%Paper: 
%From: Antonio PICH <pich@papageno.ific.uv.es>
%Date: Tue, 4 Oct 94 11:42:24 MET

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\title{\vspace*{-3.2cm}
\begin{flushright}
{\rm\normalsize FTUV/94-50\\
IFIC/94-45\\ October 1994\\}
\end{flushright}
\vspace*{1.0cm}
QCD Analysis of Inclusive $\Delta S=1,2$ Transitions:
  The $|\Delta I|=1/2$ Rule*}

\author{A Pich}

\affil{Departament de F\'{\i}sica Te\`orica and IFIC,
Universitat de Val\`encia -- CSIC, \\
Dr. Moliner 50, E-46100 Burjassot, Val\`encia, Spain}

\abstract{The interplay of QCD in $\Delta S=1,2$ non-leptonic
weak transitions can be rigorously analyzed, at the inclusive
level, by studying the 2--point functions associated
with the corresponding $\Delta S=1,2$ effective Hamiltonians.
The next-to-leading order calculation of these correlators shows
a huge ($\gsim 100\%$) gluonic enhancement of the $|\Delta I|=1/2$
channel, providing a qualitative understanding
of the $|\Delta I|=1/2$ rule within QCD.}

\twocolumn[\maketitle]

\fnm{7}{Invited talk given at ICHEP '94, Glasgow, July 1994.}

\section{Introduction}

The origin of the empirically observed enhancement of
strangeness-changing non-leptonic weak amplitudes with isospin
transfer $|\Delta I|=1/2$ is a long-standing question
in particle physics.
The short-distance analysis of the product of weak hadronic currents
results in an effective $\Delta S=1$ Hamiltonian
%
\be\label{eq:hamiltonian}
{\cal H}^{\Delta S=1}_{\mbox{\rms eff}} \;
= \; {G_F\over\sqrt{2}}\,V_{ud}^{\phantom{*}} V_{us}^{*} \sum_{i}
C_{i}(\mu^2) \, Q_{i} \,,
\ee
%
which is a sum of local four-quark operators $Q_i$,
constructed with the light ($u,d,s$) quark fields only,
modulated by Wilson coefficients $C_i(\mu^2)$ which are functions
of the heavy ($t,Z,W,b,c$) masses and an overall renormalization
scale $\mu$.

In the absence of strong interactions, $C_2(\mu^2)=1$ and all other
Wilson coefficients vanish.
The operator $Q_2$ can be decomposed as
$Q_2 = (Q_+ + Q_-)/2$, where
$Q_-\equiv Q_2-Q_1$ is a pure $|\Delta I|=1/2$ operator and
$Q_+\equiv Q_2+Q_1$ induces both $|\Delta I|=1/2$ and $|\Delta I|=3/2$
transitions.
The standard electroweak model gives then rise to $|\Delta I|=1/2$
and $|\Delta I|=3/2$ amplitudes of nearly equal size,
while experimentally the ratio between both amplitudes is a
factor of twenty.
To solve this big discrepancy, QCD effects should be enormous.

The leading $\alpha_s$ corrections indeed give, for $\mu$-values
around 1 GeV, an enhancement by a factor two to three of the
$Q_-$ Wilson coefficient with respect to the $Q_+$ one.
Moreover, the gluonic exchanges generate the additional
$|\Delta I|=1/2$ operators $Q_i$ (i=3,4,5,6), the so-called
``Penguins''.
Nevertheless, this by itself is not enough to explain the
experimentally observed rates, without simultaneously appealing to
a further enhancement in the hadronic matrix elements of at least
some of the isospin--$1/2$ four-quark operators.

The evaluation of hadronic matrix elements is unfortunately
very difficult, since it involves
non-perturbative dynamics at low energies.
The problem gets, moreover, complicated by the $\mu$-dependence
of the matrix elements, which should exactly cancel the
corresponding renormalization-scale dependence of the
Wilson coefficients.
In order to get meaningful results, a full QCD calculation
is required; this is a highly non-trivial task.

The problem becomes much easier at the inclusive level, where the
properties of
${\cal H}^{\Delta S=1}_{\mbox{\rms eff}}$
%the non-leptonic effective weak Hamiltonian
can be analyzed through the 2--point function
%
\be
%\begin{eqnarray}
\label{eq:correlator}
\eqalign{
%\Psi^{\Delta S=1}(q^2)  \equiv i \int \! dx \, e^{iqx} \,
\Psi(q^2)  &\equiv i \int \! dx \, e^{iqx} \,
\big<0\vert \, T\{\,{\cal H}^{\Delta S=1}_{\mbox{\rms eff}}(x)\,
{\cal H}^{\Delta S=1}_{\mbox{\rms eff}}(0)^\dagger\}\vert0\big>
\cr
&=  \left({G_F\over\sqrt{2}}\right)^2
\left| V_{ud}^{\phantom{*}} V_{us}^{*}\right|^2\,
\sum_{i,j} \, C_i(\mu^2) \, C_j^*(\mu^2) \,\Psi_{ij}(q^2) \,.
\cr}
\ee
%\end{eqnarray}
%
This vacuum-to-vacuum correlator can be studied with
perturbative QCD methods, allowing for a consistent combination of
Wilson coefficients $C_i(\mu^2)$ and 2--point functions of the
4--quark operators, $\Psi_{ij}$, in such a way that the
renormalization scheme and scale dependences exactly cancel (to the
computed order). The associated spectral function,
%
\be
\frac{1}{\pi}\mbox{\rm Im}\Psi(q^2)
= (2\pi)^3 \sum_\Gamma \int\, d\Gamma
\left|\langle 0| {\cal H}^{\Delta S=1}_{\mbox{\rms eff}}|
\Gamma\rangle\right|^2 \delta^4(q-p_\Gamma) ,
\ee
%
is a quantity with
definite physical information; it describes in an inclusive way how
the weak Hamiltonian couples the vacuum to physical states
$\Gamma$ of a given
invariant mass. General properties like the observed enhancement of
$|\Delta I|=1/2$ transitions can be then rigorously analyzed at the
inclusive level.

