%Paper: 
%From: Stefan Herrlich <herrl@feynman.t30.physik.tu-muenchen.de>
%Date: Wed, 31 Aug 1994 10:34:23 +0200

% tum72.tex
% Paper for conference proceedings "QCD 94" in Montpellier, France
%
% $Id:$
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\newcommand{\nn}{\nonumber}
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\newcommand{\MS}{{\rm MS}}
\newcommand{\MSbar}{\overline{\MS}}
\newcommand{\GF}{{\rm G}_{\rm F}}
\newcommand{\MW}{M_{\rm W}}
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\title{
	\hfill {\normalsize TUM-T31-72/94}\\[-2truemm]
	\hfill {\normalsize \\[-2truemm]
	\hfill {\normalsize August 94}\\
	Next-to-leading Order short distance QCD corrections to the
	effective $\Delta S = 2$ Hamiltonian, Implications for the
	${\rm K}_{\rm L}$--${\rm K}_{\rm S}$ mass difference
	\thanks{
		Invited talk presented at the conference ``QCD'94'',
		Montpellier, France, July 7 - 13, 1994.
		To appear in the proceedings.
		}
	}
\author{
	Stefan Herrlich
	\address{
		Physik-Department T31, Technische Universit\"at
		M\"unchen, D-85747 Garching, Germany
		}
	\thanks{
		E-Mail: herrl@feynman.t30.physik.tu-muenchen.de
		}
	}

\begin{document}

% \bibliographystyle{physics}

\begin{abstract}
We report on the results of a calculation of next-to leading order
short distance QCD corrections to the coefficient $\eta_1$ of the
effective $\Delta S = 2$ Lagrangian in the standard model and discuss
the uncertainties inherent in such a calculation.
As a phenomenological application we comment on the contributions of
short distance physics to the ${\rm K}_{\rm L}$--${\rm K}_{\rm S}$
mass difference.
\end{abstract}

% typeset title page
\maketitle

\section{Introduction}
This report is based on research work done in collaboration with Ulrich
Nierste \cite{HerrlichNierste:94}.

The prediction of observables in the ${\rm K^{0}}$--$\overline{\rm
K^{0}}$ system forces one to calculate an effective low-energy $\Delta
S = 2$ hamiltonian, which is difficult because of the necessary
inclusion of strong interaction effects.
Applying Wilson's operator product expansion factorizes the Feynman
amplitude into a long-distance part, to be evaluated by
non-perturbative methods, and a short-distance part, which can be
calculated in renormalization group improved perturbation theory.

The short distance part of the $\Delta S = 2$ Feynman amplitude has
first been calculated to leading order (LO) by
Vainstein et al.\ \cite{VainsteinZakharovNovikovShifman:76},
Vysotskij \cite{Vysotskij:80} and
Gilman and Wise \cite{GilmanWise:83}.
These determinations leave certain questions unanswered which we like
to summarize below.
To be specific, consider the effective $\Delta S = 2$ hamiltonian
\begin{eqnarray}
	H
	&=&
	\frac{\GF^2}{16\pi^2}\MW^2
	\biggl[
		\lambda_{\rm c}^2 \eta_1
			S\!\left(\textstyle\frac{m_{\rm c}^2}{\MW^2}\right)
		+ \lambda_{\rm t}^2 \eta_2 \times
\nn \\
	& &
			S\!\left(\textstyle\frac{m_{\rm t}^2}{\MW^2}\right)
		+ 2 \lambda_{\rm c} \lambda_{\rm t} \eta_3
			S\!\left(\textstyle\frac{m_{\rm c}^2}{\MW^2},
				\frac{m_{\rm t}^2}{\MW^2}\right)
	\biggr]
\nn \\
	& &
	\times b\!\left(\mu\right) Q_{\rm LL}\!\left(\mu\right)
\nn \\
	&=&
	H^{\rm c} + H^{\rm t} + H^{\rm ct}
\label{Heff}
\end{eqnarray}
with $\GF$ denoting Fermi's constant,
$\lambda_{j} = V_{j{\rm d}}^{*} V_{j{\rm s}}$, $j={\rm c,t}$ the
relevant combination of CKM factors and $Q_{\rm LL}$ the local
four-quark operator
\begin{eqnarray}
	Q_{\rm LL}
	&=&
	\left[\bar{\rm s} \gamma_{\mu} \left(1-\gamma_{5}\right)
		{\rm d} \right]
	\left[\bar{\rm s} \gamma^{\mu} \left(1-\gamma_{5}\right)
		{\rm d} \right]
\label{QLL}
\end{eqnarray}
In \eq{Heff} the GIM mechanism $\lambda_{\rm u}+\lambda_{\rm
c}+\lambda_{\rm t} = 0$ has been used to eliminate $\lambda_{\rm u}$.
The Inami-Lim functions $S$ are obtained in the evaluation of the
famous box diagrams depicted in \fig{fig:box}.
\begin{figure}[htb]
%\vspace{2truecm}
	\epsfysize=2truecm
	\epsffile{full-lo.pps}
\vspace{-25truept}
\caption{The box diagrams contributing to $H$ in lowest order.
	The zigzag lines represent ${\rm W}$-bosons or fictitious
	Higgs particles.
	}
\label{fig:box}
\end{figure}
The $\eta_{i}$ parametrize the short distance QCD corrections with
their explicit dependence on the renormalization scale $\mu$ factored
out in the function $b\!\left(\mu\right)$.
In absence of QCD corrections $\eta_{i} b\!\left(\mu\right) = 1$.

