member of its class, and the most
similar to cataclysmic variables, having the shortest orbital period
(5.1-hr, Chevalier \& Ilovaisky 1995) and the latest type companion
star (M2V, Casares et al.\ 1995, Filippenko et al.\ 1995).

As part of an ongoing Target-of-Opportunity programme to observe BHXRT
s in outburst with The {\it Hubble Space Telescope}, we triggered our
proposal to observe the first of the mini-outbursts.  The observations
were delayed and actually took place two years after the initial
outburst.  The system was nonetheless still active at this point,
being two magnitudes above quiescence in the V band, and we believe we
caught the end of either an otherwise unreported mini-outburst, or an
extended plateau or standstill.

In Sects.~\ref{obs} \& \ref{datred} we describe the observations we
made; they are placed within the context of the long-term light curve
in Sect.~\ref{LongLCSect} and analysed in Sects.~\ref{ContSpecSect} \&
\ref{LineSpecSect}.  Our conclusions are summarised in
Sect.~\ref{conc}.
\section{Observations}
\label{obs}
\label{hubobs}
\HST\ observed GRO\,J0422+32\ using the Faint Object Spectrograph (FOS) for
two visits, comprising a total of 11 spacecraft orbits, on 1994~August
25 and 26.  The exposures we took are listed in Table~\ref{obstab}.
To observe the vacuum UV, we invested the majority of our time in a
series of G160L exposures, interleaving these with short PRISM
exposures to capture the optical spectrum.  The G270H and G400H
spectra provide coverage overlapping that of the PRISMs at higher
spectral resolution.  All the observations used the `4.3' FOS aperture
and COSTAR.  They were obtained in RAPID mode, with a duty cycle of 77
per cent, and used standard sub-stepping, giving four pixels per
diode.
\begin{table*}
\caption{Journal of \HST\ Faint Object Spectrograph observations of
         GRO\,J0422+32.  The blank line indicates the break between the two
         visits.  The exposure times given exclude dead time, hence
         include only the 77 per cent of the time elapsed.}
\label{obstab}
\begin{tabular}{lcclrc}
                                &
\multicolumn{1}{c}{Date}        &
\multicolumn{1}{c}{UT Start}    &
\multicolumn{1}{c}{Disperser /} &
\multicolumn{1}{c}{Integration} &
\multicolumn{1}{c}{Wavelength}  \\
\multicolumn{1}{c}{ID}          &
\multicolumn{1}{c}{1994}        &
\multicolumn{1}{c}{(hh:mm)}     &
\multicolumn{1}{c}{Detector}    &
\multicolumn{1}{c}{Time (s)}    &
\multicolumn{1}{c}{Range (\AA)}  \\
\hline
0104 & Aug 25 & 21:21 & PRISM/RD &    365.4 & 1850--8950\,\AA \\
0105 & Aug 25 & 21:34 & G270H/RD &   1471.1 & 2222--3277\,\AA \\
0106 & Aug 25 & 22:56 & PRISM/RD &    365.4 & 1850--8950\,\AA \\
0107 & Aug 25 & 23:09 & G400H/RD &   1551.8 & 3235--4781\,\AA \\
0108 & Aug 26 & 00:38 & PRISM/BL &    365.4 & 1500--6000\,\AA \\
0109 & Aug 26 & 00:51 & G160L/BL &   1124.7 & 1140--2508\,\AA \\
010a & Aug 26 & 02:10 & G160L/BL &   2273.2 & 1140--2508\,\AA \\
010b & Aug 26 & 04:08 & PRISM/BL &    365.4 & 1500--6000\,\AA \\
\\
0204 & Aug 26 & 15:24 & PRISM/BL &    365.4 & 1500--6000\,\AA \\
0205 & Aug 26 & 15:38 & G160L/BL &    365.4 & 1140--2508\,\AA \\
0206 & Aug 26 & 16:37 & G160L/BL &   2273.2 & 1140--2508\,\AA \\
0207 & Aug 26 & 18:14 & G160L/BL &   1120.0 & 1140--2508\,\AA \\
0208 & Aug 26 & 18:43 & PRISM/BL &    365.4 & 1500--6000\,\AA \\
0209 & Aug 26 & 18:56 & G160L/BL &    142.4 & 1140--2508\,\AA \\
020a & Aug 26 & 19:51 & G160L/BL &   2273.2 & 1140--2508\,\AA \\
020b & Aug 26 & 21:27 & G160L/BL &   1480.6 & 1140--2508\,\AA \\
020c & Aug 26 & 22:01 & PRISM/BL &    365.4 & 1500--6000\,\AA \\
\hline
\end{tabular}
\end{table*}
\section{Data Reduction}
\label{datred}
Our \HST\ observations found GRO\,J0422+32\ in the faintest post-outburst
state then observed spectroscopically. A preliminary appraisal of the
original pipeline-calibrated data revealed an extremely noisy
spectrum, with only the 2800\,\AA\ \MgII\ emission line detected in
the UV spectrum.  Since the De-commissioning of the FOS, a
substantially revised set of calibration files has become available
\cite{K97}, and we therefore performed a re-reduction and calibration
of our data.  The pipeline reduction is also known to underestimate
the background counts significantly \cite{K95}, necessitating manual
subtraction.
\subsection{G160L Recalibration}
\label{g160lrecal}
Our average G160L spectrum is shown in Fig.~\ref{G160LFig}.  This
grating disperses the 1140--2508\,\AA\ spectrum to pixels 1275--2063.
Zeroth order light falls on the detector at around pixels 600--670.
Elsewhere in the array, the pixels monitor the background count-rate.
A cursory inspection of our data revealed that the pipeline reductions
had underestimated the background level by approximately a factor of
two.