A detailed analysis of  two-point functions
associated with $\Delta S=1$ and $\Delta S=2$ operators was
presented in Ref.~\cite{PD:91},
where the $\cO(\alpha_s)$ corrections to the
correlators $\Psi_{ij}$ were calculated.
The next-to-leading order (NLO) corrections to the
$|\Delta I|=1/2$ 2--point functions
were found to be very large, confirming the QCD enhancement
obtained in a previous approximate calculation \cite{PI:89}.
Those results were, however, incomplete because the NLO
corrections to the Wilson-coefficients of ``Penguin'' operators
were still not known.

The recent calculation of
${\cal H}^{\Delta S=1}_{\mbox{\rms eff}}$
at NLO \cite{BJLW,CFMR}
has allowed us to improve the results of Ref.~\cite{PD:91},
matching matrix elements and Wilson coefficients
consistently at NLO \cite{JP:94}.
Previously missing contributions from evanescent
operators have been also incorporated \cite{JP:94}.
In order to have a check of the results,
the calculation has been performed in two different
renormalization schemes for $\gamma_5$
(naively anticommuting $\gamma_5$ and 't Hooft--Veltman),
and the scale- and scheme-independence of the final
physical quantities has been verified.

\section{Approximate results}

The full calculation of $\Psi(q^2)$ is rather involved due to the
fact that there are several operators which mix under renormalization.
One needs to compute, at the four-loop level, all possible
2--point functions $\Psi_{ij}$; i.e. a $6\times 6$
($12\times 12$ at intermediate steps to include the contributions
of evanescent operators)
matrix correlator which must be renormalized in matrix form,
and later convoluted with the NLO Wilson coefficients
as indicated in Eq.~\ref{eq:correlator}.

It is possible to obtain some simplified results by using
two different approximations which eliminate the mixing
among operators, while keeping at the same time the important
physical effects \cite{PI:89}:

i) If ``Penguins'' are neglected, the operators $Q_\pm$ are
multiplicatively renormalizable.
The corresponding scheme- and scale-independent
spectral functions
$\Phi_{\pm\pm}(s) \equiv C_\pm^2(\mu^2)
{1\over \pi}\mbox{\rm Im}\Psi_{\pm\pm}(s,\mu^2)$
are found to be \cite{JP:94}:
%
\begin{eqnarray}
\Phi_{++}(s) \sim
\frac{8}{15}\,\frac{s^4}{(4\pi)^6}\,\alpha_s(s)^{-4/9}\,
\biggl[\,1-\frac{3649}{1620}\,\aps\,\biggr] \,,
&\label{eq:phi_pp} \\
\smvs
\Phi_{--}(s)  \sim
\frac{4}{15}\,\frac{s^4}{(4\pi)^6}\,\alpha_s(s)^{8/9\phantom{-}}\,
\biggl[\,1+\frac{9139}{810}\,\aps\,\biggr] \,.
&\label{eq:phi_mm}
\end{eqnarray}
%

ii) The interesting ``Penguin'' operator $Q_6$ can be isolated, by
noting that in the large $N_c$ limit ($N_c$ = number of colours)
the anomalous dimension matrix $\gamma_{ij}$ of the set of
operators $Q_i$ becomes zero, but for $\gamma_{66}$; i.e.
in this limit
there is no mixing among operators  and only $Q_6$ gets
renormalized.
%(Note that
%the large $N_c$ limit is not appropriate for studying the $Q_\pm$
%operators  since $\gamma_{\pm\pm}=0$ in this limit).
The $\cO(\alpha_s^2)$ correction can also be
easily computed in this limit \cite{PD:91}:
%
\be\label{eq:Phi_penguin}
\eqalign{
\fl\Phi_{66}(s)\sim {3\over 5} {s^4\over (4\pi)^6}
\alpha_s(s)^{18/11}
&\biggl[ 1 + \frac{117501}{4840}\aps \cr
&+ 470.72 \left(\aps\right)^2 \biggr] .\cr }
\ee
%

The NLO corrections to the $|\Delta I|=1/2$ correlators turn out to
be very big and positive, while for $\Phi_{++}$ the
correction is moderate and negative.
Taking $\alpha_s(s)/\pi\approx 0.1$, we find a moderate suppression
of $\Phi_{++}$ by roughly 20\%, whereas $\Phi_{--}$
acquires a huge enhancement of the order of 100\%.
The correction is even bigger for the ``Penguin'' correlator $\Phi_{66}$:
240\% at NLO and 700\% at next-to-next-to-leading order!
The perturbative calculation blows up in the
$|\Delta I|=1/2$ sector, clearly showing a dynamical gluonic
enhancement of the $|\Delta I|=1/2$ amplitudes.


\section{Exact results}

Following the notation of Refs.~\cite{BJLW}, the
Wilson-coefficient functions can be
decomposed as $C_i(s) = z_i(s) + \tau\,y_i(s)$, where
$\tau\equiv - \left(V_{td}^{\phantom{*}} V_{ts}^*\right)/
\left(V_{ud}^{\phantom{*}} V_{us}^*\right)$. The coefficients
$z_i(s)$ govern the real part of the effective Hamiltonian,
while $y_i(s)$
parametrize the imaginary part and govern, e.g., the measure for
direct CP-violation in the $K$-system, $\ve'/\ve$.
We can then form two
different scale-
and scheme-invariant spectral functions,
%
\beqn\label{eq:Phi_z}
\wh\Phi_z(s) & = & \sum_{i,j}\;
z_i(\mu^2)\,{1\over\pi}\mbox{\rm Im}\Psi_{ij}(s,\mu^2)\,z_j(\mu^2) \, ,
 \\ \label{eq:Phi_y}
\wh\Phi_y(s) & = & \sum_{i,j}\;
y_i(\mu^2)\,{1\over\pi}\mbox{\rm Im}\Psi_{ij}(s,\mu^2)\,y_j(\mu^2)\, ,
\eeqn
%
corresponding to $z_i$ and $y_i$ respectively.