The LO calculation leaves the following questions unanswered
\begin{list}{$\bullet$}{
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\item The running charm quark mass $m_{\rm c}$ enters \eq{Heff} at the
scale $\mu_{\rm c} \approx m_{\rm c}$, where the dynamic charm quark
is removed from the theory.
The LO result depends strongly on the choice of $\mu_{\rm c}$.
A similar statement applies to the scale $\mu_{\rm W} \approx \MW$,
where the ${\rm W}$-boson and the top quark are integrated out.
\item The precise definition of the QCD scale parameter $\Lambda_{\rm
QCD}$ requires at least a next-to-leading order (NLO) calculation.
\item Subleading terms may contribute sizeable, e.g.\ the LO
hamiltonian reproduces only about 60\% of the observed ${\rm K_{L}}$--
${\rm K_{S}}$ mass difference.
\end{list}
Prior to our work the only part of \eq{Heff} calculated to NLO was
$H^{\rm t}$ containing $\eta_2$ \cite{BurasJaminWeisz:90}.
This report deals with $H^{\rm c}$ and the coefficient $\eta_1$.

\section{The NLO calculation}
All calculations are carried out in the $\MSbar$ scheme using an
arbitrary $R_{\xi}$ gauge for the gluon propagator and
't\ Hooft--Feynman gauge for the ${\rm W}$-propagator.
Inspired by refs.\ \cite{BurasWeisz:90,BurasJaminWeisz:90}
we use an anticommuting $\gamma_{5}$ (NDR scheme).
Infrared singularities get regulated by small quark masses.
We only calculate the lowest nonvanishing order in $m_{\rm c}/\MW$,
therefore setting $S\!\left(m_{\rm c}^2/\MW^2\right)=m_{\rm c}^2/\MW^2$.
This turns out to be necessary, if we only want to keep operators of
the lowest twist.

The calculation is performed in the standard renormalization group
technique.
The matching of different theories at matching scales
$\mu_{\rm W}$, $\mu_{\rm b}$ and $\mu_{\rm c}$
requires the evaluation of the diagram in \fig{fig:eye} and diagrams
derived from this by dressing it with one gluon in all possible
ways.
\begin{figure}
%\vspace{2truecm}
	\epsfysize=1.5truecm
	\epsffile{eff-lo.pps}
\vspace{-25truept}
\caption{
	The lowest order diagram contributing to the effective five
	and four flavour theories between the scales $\mu_{\rm W}$ and
	$\mu_{\rm c}$.
	The crosses denote the insertion of local $\Delta S=1$
	operators.
	}
\label{fig:eye}
\end{figure}
No local $\Delta S=2$ operator appears in the effective five and four
quark theory, because the diagrams mentioned above are finite due
to the GIM mechanism.
Such an operator, it is the $Q_{\rm LL}$ of \eq{QLL}, first arises
after moving to an effective three flavour theory by removing the
${\rm c}$-quark.
For details, we refer the reader to \cite{HerrlichNierste:94}.