\begin{figure}
\begin{center}
\epsfig{width=2in,height=3in,angle=90,file=fig1a.ps}
\epsfig{width=2in,height=3in,angle=90,file=fig1b.ps}
\epsfig{width=2in,height=3in,angle=90,file=fig1c.ps}
\caption{a) Raw count rates in our averaged G160L spectrum, over the
         whole detector, smoothed with a 5 pixel boxcar.  We see the
         zeroth order spectrum centred at pixel 640.  The first order
         spectrum runs from pixels 1275--2063.  We clearly see the
         geocoronal Ly$\alpha$ emission around pixel 1320.  We also
         show with the smooth solid line the model background
         calculated based upon the geomagnetic position, and with the
         dashed lines the same scaled up to match the unexposed
         regions (MHR95).  The dotted line shows the model background
         with a fixed (wavelength independent) constant added (KB93).
         The adopted unexposed regions is indicated. b) Rebinned
         ($\times16$) count rates after subtraction using the MHR95
         recipe. c) Rebinned count rates after subtraction using the
         KB93 recipe.}
\label{G160LFig}
\end{center}
\end{figure}
McClintock, Horne \& Remillard (1995, hereafter MHR95) and Kinney \&
Bohlin (1993, hereafter KB93) disagree about the source of the extra
background light.  MHR95 adopted the hypothesis that it is due to
cosmic rays, whereas KB93 assert that it is due to scattered light.
This disagreement is significant because the {\it shape} of the extra
background spectrum has an effect on the UV continuum flux attributed
to the target.  We believe that in our case the background is
dominated by particle events, in line with the MHR95 interpretation,
for the following reasons:
\begin{enumerate}
\item
MHR95 explicitly ruled out scattered {\em optical} light in their
G160L observations of A0620--00.  At the time of our observations,
GRO\,J0422+32\ was two visual magnitudes fainter than the quiescent
magnitude of A0620--00, thus scattered optical light from the source is
surely also negligible in our case.
\item
There is very little UV flux from the source so scattering of this is
utterly implausible.
\item
The correction needed for the PRISM/BL spectrum, which used the same
detector, but a different disperser, was of comparable size to that
for G160L.  This suggests that the background level is a property of
the detector, independent of which disperser is used and so supports a
model based on particle events rather than scattered light.  This is
in contrast to the \HST/FOS observations of the highly reddened BHXRT\
GRO\,J1655--40\ \cite{Hy98} which probably were influenced by scattered red
light leading to G160L corrections significantly greater than for the
other dispersers.
\end{enumerate}

The one remaining possible source of background counts is scattered
geocoronal light.  Since we used a larger aperture than did MHR95,
this cannot readily be ruled out, but argument (iii) suggests it does
not dominate.