Since we are mainly interested in the size of the
radiative corrections, let us write
$\wh\Phi_{z,\,y}(s)$ as
%
\begin{equation}
\wh\Phi_{z,\,y}(s)\;=\;
\wh\Phi^{(0)}_{z,\,y}(s)+\wh\Phi^{(1)}_{z,\,y}(s)
\,,
\label{eq:Phi_z_y}
\end{equation}
%
where the superscripts $(0)$ and $(1)$ refer to the
leading and next-to-leading order respectively.
The exact results obtained \cite{JP:94} for the ratios
$\wh\Phi^{(1)}_{z}/\wh\Phi^{(0)}_{z}$ and
$\wh\Phi^{(1)}_{y}/\wh\Phi^{(0)}_{y}$
are plotted in Fig.~1, for $\La=200,\,300$, and $400$
MeV.

%%%%%%%%%%%%%%%%%%  FIGURE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[bht]
\centerline{
\rotate[r]{
\epsfysize=8.5cm
\epsffile{cpc.ps}
}}
\caption[]{The ratios $\wh\Phi^{(1)}_{z}/\wh\Phi^{(0)}_{z}$ and
$\wh\Phi^{(1)}_{y}/\wh\Phi^{(0)}_{y}$.\label{figure}}
\end{figure}

%%%%%%%%%%%%%%%%% END FIGURE  %%%%%%%%%%%%%%%%%%%%%%%%%


{}From Fig.~1, we can see that in the region $Q=1-3\,\mbox{\rm GeV}$,
and for a central value $\La=300\,\mbox{\rm MeV}$, the radiative QCD
correction to $\wh\Phi_z$ ranges approximately between 40\% and 120\%,
whereas in the case of $\wh\Phi_y$ we find a correction of the order
of 100\%--240\%.
As explicitly shown by the approximate results of the previous
section,
the large $\al$ corrections
correspond to the $|\Delta I|=1/2$ part of the effective weak
Hamiltonian.
In fact, the corrections to the $|\Delta I|=3/2$
correlator are exactly given by Eq.~\ref{eq:phi_pp}
(``Penguins'' only give $\Delta I=1/2$ contributions), and
therefore are quite moderate.

In the case of $\Delta S=2$ transitions, there is only one 4--quark
operator.
% in the effective Hamiltonian.
Since the $\Delta S=2$ and $|\Delta I|=3/2$
operators belong to the same representation  of the
(flavour) $SU(3)_L\otimes SU(3)_R$ group, the NLO corrections
to the $\Delta S=2$ correlator are also exactly given by
Eq.~\ref{eq:phi_pp}.

\section{Summary}

The short-distance behaviour of the $\Delta S=1$ correlators clearly
shows a dynamical enhancement of the $|\Delta I|=1/2$
channel, as a consequence of the interplay of gluonic corrections.
The structure of the radiative corrections also allows for a deeper
understanding of the underlying dynamical mechanism \cite{JP:94}:
large corrections appear wherever quark-quark correlations can contribute.
This explains why the phenomenological description
of the $|\Delta I|=1/2$ rule
in terms of intermediate
effective diquarks \cite{NS:91} was so successful.

A full QCD calculation has been possible because of the inclusive
character of the defined 2--point functions. Although only qualitative
conclusions can be directly extracted from these results, they are
certainly important since they rigorously point to the
QCD origin of the infamous $|\Delta I|=1/2$ rule.
%and, moreover, provide valuable information on the relative importance
%of the different operators, which can be very helpful to
%attempt more pragmatic calculations.

\section*{Acknowledgments}

I would like to thank M. Jamin for a very enjoyable collaboration.
This work has been supported in part by  CICYT (Spain) under
Grant No. AEN-93-0234.

%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%

\Bibliography{9}

\bibitem{PD:91}
   A.~Pich and  E.~de~Rafael, \np{B358}{91}{311}.

\bibitem{PI:89}
   A.~Pich, Nucl. Phys. B (Proc. Suppl.) {\bf 7A} (1989) 194.

\bibitem{BJLW}
   A.~J. Buras {\it et al.}, \np{408}{93}{209}; ibid {\bf 400} (1993) 37, 75;
   ibid {\bf B370} (1992) 69; add. ibid.~{\bf B375} (1992) 501;
   ibid {\bf 347} (1990) 491; ibid {\bf 333} (1990) 66.

\bibitem{CFMR}
   M.~Ciuchini {\it et al.}, \np{B415}{94}{403};
    \pl{B301}{93}{263}.

\bibitem{JP:94}
   M. Jamin and A. Pich, \np{B425}{94}{15}.

\bibitem{NS:91}
   M. Neubert and B. Stech, \prev{D44}{91}{775}, and references therein.

\end{thebibliography}

\end{document}

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      \Alph{section}.\arabic{equation}\else
      \Alph{section}\arabic{equation}\fi}
 \def\thetable{\ifnumbysec
      \Alph{section}\arabic{table}\else
      A\arabic{table}\fi}
 \def\thefigure{\ifnumbysec
      \Alph{section}\arabic{figure}\else
      A\arabic{figure}\fi}}



\labelsep 4\p@

\leftmargini 16\p@
\leftmarginii 18\p@
\leftmarginiii 16\p@
\leftmarginiv 14\p@
\leftmarginv 10\p@
\leftmarginvi 10\p@
\leftmargin\leftmargini
\labelwidth\leftmargini\advance\labelwidth-\labelsep
\parsep 0\p@ plus 1\p@
\def\@listI{\leftmargin\leftmargini \parsep 4\p@ plus2\p@ minus\p@
\topsep 8\p@ plus2\p@ minus4\p@
\itemsep 4\p@ plus2\p@ minus\p@}

\let\@listi\@listI
\@listi

\def\@listii{\leftmargin\leftmarginii
 \labelwidth\leftmarginii\advance\labelwidth-\labelsep
 \topsep 3\p@ plus 1\p@ minus 1\p@
 \parsep 0\p@ plus 1\p@
 \itemsep \parsep}
\def\@listiii{\leftmargin\leftmarginiii
 \labelwidth\leftmarginiii\advance\labelwidth-\labelsep
 \topsep 2\p@ plus 1\p@ minus 1\p@
 \parsep \z@ \partopsep 1\p@ plus 0\p@ minus 1\p@
 \itemsep \topsep}
\def\@listiv{\leftmargin\leftmarginiv
 \labelwidth\leftmarginiv\advance\labelwidth-\labelsep}
\def\@listv{\leftmargin\leftmarginv
 \labelwidth\leftmarginv\advance\labelwidth-\labelsep}
\def\@listvi{\leftmargin\leftmarginvi
 \labelwidth\leftmarginvi\advance\labelwidth-\labelsep}