To check our calculation, we performed various consistency tests.
\begin{list}{$\bullet$}{
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\item
The Wilson coefficient functions turn out to be independent of the
gluon gauge parameter $\xi$ and the small quark masses used as a
regulator for the infrared singularities.
\item
Setting $\mu_{\rm W}=\mu_{\rm b}=\mu_{\rm c}$, we recover the result
obtained in naive perturbation theory up to the first order in the
strong coupling constant $\alpha_{\rm s}$.
\item
The dependence of the final result on the matching scales $\mu_{\rm
W}$, $\mu_{\rm b}$, $\mu_{\rm c}$ vanishes up to first order in
$\alpha_{\rm s}$.
\end{list}

\section{Numerical results}
In the analytical calculation the running charm quark mass
renormalized at the scale $\mu_{\rm c}$, where it gets incorporated
into the Wilson coefficient function, enters quite naturally.
For the numerical analysis and the discussion of renormalization scale
dependence it is better to use
$m_{\rm c}^{\star} = m_{\rm c}\!\left(m_{\rm c}\right)$.
We therefore define $\eta_1^{\star}$ by
\begin{eqnarray}
	\eta_{1}\!\left(\mu_{\rm W},\mu_{\rm c}\right)
	m_{\rm c}^2\!\left(\mu_{\rm c}\right)
	&=&
	\eta_{1}^{\star}\!\left(\mu_{\rm W},\mu_{\rm c}\right)
	\left(m_{\rm c}^{\star}\right)^2
	.
\label{EtaStar}
\end{eqnarray}
In \tab{tab:eta} we listed $\eta_1^{\star}$ for different values of
the QCD scale parameter in the effective four flavour theory
$\Lambda_{\overline{\rm MS}}=\Lambda_{4}$ and $m_{\rm c}^{\star}$.
\begin{table*}[t]
\caption{
	$\eta_1^{\star}$ for different values of $m_{\rm c}^{\star}$
	and $\Lambda_{\overline{\rm MS}}=\Lambda_{4}$ (both given in
	GeV).
	The matching scales have been taken to be
	$\mu_{\rm W}=\MW=80{\rm GeV}$, $\mu_{\rm b}=4.8{\rm GeV}$ and
	$\mu_{\rm c}=m_{\rm c}^{\star}=m_{\rm c}\!\left(m_{\rm
	c}\right)$.
	}
% \input{eta1-la-mc}
%%%%%%% include file eta1-la-mc.tex
% warning:
% table automatically generated by eta1-la-mc!
% modified by hand Wed Aug 24 17:40:52 MET DST 1994
%
% matching scales: b: 4.800000, t: 80.000000;
% mw = 80.000000;
%
\begin{tabular*}{\textwidth}{@{}c@{\extracolsep{\fill}}cccccccccc}
\hline
$\Lambda_{\overline {\rm MS}}$
 & \multicolumn{2}{c}{0.150}
 & \multicolumn{2}{c}{0.200}
 & \multicolumn{2}{c}{0.250}
 & \multicolumn{2}{c}{0.300}
 & \multicolumn{2}{c}{0.350}
\\
\cline{2-3} \cline{4-5} \cline{6-7} \cline{8-9} \cline{10-11}
$m_c^{\star}$ & LO & NLO & LO & NLO & LO & NLO & LO & NLO & LO & NLO\\
\hline
1.25 &  0.809 &  0.885 &  0.895 &  1.007 &  0.989 &  1.154 &  1.096 &  1.334 &
1.216 &  1.562 \\
1.30 &  0.797 &  0.868 &  0.877 &  0.982 &  0.965 &  1.117 &  1.064 &  1.281 &
1.175 &  1.485 \\
1.35 &  0.786 &  0.854 &  0.861 &  0.960 &  0.944 &  1.085 &  1.035 &  1.235 &
1.138 &  1.419 \\
1.40 &  0.775 &  0.840 &  0.847 &  0.940 &  0.924 &  1.056 &  1.010 &  1.194 &
1.105 &  1.361 \\
1.45 &  0.766 &  0.828 &  0.834 &  0.922 &  0.907 &  1.030 &  0.987 &  1.157 &
1.075 &  1.310 \\
1.50 &  0.757 &  0.817 &  0.822 &  0.905 &  0.890 &  1.006 &  0.966 &  1.125 &
1.048 &  1.265 \\
1.55 &  0.749 &  0.806 &  0.810 &  0.890 &  0.876 &  0.985 &  0.946 &  1.095 &
1.024 &  1.225 \\
\hline
\end{tabular*}
%%%%%%% end of included file eta1-la-mc.tex
\label{tab:eta}
\end{table*}
\begin{table*}[t]
\caption{
	$\Delta m_{\rm K}^{\rm SD,c}$ in NLO in units of $10^{-15}{\rm
	GeV}$ for different values of $\Lambda_{\overline{\rm
	MS}}=\Lambda_{4}$ and $\mu_{\rm c}=m_{\rm c}^{\star}$ (both
	given in GeV).
	$B_{\rm K} = 0.7$
	}
% \input{dm-la-mc}
%%%%%% included file dm-la-mc.tex
% warning:
% table automatically generated by dm-la-mc!
% modified by hand Wed Aug 24 19:16:41 MET DST 1994
%
% matching scales: b: 4.800000, t: 80.000000;