In our grand sum G160L exposure, the background level, determined from
pixels 900--1199, was $71\pm2$ per cent higher than the pipeline
value, as shown in Fig.~\ref{G160LFig}.  The panels below show the
rebinned spectrum obtained by subtracting the revised backgrounds:
Fig.~\ref{G160LFig}(b) uses the MHR95 background recipe,
Fig.~\ref{G160LFig}(c) uses KB93's recipe.  At the left of these
figures can be seen the background subtracted count-rate for the first
546 pixels in the array. These pixels are exposed to background only,
but were not used in the calculation of the background level.
Therefore they provide an assessment of the accuracy of the revised
background.  For both recipes we see significantly lower counts in
this region than from the source region at the right, (pixels
1400--2000) therefore we cautiously conclude that we have detected
vacuum UV emission from GRO\,J0422+32\ in early quiescence.  The revised
background subtraction shown in panel (b) gives a mean count rate per
pixel in pixels 0--545 of $(2.8\pm0.9)\times10^{-4}$\,s$^{-1}$, about
half that in panel (c), $(5.4\pm0.9)\times10^{-4}$\,s$^{-1}$.  While
the particle model seems to give a better estimate of the shape of the
background count rate than a scattered light model, neither model
gives a mean count in pixels 0--545 within $3\sigma$ of zero so we
should view these results with considerable caution, as this may
indicate a deficiency in {\em both} models.
\subsection{PRISM Recalibration}
\label{PrismRecalSect}
As for G160L, for both PRISM configurations (BL and RD) we combined
exposures into a grand sum before attempting to characterise the
background.  Both PRISM configurations give many pixels exposed to
background only: we used pixels 1400--1999 for PRISM/BL and 100--899
for PRISM/RD.  For the former case, the observed background count rate
was higher than the background model by $63\pm3$ per cent, a similar
value to that found with G160L which also used the BL detector.  For
PRISM/RD no significant correction was needed; the background model
background was $2\pm3$ per cent below that observed.

For the PRISM/RD combination, there appeared to be an offset in the
dispersion direction, probably due to miscentering of the target in the
aperture.  We detected this by comparison of the position of the
\MgII\ (2798\,\AA) line with its position in the higher resolution
G270H spectrum.  The offset was the same for both PRISM/RD exposures,
so the miscentering was probably also the same for the G270H exposure
taken between them.  We were able to correct the offset by shifting
both G270H and PRISM/RD spectra by 12 pixels after flat-fielding.
This also brought the H$\alpha$ and H$\beta$ emission lines in the
PRISM/RD spectrum to within one pixel of their rest wavelengths.  Since
the PRISM/BL spectrum contains no prominent emission lines (the \MgII\
line is undetected), we cannot determine if there is a similar offset
in this case.  This makes our calibration of the PRISM/BL spectra less
certain than PRISM/RD.  Since the BL detector is also less sensitive
than the RD, we focussed on the PRISM/RD spectrum in our model
fitting.