\pretolerance=5000
\tolerance=8000
\hbadness=5000
\vbadness=5000
%
\def\labelenumi{\theenumi}
\def\theenumi{\arabic{enumi}}
\def\labelenumii{\theenumii}
\def\theenumii{\alpha{enumii}}
\def\p@enumii{\theenumi.}
\def\labelenumiii{\theenumiii.}
\def\theenumiii{\arabic{enumiii}}
\def\p@enumiii{\p@enumii.\theenumii}
\def\labelenumiv{\theenumiv.}
\def\theenumiv{\arabic{enumiv}}
\def\p@enumiv{\p@enumiii.\theenumiii}

\def\labelitemi{$\m@th\bullet$}
\def\labelitemii{\bf --}
\def\labelitemiii{$\m@th\ast$}
\def\labelitemiv{$\m@th\cdot$}

\def\verse{\let\\=\@centercr
 \list{}{\itemsep\z@ \itemindent -1.5em\listparindent \itemindent
 \rightmargin\leftmargin\advance\leftmargin 1.5em}\item[]}
\let\endverse\endlist
\def\quotation{\list{}{\listparindent 1.5em
 \itemindent\listparindent
 \rightmargin\leftmargin\parsep 0\p@ plus 1\p@}\item[]}
\let\endquotation=\endlist
\def\quote{\list{}{\rightmargin\leftmargin}\item[]}
\let\endquote=\endlist

\def\descriptionlabel#1{\hspace\labelsep \bf #1}
\def\description{\list{}{\labelwidth\z@ \itemindent-\leftmargin
 \let\makelabel\descriptionlabel}}
\let\enddescription\endlist
%
\def\enumerate{\ifnum \@enumdepth >3 \@toodeep\else
      \advance\@enumdepth \@ne
      \edef\@enumctr{enum\romannumeral\the\@enumdepth}\list
      {\csname label\@enumctr\endcsname}{\usecounter
        {\@enumctr}\def\makelabel##1{##1\hss}}\fi}
%
\def\itemize{\ifnum \@itemdepth >3 \@toodeep\else \advance\@itemdepth \@ne
\edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
\list{\csname\@itemitem\endcsname}{\def\makelabel##1{##1\hss}\topsep=3pt
  \parsep=0pt\listparindent=0pt\itemsep=0pt\partopsep=0pt\rightmargin=0pt
  }\fi}
%
\newenvironment{leqnarray}{\begin{leqnarray}}{\end{leqnarray}}
\def\leqnarray{\stepcounter{equation}\let\@currentlabel=\theequation
\global\@eqnswtrue
\global\@eqcnt\z@\tabskip\mathindent\let\\=\@eqncr
\abovedisplayskip\topsep\ifvmode\advance\abovedisplayskip\partopsep\fi
\belowdisplayskip\abovedisplayskip
\belowdisplayshortskip\abovedisplayskip
\abovedisplayshortskip\abovedisplayskip
$$\halign to
\columnwidth\bgroup\@eqnsel$\displaystyle\tabskip\z@
 {##{}}$&\global\@eqcnt\@ne
                    $\displaystyle{{}##{}}$\hfil    %\hfil delete before 2nd $
 &\global\@eqcnt\tw@ $\displaystyle{{}##}$\hfil
 \tabskip\@centering&\llap{##}\tabskip\z@\cr}
%
\def\endleqnarray{\@@eqncr\egroup
 \global\advance\c@equation\m@ne$$\global\@ignoretrue }
%
\arraycolsep 5\p@
\tabcolsep=6\p@
\arrayrulewidth .4\p@
\doublerulesep 2\p@
\tabbingsep \labelsep
\skip\@mpfootins = \skip\footins
\fboxsep = 3\p@
\fboxrule = .4\p@
\def\titlepage{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
     \else \newpage \fi \thispagestyle{myheadings}\c@page\z@}

\def\endtitlepage{\if@restonecol\twocolumn \else \newpage \fi}

\newcounter {section}
\newcounter {subsection}[section]
\newcounter {subsubsection}[subsection]
\newcounter {paragraph}[subsubsection]
\newcounter {subparagraph}[paragraph]



\def\thesection {\arabic{section}}
\def\thesubsection {\thesection.\arabic{subsection}}
\def\thesubsubsection {\thesubsection .\arabic{subsubsection}}
\def\theparagraph {\thesubsubsection.\arabic{paragraph}}
\def\thesubparagraph {\theparagraph.\arabic{subparagraph}}
\def\@chapapp{Section}


\def\@pnumwidth{1.55em}
\def\@tocrmarg {2.55em}
\def\@dotsep{4.5}
\setcounter{tocdepth}{2}


\def\tableofcontents{\@restonecolfalse\if@twocolumn\@restonecoltrue
 \onecolumn\fi\section*{Contents}{}\thispagestyle{empty}
 \@starttoc{toc}\if@restonecol\twocolumn\fi}
%
\def\l@section{\@dottedtocline{1}{1.5em}{2.3em}}
\def\l@subsection{\@dottedtocline{2}{3.8em}{3.2em}}
\def\l@subsubsection{\@dottedtocline{3}{7.0em}{4.1em}}
\def\l@paragraph{\@dottedtocline{4}{10em}{5em}}
\def\l@subparagraph{\@dottedtocline{5}{12em}{6em}}
\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
 \fi\section*{List of Figures\@mkboth
 {LIST OF FIGURES}{LIST OF FIGURES}}\@starttoc{lof}\if@restonecol\twocolumn
 \fi}
\def\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}
\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
 \fi\section*{List of Tables\@mkboth
 {LIST OF TABLES}{LIST OF TABLES}}\@starttoc{lot}\if@restonecol\twocolumn
 \fi}
\let\l@table\l@figure
%
% Redefinition to remove dotted lines from \@dottedtocline
%
\def\@dottedtocline#1#2#3#4#5{\ifnum #1>\c@tocdepth \else
  \vskip \z@ plus .2\p@
  {\leftskip #2\relax \rightskip \@tocrmarg \parfillskip -\rightskip
    \parindent #2\relax\@afterindenttrue
   \interlinepenalty\@M
   \leavevmode
   \@tempdima #3\relax \advance\leftskip \@tempdima
   \hbox{}\hskip -\leftskip
    #4\nobreak\hfill \nobreak \hbox to\@pnumwidth{\hfil
   \rm #5}\@@par}\fi}