% mw = 80.000000;
% mK = 0.498000, fK = 0.161000;
% Re[la_c] = -0.215000;
% BK = 0.700000;
%
% fore = 6141.473865;
%
\begin{tabular*}{\textwidth}{@{}c@{\extracolsep{\fill}}cccccccccc}
\hline
$\Lambda_{\overline {\rm MS}}$
 & \multicolumn{2}{c}{0.150}
 & \multicolumn{2}{c}{0.200}
 & \multicolumn{2}{c}{0.250}
 & \multicolumn{2}{c}{0.300}
 & \multicolumn{2}{c}{0.350}
\\
\cline{2-3} \cline{4-5} \cline{6-7} \cline{8-9} \cline{10-11}
$m_c^{\star}$ & $\Delta m$ & $\frac{\Delta m}{\Delta m_{\rm EXP}}$ & $\Delta m$
& $\frac{\Delta m}{\Delta m_{\rm EXP}}$ & $\Delta m$ & $\frac{\Delta m}{\Delta
m_{\rm EXP}}$ & $\Delta m$ & $\frac{\Delta m}{\Delta m_{\rm EXP}}$ & $\Delta m$
& $\frac{\Delta m}{\Delta m_{\rm EXP}}$\\
\hline
1.25 &  1.327 &  0.377 &  1.510 &  0.429 &  1.730 &  0.491 &  2.000 &  0.568 &
2.342 &  0.665 \\
1.30 &  1.408 &  0.400 &  1.593 &  0.452 &  1.812 &  0.514 &  2.078 &  0.590 &
2.409 &  0.684 \\
1.35 &  1.493 &  0.424 &  1.679 &  0.477 &  1.897 &  0.539 &  2.159 &  0.613 &
2.481 &  0.705 \\
1.40 &  1.580 &  0.449 &  1.768 &  0.502 &  1.986 &  0.564 &  2.245 &  0.637 &
2.560 &  0.727 \\
1.45 &  1.670 &  0.474 &  1.860 &  0.528 &  2.078 &  0.590 &  2.335 &  0.663 &
2.643 &  0.751 \\
1.50 &  1.763 &  0.501 &  1.955 &  0.555 &  2.173 &  0.617 &  2.428 &  0.689 &
2.731 &  0.776 \\
1.55 &  1.859 &  0.528 &  2.052 &  0.583 &  2.271 &  0.645 &  2.525 &  0.717 &
2.824 &  0.802 \\
\hline
\end{tabular*}
%%%%%% end of included file dm-la-mc.tex
\label{tab:deltaM}
\end{table*}
For $m_{\rm c}^{\star}=1.4{\rm GeV}$ and $\Lambda_{\overline{\rm
MS}}=300{\rm MeV}$ the NLO result is by about 20\% larger than the LO
one.
So the correction turns out to be quite sizable.
To estimate something like a ``theoretical error'', we look at
$\eta_1^{\star}$'s dependence on the three matching scales for fixed
values of $m_{\rm c}^{\star}$ and $\Lambda_{\overline{\rm MS}}$.
The variation with $\mu_{\rm c}$ appears to be the largest one, it is
plotted in \fig{fig:scaledep}.
\begin{figure}[htb]
	\rotate[r]{
	\epsfysize=7truecm
	\epsffile{muc-0.123.ps}
	}
\vspace{-25truept}
\caption{
	The variation of $\eta_1^{\star}$ with the matching scale
	$\mu_{\rm c}$ for different values of $\Lambda_{\overline{\rm
	MS}}=\Lambda_{4}$ and $\mu_{\rm c}$.
	$\mu_{\rm W}=\MW=80{\rm GeV}$, $\mu_{\rm b}=4.8{\rm GeV}$ and
	$m_{\rm c}^{\star}=1.4{\rm GeV}$ have been kept fixed.
	}
\label{fig:scaledep}
\end{figure}
It is clearly seen, that the scale dependence in NLO is about 50\%
smaller than in LO.
E.g.\ for the reference values of $\Lambda_{\overline{\rm MS}}=0.3{\rm
GeV}$, $m_{\rm c}^{\star}=1.4{\rm GeV}$, $\mu_{\rm b}=4.8{\rm GeV}$,
$\mu_{\rm W}=\MW=80{\rm GeV}$ the variation of $\mu_{\rm c}$ from 1.1
to 1.7 GeV amounts to a change of 63\% in the LO case and 34\% in the
NLO result.
\begin{eqnarray}
	\eta_{1}^{\star} &=& 1.01
	\begin{array}{l}
	\scriptstyle+0.42\\[-2.1truemm]\scriptstyle-0.21
	\end{array}
	\mbox{(LO)}
\nn \\
	\eta_{1}^{\star} &=& 1.19
	\begin{array}{l}
	\scriptstyle+0.23\\[-2.1truemm]\scriptstyle-0.17
	\end{array}
	\mbox{(NLO)}
\end{eqnarray}
Fixing $\mu_{\rm c}=m_{\rm c}^{\star}=1.4{\rm GeV}$, we further
analyze the dependence of $\eta_{1}^{\star}$ on the scale $\mu_{\rm
W}$.
Varying $\mu_{\rm W}$ from 60 to 100 GeV, we find a change of
$\eta_{1}^{\star}$ by 7\% in LO and 4\% in NLO.
This residual scale dependence is therefore much weaker than the one
for $\mu_{\rm c}$.
The residual dependence on the matching scale $\mu_{\rm b}$ turns out
to be completely negligible, the extreme choice $\mu_{\rm b}=\mu_{\rm
W}=\MW$, which means neglecting completely the effects from an
effective five flavour theory, leads to an error of the order of 1\%.