We note that a 12 pixel error would correspond to 4 diodes or 1.22
arcsec in the mode used.  Since the aperture width was 4.3 arcsec, the
stellar image should still have been $\sim1$ arcsec, or $10\sigma$,
from the aperture edge and so aperture losses will not significantly
affect the accuracy of the spectrophotometry.  We also note that the
revised calibration files \cite{K97} made a significant difference for
the extreme long wavelength end of the PRISM/RD spectrum, with the
flux longwards of 7500\,\AA\ being increased by 20--80 per cent.  Our
recalibrated G160L and PRISM/RD spectra, spanning 1300--9000\,\AA\ are
shown in Fig.~\ref{CompSpecFig}.
\begin{figure}
\begin{center}
\epsfig{width=2in,height=3in,angle=90,file=fig2.ps}
\caption{G160L and PRISM/RD spectra.  The G160L spectrum has been
         averaged into 100\,\AA\ bins.  Also shown dotted is our
         estimate of the spectrum of the companion star (see
         Sect.~\ref{SecSect}), reddened by $E(B-V)=0.3$.}
\label{CompSpecFig}
\end{center}
\end{figure}
\subsection{G270H and G400H Recalibration}
\label{G270HRecalSect}
Because of the high dispersion, the count rates are relatively low and
the background is a serious problem for both G270H and G400H.  The
problem is exacerbated because these gratings both completely fill the
diode array with the source spectrum, leaving no region to monitor the
background.  Comparing these spectra with those from the PRISMs, the
pipeline does seem to have subtracted the background reasonably well,
but we cannot attempt a precise recalibration of the high resolution
spectra or quantify the uncertainty in the background level.  We
therefore use the G270H and G400H spectra primarily to search for and
examine spectral lines.

Our combined G270H and G400H spectrum is shown in Fig.~\ref{HiResFig}.
For clarity it has been convolved with a Gaussian of the same width as
the estimated line spread function, i.e.\ one diode.  This corresponds
to 2.05\,\AA\ for the G270H grating and 3.00\,\AA\ for the G400H.  The
smoothed spectrum was then rebinned to give one pixel per resolution
element.  The spectrum is largely featureless with only the \MgII\
(2798\,\AA) emission line unambiguously detected.
\begin{figure}
\begin{center}
\epsfig{width=2in,height=3in,angle=90,file=fig3.ps}
\caption{Combined G270H and G400H spectrum.  These have been convolved
         with a Gaussian of width equal to the line spread function
         and rebinned for clarity.  The spectrum is largely
         featureless, with the \MgII\ (2798\,\AA) emission line the
         only prominent feature.}
\label{HiResFig}
\end{center}
\end{figure}
\section{The Photometric State of GRO\,J0422+32}
\label{LongLCSect}
The R band light curve of the outburst of GRO\,J0422+32\ is shown in
Fig.~\ref{PhotFig}.  On 1994 September 5, approximately 10 days after
our observations, Zhao et al.\ \shortcite{Z94b} measured magnitudes of
$V=22.39\pm0.27$, $R=21.06\pm0.10$.  The system remained at this level
and this is representative of its subsequent quiescent state
($V=22.24\pm0.14$, $R=20.97\pm0.10$, Casares et al.\ 1995.)  We can
also calculate approximate magnitudes from our PRISM/RD spectrum
however, and obtain $U=20.75\pm0.04$, $B=20.84\pm0.03$,
$V=20.42\pm0.04$, $R=20.08\pm0.04$, brighter by one magnitude in the R
band and two magnitudes in V.  This is a significant difference, and
it is very unlikely that the \HST\ calibration could be this much in
error; our error estimates include statistical errors, the documented