\def\footnoterule{}%
\setcounter{footnote}{0}
\@addtoreset{footnote}{page}
\long\def\@makefntext#1{\parindent 1em\noindent
 \makebox[1em][l]{\footnotesize\rm$\m@th{\fnsymbol{footnote}}$}%
 \footnotesize\rm #1}
\def\@makefnmark{\hbox{${\fnsymbol{footnote}}\m@th$}}
\def\@thefnmark{\fnsymbol{footnote}}
\def\footnote{\@ifnextchar[{\@xfootnote}{\stepcounter{\@mpfn}%
     \begingroup\let\protect\noexpand
       \xdef\@thefnmark{\thempfn}\endgroup
     \@footnotemark\@footnotetext}}
\def\@fnsymbol#1{\ifcase#1\or \dagger\or \ddagger\or \S\or
   \|\or \P\or ^{+}\or ^{\tsty *}\or \sharp
   \or \dagger\dagger \else\@ctrerr\fi\relax}
\newcommand\ftnote[1]{\setcounter{footnote}{#1}%
   \addtocounter{footnote}{-1}\footnote}
\newcommand{\fnm}[1]{\setcounter{footnote}{#1}\footnotetext}
%
\def\center{\trivlist\topsep=0\p@\partopsep=0\p@
   \parsep=0\p@\itemsep=0\p@\centering\item[]}
%
\newenvironment{indented}{\begin{indented}}{\end{indented}}
\def\indented{\list{}{\itemsep=0\p@\labelsep=0\p@\itemindent=0\p@
   \labelwidth=0\p@\leftmargin=1.5cm\rightmargin=1.5cm
   \topsep=0\p@\partopsep=0\p@
   \parsep=0\p@\listparindent=0\p@}\rm}

\let\endindented=\endlist
%
\def\catchline{\hfill}

\def\cpyrtline{\hfill}
%
\def\maketitle{\thispagestyle{myheadings}%
   \vspace*{1.8cm}
   \begin{center}\@title\end{center}
   \vspace*{1.1cm}
   \normalsize\rm
   \begin{center}\@author\end{center}
   \begin{center}\@address\end{center}
   \@collab
   \@abstract}
%
%  Title
%
\def\title#1{\def\@title{\exhyphenpenalty=10000\hyphenpenalty=10000
    \Large\bf#1\par}}
\def\shortitle#1{\def\@shorttitle{#1}}
\let\paper=\title
%
% Authors
%
\renewcommand{\author}[1]{\def\@author{{\large #1\par}}}
%
% Affiliation
%
\newcommand{\address}[1]{\def\@address{\rm #1\par}}
\let\affil=\address
%
\newcommand{\collab}[1]{\def\@collab{\begin{center}%
   {\large\rm #1}\par
   \end{center}}}
%
% Default
%
\def\@collab{}
%
% Abstract
%
\def\abstract#1{\def\@abstract{\begin{center}
{\bf\abstractname}\end{center}%
\begin{indented}
\item[]#1\par
\end{indented}
\vspace{2cm minus1cm}}}%
%
\def\endabstract{}
%
% Command for second and subsequent paragraphs in abstract
%
\def\cabs{\\\hspace*{16\p@}}
%
\def\nosections{\vspace{30\p@ plus12\p@ minus12\p@}
    \noindent\ignorespaces}
%
\def\ack{\ifletter\bigskip\noindent\ignorespaces\else
    \section*{Acknowledgments}\fi}
\def\ackn{\ifletter\bigskip\noindent\ignorespaces\else
    \section*{Acknowledgment}\fi}
%
\newif\ifnumbysec
\def\theequation{\ifnumbysec
      \arabic{section}.\arabic{equation}\else
      \arabic{equation}\fi}
\def\eqnobysec{\numbysectrue\@addtoreset{equation}{section}}
%
\def\eqalign#1{\null\vcenter{\def\\{\cr}\openup\jot\m@th
  \ialign{\strut\hfil$\displaystyle{##{}}$&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}
%
\def\eqalignno#1{\displ@y \tabskip\z@skip
  \halign to\if@twocolumn\columnwidth\else\displaywidth\fi
   {\hfil$\@lign\displaystyle{##}$%
    \tabskip\z@skip
    &$\@lign\displaystyle{{}##}$\hfill\tabskip\@centering
    &\llap{$\@lign\hbox{\rm##}$}\tabskip\z@skip\crcr
    #1\crcr}}
%
\def\numparts{\addtocounter{equation}{1}%
     \setcounter{eqnval}{\value{equation}}%
     \setcounter{equation}{0}%
     \def\theequation{\ifnumbysec
     \arabic{section}.\arabic{eqnval}{\it\alph{equation}}%
     \else\arabic{eqnval}{\it\alph{equation}}\fi}}