3--4 per cent photometric accuracy of the FOS \cite{K97} and allow for
a miscentering uncertainty $\pm1$ pixel.  We conclude that GRO\,J0422+32\
was not completely quiescent at the time of our observations.
Instead, we suggest that we saw the final stages of a last, previously
undetected, mini-outburst.  If this is the case, then this
mini-outburst occurred approximately 240 days after the last recorded
one, consistent with the recurrence time of 120 days originally
suggested for the 1993--94 mini-outbursts (Augusteijn, Kuulkers \&
Shaham 1993, Chevalier \& Ilovaisky 1995) and subsequently detected in
quiescence (Iyudin \& Haberl 1997a,b).  An alternative interpretation
is that after the last observations around day 600, when Zhao et
al.\ \shortcite{Z94a} measured $R=20.03\pm0.11$, $V=20.67\pm0.22$, the
system remained on a plateau, finally dropping to quiescence just
after our observation.  This is plausible, as both R and V band
magnitudes around day 600 are similar to values we observe.
\begin{figure}
\begin{center}
\epsfig{width=2in,height=3in,angle=90,file=fig4.ps}
\caption{The R band lightcurve of the outburst, adapted from Fig.\ 1
         of Garcia et al.\ (1996).  Vertical arrows indicate observed
         or extrapolated times of mini-outburst.  Our measurement
         clearly lies significantly above the subsequent photometry,
         and is consistent with the expected time of a mini-outburst.}
\label{PhotFig}
\end{center}
\end{figure}
\section{The Continuum Spectrum}
\label{ContSpecSect}
\subsection{Interstellar Reddening}
Due to the low signal-to-noise ratio and uncertain background level of
our G160L spectrum, we cannot reliably attempt our own reddening determination
using the 2175\,\AA\ feature.  Existing reddening estimates are
summarised in Table \ref{ReddenTable}.  We adopt a compromise value of
$E(B-V)=0.3\pm0.1$ and deredden our spectra using the galactic average
extinction curve of Seaton \shortcite{S79}.  Since the only UV
determination lies at the extreme end of the spread of estimates,
however, it may be that the extinction curve actually diverges from
the galactic average; our dereddened UV spectrum should therefore be
viewed with some caution.
\begin{table}
\caption{Estimates of the reddening of GRO\,J0422+32.  References are 1.\
         Shrader et al.\ \shortcite{S94}, 2.\ Callanan et al.\
         \shortcite{C95}, 3.\ Harlaftis \& Charles \shortcite{HC93},
         4.\ Chevalier \& Ilovaisky \shortcite{CI95}, 5. Castro-Tirado
         et al.\ \shortcite{Castro93}}
\label{ReddenTable}
\begin{tabular}{llc}
Method                   & Value         & Ref. \\
\hline
2175\,\AA\ feature       & $0.40\pm0.06$ & 1.\\
EW of interstellar lines & $0.2\pm0.1$   & 1.\\
                         & $0.3\pm0.1$   & 2.\\
                         & $0.2$         & 3.\\
Surrounding field stars  & $0.40\pm0.07$ & 4.\\
Optical continuum shape  & $0.23\pm0.02$ & 1.\\
Unstated                 & $0.3\pm0.1$   & 5.\\
\hline
\end{tabular}
\end{table}
\subsection{The Companion Star Contribution}
\label{SecSect}
Since GRO\,J0422+32\ was near quiescence when we observed it, the spectrum
of the companion star can be expected to be a significant fraction of
the total, at least at
longer wavelengths.  Unfortunately, the resolution of the PRISM/RD
spectrum at long wavelengths is too low to detect spectral features
from the companion, so we cannot perform an independent analysis of
veiling to estimate how strong the contribution is.  Instead we adopt
results from the literature.

For the spectral type of the companion star we assume M2V as favoured
by Casares et al.\ \shortcite{C95} and Filippenko et al.\
\shortcite{F95} and use the spectrum of the M2V star GL49 \cite{GS83}
as the template for our companion contribution.  Casares et al.\
\shortcite{C95} give an R band (quiescent) magnitude of $20.96\pm0.10$
and estimate the fraction of the light due to the secondary to be
$52\pm8$ per cent in the 6700--7500\,\AA\ range.  Their results are
consistent with those of Filippenko et al.\ \shortcite{F95}.  We
therefore estimate an R band magnitude for the companion star alone of
$21.68\pm0.11$ and normalise the spectrum of GL49 accordingly; the
resulting spectrum, reddened by $E(B-V)=0.3$, is shown in
Fig.~\ref{CompSpecFig}.  We then subtract this normalised spectrum,
smoothed to the instrumental resolution of $1\,{\rm diode}=4\,{\rm
pixels}$ \cite{K97}, to obtain the spectrum of the accretion flow
shown in Fig.~\ref{SynchFitFig}.

\begin{figure}
\begin{center}
\epsfig{width=2in,height=3in,angle=90,file=fig5.ps}
\caption{Dereddened PRISM/RD spectrum with the companion contribution
         subtracted.  Points with error bars correspond to the
         $E(B-V)=0.3$ solution; the effect of instead assuming
         $E(B-V)=0.2$ or 0.4 is indicated by dotted lines.  The data
         below 4000\,\AA\ ($\log\nu>14.88$) has been averaged into
         25\,\AA\ bins for clarity.  The large error bars at low
         frequencies are systematic and arise from assuming a $\pm1$
         pixel uncertainty in the miscentering correction.  The best
         fitting black body (dashed) and self-absorbed synchrotron
         (solid) spectra are overplotted.}
\label{SynchFitFig}
\end{center}
\end{figure}

GRO\,J0422+32\ presents a unique opportunity to study the spectrum of the
quiescent or near quiescent accretion flow in a BHXRT.  Because the
companion star is an M-type dwarf, rather than the more typical K, G
or even F-type subgiant, it is redder than in the other systems and
contaminates the optical spectrum much less.  Furthermore the late
type companion shows very strong molecular bands, so its spectral type
and brightness can be better determined, and so the uncertainty in the
contamination is smaller.  The deduced spectrum of the accretion flow
is therefore much less affected by uncertainties in the companion's
contribution.
\subsection{Black Body Models}
We began by trying to characterise the spectrum of the accretion flow
with a black body fit.  The best fitting black body spectrum (in the
$\chi^2$ sense) is shown by the dashed line in Fig.~\ref{SynchFitFig}.
This has $T \sim 7600$\,K and a reduced $\chi^2$ of 3.6 after masking
out emission lines.  It is clear that this is not a satisfactory fit
and allowing for a $\pm0.1$ uncertainty in $E(B-V)$ does not resolve
this.  The observed spectrum is sharper peaked than a black body and
tends to a power-law form at short wavelengths.  Adopting a
temperature distribution in a disc instead of a single temperature
black body can produce a power-law at short wavelengths, but will
always produce a broader spectrum than a single temperature model.  In
principle, a thermal spectrum from an accretion disc might be expected
to deviate from a black body, but we would expect deviations in the
form of edges or structure into the spectrum, in particular a Balmer
jump in emission or absorption.  Apart from the sharp, and uncertain,
upturn below 2500\,\AA\ (Fig.~\ref{CompSpecFig}), there is no evidence
for this.  We encountered similar problems in trying to fit the
spectrum of GRO\,J1655--40\ in outburst, which also seemed too sharply
peaked to be fit by a black body model \cite{Hy98}.  There, however,
the situation was complicated by the much larger reddening.  In this
case we have much less uncertainty in isolating the intrinsic spectrum
of the accretion flow, and hence we can say with more confidence that
it cannot be modelled by purely thermal emission.
\subsection{Non-thermal Models}
\label{AdvectionSect}
A more promising explanation for the observed spectrum is that it is
non-thermal in origin: self-absorbed synchrotron emission.  Such a
mechanism can readily produce a spectrum with a sharper peak than a
black body and, as illustrated in Fig.~\ref{SynchFitFig}, can fit the
spectrum rather well.  The spectrum plotted is based on a free-fit of
a synchrotron emitting slab containing a magnetic field of uniform
strength and electrons with a power-law energy spectrum.  This fit is
intended purely to demonstrate that even a very simple synchrotron
emitting model can fit the accretion spectrum well.  More realistic
models of self-absorbed synchrotron emission in X-ray binaries invoke
thermal electron distributions, but over the limited spectral range
covered, a power-law model leads to a reasonable approximation to the
expected spectrum.

We suggest two more sophisticated models that could lead to such a
spectrum.  The first is that the accretion disc extends inwards close
to the compact object but has a magnetically dominated corona above
it.  Such a model was considered by Di Matteo, Celotti \& Fabian
\shortcite{DMCF97} in the context of both active galactic nuclei and
galactic black hole candidates.  In their model there are at any time
a number of localised active coronal regions in which electrons are
energised by magnetic reconnection (a possible mechanism for this is
described by Haswell, Tajima \& Sakai 1992) and subsequently, cool
emitting cyclo-synchrotron radiation which becomes self-absorbed below
some critical frequency.  These models predict that for solar mass
black holes, the critical frequency, and hence the peak of the
spectrum, should lie in the extreme UV, i.e.\ at higher frequencies
than we observe.  We are observing GRO\,J0422+32\ in a much lower state
than was considered by Di Matteo et al.\ \shortcite{DMCF97}, however,
and this could move the peak to lower energies.