\def\endnumparts{\def\theequation{\ifnumbysec
     \arabic{section}.\arabic{equation}\else
     \arabic{equation}\fi}%
     \setcounter{equation}{\value{eqnval}}}
%
\def\cases#1{%
     \left\{\,\vcenter{\def\\{\cr}\normalbaselines\openup1\jot\m@th%
     \ialign{\strut$\displaystyle{##}\hfil$&\tqs
     \rm##\hfil\crcr#1\crcr}}\right.}%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Floats
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  \c@topnumber            : Number of floats allowed at the top of a column.
%
\setcounter{topnumber}{4}
%
%  \topfraction            : Fraction of column that can be devoted to floats.
%
\def\topfraction{1}
%
%  \c@dbltopnumber, \dbltopfraction : Same as above, but for double-column
%                          floats.
%
\setcounter{dbltopnumber}{4}
\def\dbltopfraction{1}
%
%  \c@bottomnumber, \bottomfraction : Same as above for bottom of page.
%
\setcounter{bottomnumber}{2}
\def\bottomfraction{.8}
%
%  \c@totalnumber          : Number of floats allowed in a single column,
%                          including in-text floats.
%
\setcounter{totalnumber}{5}
%
%  \textfraction         : Minimum fraction of column that must contain text.
%
\def\textfraction{0}
%
%  \floatpagefraction    : Minimum fraction of page that must be taken
%                          up by float page.
%
\def\floatpagefraction{.8}
%
%  \dblfloatpagefraction : Same as above, for double-column floats.
%
\def\dblfloatpagefraction{.8}
%
\newcounter{figure}
\def\thefigure{\@arabic\c@figure}
\def\figure{\let\@makecaption\@makeonecolcaption\@float{figure}}
\let\endfigure\end@float
%
\@namedef{figure*}{\let\@makecaption\@makewidecaption
      \@dblfloat{figure}}
\@namedef{endfigure*}{\end@dblfloat}
%
\def\@makewidecaption#1#2{\vspace{10\p@}%
     \sbox{\captionbox}{\noindent\footnotesize\rm\raggedright{\bf #1.} #2}%
     \ifdim\wd\captionbox > \indentedwidth
     \begin{indented}
     \item[]\footnotesize\rm\raggedright{\bf #1.} #2\par
     \end{indented}%
     \else
     \hbox to \hsize{\hfil\box\captionbox\hfil}\fi}
%
\def\@makeonecolcaption#1#2{\vspace{10pt}%
     \parbox{\columnwidth}{\noindent
     \footnotesize\rm\raggedright{\bf #1.} #2}\par}
%
%
% The document style must define the following.
%
%    \fps@TYPE   : The default placement specifier for floats of type TYPE.
%
\def\fps@figure{tb}
\def\fps@table{tb}
%
%    \ftype@TYPE : The type number for floats of type TYPE.
%
\def\ftype@figure{1}
\def\ftype@table{2}
%
%    \ext@TYPE   : The file extension indicating the file on which the
%                  contents list for float type TYPE is stored.  For example,
%                  \ext@figure = 'lof'.
%
\def\ext@table{aux}
\def\ext@figure{aux}
%
%    \fnum@TYPE  : A macro to generate the figure number for a caption.
%                  For example, \fnum@TYPE == Figure \thefigure.
%
\def\fnum@table{\tablename~\thetable}
\def\fnum@figure{\figurename~\thefigure}
%
%    \@makecaption{NUM}{TEXT} : A macro to make a caption, with NUM the value
%                  produced by \fnum@... and TEXT the text of the caption.
%                  It can assume it's in a \parbox of the appropriate width.
%
\newcommand{\Figure}[2]{\def\figspace{\vspace*{#1}}%
    \def\figcap{\caption{#2}}%
    \futurelet\next\@figplace}
\def\@figplace{\ifx\next[\let\next=\@figpl
                 \else\let\next=\@fignopl\fi\next}
\def\@figpl[#1]{\begin{figure}[#1]
   \figspace
   \figcap
   \end{figure}}
\def\@fignopl{\begin{figure}
   \figspace
   \figcap
   \end{figure}}
%
\newcommand{\widefigure}[2]{\def\figspace{\vspace*{#1}}%
    \def\figcap{\caption{#2}}%
    \futurelet\next\@wfigplace}
\def\@wfigplace{\ifx\next[\let\next=\@wfigpl
                 \else\let\next=\@wfignopl\fi\next}
\def\@wfigpl[#1]{\begin{figure*}[#1]
   \figspace
   \figcap
   \end{figure*}}
\def\@wfignopl{\begin{figure*}
   \figspace
   \figcap
   \end{figure*}}
%
% \@float{TYPE}[PLACEMENT] : This macro begins a float environment for a
%     single-column float of type TYPE with PLACEMENT as the placement
%     specifier.  The default value of PLACEMENT is defined by \fps@TYPE.
%     The environment is ended by \end@float.
%     E.g., \figure == \@float{figure}, \endfigure == \end@float.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Tables
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcounter{table}
%
\def\thetable{\@arabic\c@table}
\def\table{\let\@makecaption\@makeonecolcaption
    \footnotesize\rm\@float{table}}
\let\endtable\end@float
%
\@namedef{table*}{\let\@makecaption\@makewidecaption
   \footnotesize\rm
   \@dblfloat{table}}
\@namedef{endtable*}{\end@dblfloat}
%
\def\tabular{\def\@halignto{}\@tabular}
\def\endtabular{\crcr\egroup\egroup $\egroup}
\expandafter \let \csname endtabular*\endcsname = \endtabular
%
\newsavebox{\tablebox}
%
\newcommand{\Table}[2]{\begin{center}
    \lineup
    \begin{tabular}{#1}%
    \hline
    #2
    \hline
    \end{tabular}
    \end{center}}
%
\newcommand{\tabnote}[1]{\begin{indented}
     \item[]\footnotesize\rm\raggedright #1\par
     \end{indented}}
%
% Definitions for centring headings over several columns
% \centre{4}{Results for helium} will centre
% Results for helium over four columns
% \crule{4} will produce a rule centred over four columns
% to go below a centred heading
%
\newcommand{\centre}[2]{\multicolumn{#1}{c}{#2}}
\newcommand{\crule}[1]{\multispan{#1}{\hrulefill}}
%
\def\lineup{\def\0{\hbox{\phantom{\footnotesize\rm 0}}}%
    \def\m{\hbox{$\phantom{-}$}}%
    \def\-{\llap{$-$}}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% References
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\Bibliography}[1]{\section*{References}\par\numrefs{#1}}
\newcommand{\References}[1]{\section*{References}\footnotesize\rm}
%
\def\thebibliography#1{\list
 {\hfil[\arabic{enumi}]}{\topsep=0\p@\parsep=0\p@
 \partopsep=0\p@\itemsep=0\p@
 \labelsep=5\p@\itemindent=0\p@                %-10
 \settowidth\labelwidth{\footnotesize[#1]}%
 \leftmargin\labelwidth
 \advance\leftmargin\labelsep
% \advance\leftmargin -\itemindent
 \usecounter{enumi}}%
 \def\newblock{\ }
 \sloppy\clubpenalty4000\widowpenalty4000
 \sfcode`\.=1000\footnotesize\rm\relax}
\let\endthebibliography=\endlist
%
\def\numrefs#1{\begin{thebibliography}{#1}}
\def\endnumrefs{\end{thebibliography}}
\let\endbib=\endnumrefs