An alternative set of models, which have seen much discussion
recently, are the so called advective accretion flows.  These have
been put forward as an explanation of many properties of BHXRT s in
quiescence (Narayan, McClintock \& Yi 1996, Hameury et al.\ 1997,
Narayan, Barret \& McClintock 1997), and subsequently the models have
been extended to consider the outburst and decline phases
\cite{EMN97}.  The essence of these models is that the inner region of
the disc becomes evaporated into a geometrically thick, optically thin
region.  This flow radiates less efficiently than an optically thick disc
and so there is a bulk inflow of thermal energy (i.e.\ the flow is
advective), ultimately through the event horizon.  The models predict
that the optical and UV accretion light should be dominated by
self-absorbed synchrotron emission from this advective inner region.  Since our
spectrum was obtained at a very low luminosity, shortly before
GRO\,J0422+32\ reached its quiescent state, we should consider if these
models agree with our observations.  A detailed comparison is beyond
the scope of this paper, requiring some refinement of the models, but
a preliminary analysis appears promising (A. Esin, priv.\ comm.).  Both
the frequency of the predicted synchrotron peak and its luminosity are
strikingly consistent with our observations.  We therefore suggest it
is possible that the optical continuum emission at the time of our
observations was dominated by an advective flow with mass transfer
rate somewhat above quiescence.
\subsection{The Far Ultraviolet Spectrum}
As discussed in Sect.~\ref{g160lrecal}, we appear to have detected
source counts in the far UV from GRO\,J0422+32.  For a range of extinctions
$E(B-V)=0.2-0.42$, our average flux over the range 1650--1950\,\AA, is
in the range $f_{\lambda} = (0.4-1.9) \times
10^{-16}$\,erg\,s$^{-1}$\,cm$^{-2}$\,\AA$^{-1}$.  Normalised to what
would be seen at a distance of 1\,kpc (assuming a distance of 2.4\,kpc
for GRO\,J0422+32, Shrader et al.\ 1994), $f_{\lambda} = (3-11)
\times10^{-16}$\,erg\,s$^{-1}$\,cm$^{-2}$\,\AA$^{-1}$.  Uncertainties
both in the {\em shape} of the background spectrum and its
normalisation will increase this range and our results do not
conclusively rule out zero flux in the far UV.  Comparing this with
the normalised far UV fluxes tabulated by MHR95 for A0620--00\ and
selected dwarf novae, our measurement agrees with the latter, but is
ten times greater than in A0620--00.  This may be because GRO\,J0422+32\ was
not completely quiescent at the time of our observation, but since our
detection is marginal, the disagreement may not be significant.  Our
deduced far UV spectrum is strongly dependent on the exact form and
level of background spectrum adopted.  The data indicate a sharp
upturn in the UV, however the reliability of this is far from certain
(Fig.\ \ref{G160LFig}).  If it is indeed a real upturn, it would not
readily come from advective models in their current form (e.g.\ Esin
et al.\ 1997).  While these do predict an upturn due to Comptonisation
of the synchrotron peak, this is expected to occur at shorter
wavelengths and to be smoother than we seem to observe.
\section{Emission Lines}
\label{LineSpecSect}
\begin{table}
\caption{Dereddened fluxes and full width half maxima of detected
         emission lines.  The measured width of H$\alpha$
         is comparable to the width of the line spread function (LSF),
         so is not meaningful.  The fit to H$\beta$ was very poor.}
\label{LineTable}
\begin{tabular}{lccc}
 & Line flux                  & Line & \multicolumn{1}{c}{LSF} \\
 & (erg\,s$^{-1}$\,cm$^{-2}$) & FWHM (\AA) &
\multicolumn{1}{c}{FWHM (\AA)} \\
\hline
\MgII (2798) & $2.8 \pm 0.4\times10^{-15}$ & $18.5\pm2.4$ & 2 \\
H$\beta$          & $3.4 \pm 0.6\times10^{-15}$ & $240\pm50$   & 120\\
H$\alpha$         & $3.8 \pm 0.7\times10^{-15}$ & ($200\pm30$) & 270\\
\hline
\end{tabular}
\end{table}
In the optical, we detect unresolved H$\alpha$ and barely resolved
H$\beta$ emission in our PRISM/RD spectrum.  The only emission line
detected in the ultraviolet was \MgII\ (2798\,\AA), which was also
seen to be prominent in A0620--00\ (MHR95).  The deduced line
parameters are given in Table \ref{LineTable}.  The \MgII\ line is
well resolved by the G270H grating and has a FWHM in velocity space of
$2000\pm300$\,km\,s$^{-1}$, comparable to that observed in A0620--00.
We estimate that emission is detectable to velocities of $\sim
1500$\,km\,s$^{-1}$ (i.e.\ ${\rm FWZI}\sim3000$\,km\,s$^{-1}$.  This
is similar to the quiescent H$\alpha$ and H$\beta$ lines observed by
Orosz \& Bailyn \shortcite{OB95}, for which we estimate FWHM
1500\,km\,s$^{-1}$ and 1800\,km\,s$^{-1}$ respectively, with FWZI
$\sim 3000$\,km\,s$^{-1}$ in both cases.