\mark{{}{}}

\def\ps@headings{\let\@mkboth\markboth
 \def\@oddfoot{}%
 \def\@evenfoot{}%
 \def\@evenhead{\makebox[\mathindent][l]{\normalsize\rm \thepage}%
  \normalsize\it\rightmark\hfill}%
 \def\@oddhead{\makebox[\mathindent][r]{\hfill}{\normalsize\it\leftmark}\hfill
  \normalsize\rm\thepage}%
}%

\def\ps@myheadings{\let\@mkboth\markboth
 \def\@oddhead{\catchline}%
 \def\@oddfoot{\cpyrtline}%
 \def\@evenhead{}%
 \def\@evenfoot{}%
}


\def\today{\ifcase\month\or
 January\or February\or March\or April\or May\or June\or
 July\or August\or September\or October\or November\or December\fi
 \space\number\day, \number\year}

\def\@begintheorem#1#2{\rm \trivlist \item[\hskip \labelsep{\it #1\ #2.}]}
\def\@opargbegintheorem#1#2#3{\rm \trivlist
      \item[\hskip \labelsep{\it #1\ #2\ (#3).}]}

\let\scap=\sc
\renewcommand{\sc}{\protect\scriptsize}
\newcommand{\itsc}{\protect\scriptsize\it}
\newcommand{\bfsc}{\protect\scriptsize\bf}
\def\p@LaTeX{{L\kern-.3em\lower.1em\hbox{$^{\rm A}$}\kern-.15em%
    T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
%
\newcommand{\nohyphens}{\hyphenpenalty=10000\exhyphenpenalty=10000}
\newcommand{\fl}{\hspace*{-\mathindent}}
\newcommand{\Tr}{\mathop{\rm Tr}\nolimits}
\newcommand{\tr}{\mathop{\rm tr}\nolimits}
\newcommand{\Or}{\mathop{\rm O}\nolimits}
\newcommand{\lshad}{[\![}
\newcommand{\rshad}{]\!]}
\newcommand{\case}[2]{{\textstyle\frac{#1}{#2}}}
\def\pt(#1){({\it #1\/})}
\newcommand{\dsty}{\displaystyle}
\newcommand{\tsty}{\textstyle}
\newcommand{\ssty}{\scriptstyle}
\newcommand{\sssty}{\scriptscriptstyle}
\def\lo#1{\llap{${}#1{}$}}
\def\eql{\llap{${}={}$}}
\def\lsim{\llap{${}\sim{}$}}
\def\lsimeq{\llap{${}\simeq{}$}}
\def\lequiv{\llap{${}\equiv{}$}}
%
\def\;{\protect\psemicolon}
\def\psemicolon{\relax\ifmmode\mskip\thickmuskip\else\kern .3333em\fi}
%
\newcommand{\eref}[1]{(\ref{#1})}
\newcommand{\sref}[1]{section~\ref{#1}}
\newcommand{\fref}[1]{figure~\ref{#1}}
\newcommand{\tref}[1]{table~\ref{#1}}
\newcommand{\Eref}[1]{Equation~(\ref{#1})}
\newcommand{\Sref}[1]{Section~\ref{#1}}
\newcommand{\Fref}[1]{Figure~\ref{#1}}
\newcommand{\Tref}[1]{Table~\ref{#1}}