If we assume the system parameters of Filippenko et al.\
\shortcite{F95}, then at the circularisation radius in the disc we
would expect $v_{\phi} \sin i \sim 700$\,km\,s$^{-1}$.  The observed
velocities of the \MgII\ emission (and the quiescent H$\alpha$
emission) therefore suggest emission from a disc extending inwards to
at least one quarter the circularisation radius.  This is consistent
with the advective accretion models discussed in
Sect.~\ref{AdvectionSect}, which assume evaporation of the accretion
disc at a few times less than the circularisation radius \cite{EMN97},
but obviously this does not prove that the disc is evaporated.
\section{Summary and Conclusions}
\label{conc}
We have presented a study of GRO\,J0422+32\ near quiescence based on UV and
optical spectroscopic observations, and suggest that these took place
during the decline from a previously unreported mini-outburst.  Our
observations show similarities to other BHXRT s in {\em quiescence},
and in particular to A0620--00, in two respects:
\begin{enumerate}
\item
We find that this accretion spectrum cannot be well fitted by black
body models, but can be better characterised by a self-absorbed
synchrotron spectrum.  Two possible explanations have been advanced:
either a magnetically-dominated corona above a thin disc, or an
advection dominated flow.  Further modelling beyond the scope of this
paper will be required to discriminate between these possibilities.
\item
The emission line spectrum is dominated by H$\alpha$, H$\beta$ and
\MgII\ (2798\,\AA), as is found for A0620--00\ in quiescence.  The
\MgII\ width is similar to that in A0620--00, suggesting a similar
origin.
\end{enumerate}

The similarity on both counts to other quiescent BHXRT s suggests a
similar structure.  The brightness of GRO\,J0422+32\ at the time of our
observations can be readily explained if the mass transfer rate had
not completely dropped to its quiescent level.
\section*{Acknowledgements}
RIH is supported by a PPARC Research Studentship.  Thanks to our
referee, Tim Naylor, and the editor, Andy Fabian, for constructive
criticism and advice, and to Phil Charles, Erik Kuulkers and Tariq
Shahbaz for hospitality, and helpful suggestions and discussion on
this work.  The advice of Tony Keyes is appreciatively acknowledged,
as is the generously shared expertise of the STSDAS team at STScI.  We
thank Ping Zhao, Paul Callanan, Mike Garcia and Jeff McClintock for
obtaining photometry shortly after our observations and providing
their full set of photometry for inclusion in Fig.~\ref{PhotFig}.  We
used the NASA Astrophysics Data System Abstract Service and support
for this work was provided by NASA through grant number GO-4377-02-92A
from the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Incorporated,
under NASA contract NAS5-26555.