\newcommand{\opencirc}{\raisebox{2\p@}{\;\circle{5}}}
\newcommand{\opensqr}{\mbox{$\Box$}}
\newcommand{\fullcirc}{\raisebox{-2\p@}{\Large$\bullet$}}
\newcommand{\fullsqr}{\mbox{\vrule height6pt width6pt}}
\newcommand{\dotted}
                 {\mbox{${\mathinner{\cdotp\cdotp\cdotp\cdotp\cdotp\cdotp}}$}}
\newcommand{\dashed}{\mbox{-\; -\; -\; -}}
\newcommand{\broken}{\mbox{-- -- --}}
\newcommand{\longbroken}{\mbox{--- --- ---}}
\newcommand{\chain}{\mbox{--- $\cdot$ ---}}
\newcommand{\dashddot}{\mbox{--- $\cdot$ $\cdot$ ---}}
\newcommand{\full}{\mbox{------}}
%
\newcommand{\etal}{{\it et al\/}\ }
\newcommand{\nonum}{\item[]}
%
% abbreviations for IOPP journals
%
\newcommand{\CQG}{{\em Class. Quantum Grav.} }
\newcommand{\HPP}{{\em High Perform. Polym.} }              % added 4/5/93
\newcommand{\IP}{{\em Inverse Problems\/} }
\newcommand{\JHM}{{\em J. Hard Mater.} }                    % added 4/5/93
\newcommand{\JPA}{{\em J. Phys. A: Math. Gen.} }
\newcommand{\JPB}{{\em J. Phys. B: At. Mol. Phys.} }      %1968-87
\newcommand{\jpb}{{\em J. Phys. B: At. Mol. Opt. Phys.} } %1988 and onwards
\newcommand{\JPC}{{\em J. Phys. C: Solid State Phys.} }   %1968--1988
\newcommand{\JPCM}{{\em J. Phys.: Condens. Matter\/} }    %1989 and onwards
\newcommand{\JPD}{{\em J. Phys. D: Appl. Phys.} }
\newcommand{\JPE}{{\em J. Phys. E: Sci. Instrum.} }
\newcommand{\JPF}{{\em J. Phys. F: Met. Phys.} }
\newcommand{\JPG}{{\em J. Phys. G: Nucl. Phys.} }         %1975--1988
\newcommand{\jpg}{{\em J. Phys. G: Nucl. Part. Phys.} }   %1989 and onwards
\newcommand{\MSMSE}{{\em Modelling Simulation Mater. Sci. Eng.} }
\newcommand{\MST}{{\em Meas. Sci. Technol.} }              %1990 and onwards
\newcommand{\NET}{{\em Network\/} }
\newcommand{\NL}{{\em Nonlinearity\/} }
\newcommand{\NT}{{\em Nanotechnology} }
\newcommand{\PAO}{{\em Pure Appl. Optics\/} }
\newcommand{\PM}{{\em Physiol. Meas.} }                        % added 4/5/93
\newcommand{\PMB}{{\em Phys. Med. Biol.} }
\newcommand{\PPCF}{{\em Plasma Phys. Control. Fusion\/} }      % added 4/5/93
\newcommand{\PSST}{{\em Plasma Sources Sci. Technol.} }
\newcommand{\QO}{{\em Quantum Opt.} }
\newcommand{\RPP}{{\em Rep. Prog. Phys.} }
\newcommand{\SLC}{{\em Sov. Lightwave Commun.} }               % added 4/5/93
\newcommand{\SST}{{\em Semicond. Sci. Technol.} }
\newcommand{\SUST}{{\em Supercond. Sci. Technol.} }
\newcommand{\WRM}{{\em Waves Random Media\/} }
%
% Other commonly quoted journals
%
\newcommand{\AC}{{\em Acta Crystallogr.} }
\newcommand{\AM}{{\em Acta Metall.} }
\newcommand{\AP}{{\em Ann. Phys., Lpz.} }
\newcommand{\APNY}{{\em Ann. Phys., NY\/} }
\newcommand{\APP}{{\em Ann. Phys., Paris\/} }
\newcommand{\CJP}{{\em Can. J. Phys.} }
\newcommand{\JAP}{{\em J. Appl. Phys.} }
\newcommand{\JCP}{{\em J. Chem. Phys.} }
\newcommand{\JJAP}{{\em Japan. J. Appl. Phys.} }
\newcommand{\JP}{{\em J. Physique\/} }
\newcommand{\JPhCh}{{\em J. Phys. Chem.} }
\newcommand{\JMMM}{{\em J. Magn. Magn. Mater.} }
\newcommand{\JMP}{{\em J. Math. Phys.} }
\newcommand{\JOSA}{{\em J. Opt. Soc. Am.} }
\newcommand{\JPSJ}{{\em J. Phys. Soc. Japan\/} }
\newcommand{\JQSRT}{{\em J. Quant. Spectrosc. Radiat. Transfer\/} }
\newcommand{\NC}{{\em Nuovo Cimento\/} }
\newcommand{\NIM}{{\em Nucl. Instrum. Methods\/} }
\newcommand{\NP}{{\em Nucl. Phys.} }
\newcommand{\PL}{{\em Phys. Lett.} }
\newcommand{\PR}{{\em Phys. Rev.} }
\newcommand{\PRL}{{\em Phys. Rev. Lett.} }
\newcommand{\PRS}{{\em Proc. R. Soc.} }
\newcommand{\PS}{{\em Phys. Scr.} }
\newcommand{\PSS}{{\em Phys. Status Solidi\/} }
\newcommand{\PTRS}{{\em Phil. Trans. R. Soc.} }
\newcommand{\RMP}{{\em Rev. Mod. Phys.} }
\newcommand{\RSI}{{\em Rev. Sci. Instrum.} }
\newcommand{\SSC}{{\em Solid State Commun.} }
\newcommand{\ZP}{{\em Z. Phys.} }
%

\def\ap#1#2#3 {Ann. Phys. (NY) {\bf#1} (19#2) #3}
\def\apj#1#2#3 {Astrophys. J. {\bf#1} (19#2) #3}
\def\apjl#1#2#3 {Astrophys. J. Lett. {\bf#1} (19#2) #3}
\def\app#1#2#3 {Acta. Phys. Pol. {\bf#1} (19#2) #3}
\def\ar#1#2#3 {Ann. Rev. Nucl. Part. Sci. {\bf#1} (19#2) #3}
\def\cpc#1#2#3 {Computer Phys. Comm. {\bf#1} (19#2) #3}
\def\err#1#2#3 {{\it Erratum} {\bf#1} (19#2) #3}
\def\ib#1#2#3 {{\it ibid.} {\bf#1} (19#2) #3}
\def\jmp#1#2#3 {J. Math. Phys. {\bf#1} (19#2) #3}
\def\ijmp#1#2#3 {Int. J. Mod. Phys. {\bf#1} (19#2) #3}
\def\jetp#1#2#3 {JETP Lett. {\bf#1} (19#2) #3}
\def\jpg#1#2#3 {J. Phys. G. {\bf#1} (19#2) #3}
\def\mpl#1#2#3 {Mod. Phys. Lett. {\bf#1} (19#2) #3}
\def\nat#1#2#3 {Nature (London) {\bf#1} (19#2) #3}
\def\nc#1#2#3 {Nuovo Cim. {\bf#1} (19#2) #3}
\def\nim#1#2#3 {Nucl. Instr. Meth. {\bf#1} (19#2) #3}
\def\np#1#2#3 {Nucl. Phys. {\bf#1} (19#2) #3}
\def\pcps#1#2#3 {Proc. Cam. Phil. Soc. {\bf#1} (#2) #3}
\def\pl#1#2#3 {Phys. Lett. {\bf#1} (19#2) #3}
\def\prep#1#2#3 {Phys. Rep. {\bf#1} (19#2) #3}
\def\prev#1#2#3 {Phys. Rev. {\bf#1} (19#2) #3}
\def\prl#1#2#3 {Phys. Rev. Lett. {\bf#1} (19#2) #3}
\def\prs#1#2#3 {Proc. Roy. Soc. {\bf#1} (19#2) #3}
\def\ptp#1#2#3 {Prog. Th. Phys. {\bf#1} (19#2) #3}
\def\ps#1#2#3 {Physica Scripta {\bf#1} (19#2) #3}
\def\rmp#1#2#3 {Rev. Mod. Phys. {\bf#1} (19#2) #3}
\def\rpp#1#2#3 {Rep. Prog. Phys. {\bf#1} (19#2) #3}
\def\sjnp#1#2#3 {Sov. J. Nucl. Phys. {\bf#1} (19#2) #3}
\def\spj#1#2#3 {Sov. Phys. JEPT {\bf#1} (19#2) #3}
\def\spu#1#2#3 {Sov. Phys.-Usp. {\bf#1} (19#2) #3}
\def\zp#1#2#3 {Zeit. Phys. {\bf#1} (19#2) #3}
%
\ps@headings \pagenumbering{arabic} \onecolumn