We now want to discuss the implications of $\eta_{1}^{\star}$ on the
short distance contribution to the ${\rm K}_{\rm L}$-${\rm K}_{\rm S}$
mass difference.
To a very good approximation the part stemming only from the first two
generations is given by
\cite{BurasSlominskiSteger:84}
\begin{eqnarray}
	\Delta m_{\rm K}^{\rm SD,c}
	&=&
	\frac{\GF^2}{6\pi^2}m_{\rm K} f_{\rm K}^2 B_{\rm K}
	\left({\rm Re} \lambda_{\rm c}\right)^2 m_{\rm c}^{\star 2}
	\eta_{1}^{\star}
	.
\label{MassDiff}
\end{eqnarray}
For the input parameters $m_{\rm K}=0.498{\rm GeV}$, $f_{\rm
K}=0.161{\rm GeV}$, ${\rm Re}\lambda_{\rm c}=0.215$, $\GF=1.167\times
10^{-5}{\rm GeV}^{-2}$ and $B_{\rm K}=0.7$ the mass difference $\Delta
m_{\rm K}^{\rm SD,c}$ is given in \tab{tab:deltaM}.
Since $\Delta m_{\rm K}^{\rm SD}$ depends linearly upon the
nonperturbative parameter $B_{\rm K}$ the result may be easily
rescaled to other values of this parameter.
Note, that if we take $\Lambda_{\overline{\rm MS}}=0.3{\rm GeV}$ and
$m_{\rm c}^{\star}=1.4{\rm GeV}$, the short distance contribution of
the charm sector to the total mass difference is as much as 64\%.
The terms containing the top quark contribute another 6\%, therefore
the short distance physics is able to reproduce about 70\% of the
observed mass difference, which is much more than previously thought.

% \bibliography{../diss/litstd,../diss/litown,../diss/litpap}
%%%%%%%%% included bib file
\begin{thebibliography}{1}

\bibitem{HerrlichNierste:94}
{\sc S.~Herrlich} and {\sc U.~Nierste},
\newblock {\em Nucl. Phys. B} {\bf 419} (1994) 292.

\bibitem{VainsteinZakharovNovikovShifman:76}
{\sc A.~I. Vainstein}, {\sc V.~I. Zakharov}, {\sc V.~A. Novikov}, and {\sc
  H.~A. Shifman},
\newblock {\em Sov. J. Nucl. Phys.} {\bf 23} (1976) 50.

\bibitem{Vysotskij:80}
{\sc M.~I. Vysotskij},
\newblock {\em Sov. J. Nucl. Phys.} {\bf 31} (1980) 797.

\bibitem{GilmanWise:83}
{\sc F.~J. Gilman} and {\sc M.~B. Wise},
\newblock {\em Phys. Rev. D} {\bf 27} (1983) 1128.

\bibitem{BurasJaminWeisz:90}
{\sc A.~J. Buras}, {\sc M.~Jamin}, and {\sc P.~H. Weisz},
\newblock {\em Nucl. Phys. B} {\bf 347} (1990) 491.

\bibitem{BurasWeisz:90}
{\sc A.~J. Buras} and {\sc P.~H. Weisz},
\newblock {\em Nucl. Phys. B} {\bf 333} (1990) 66.

\bibitem{BurasSlominskiSteger:84}
{\sc A.~J. Buras}, {\sc W.~S{\l}ominski}, and {\sc H.~Steger},
\newblock {\em Nucl. Phys. B} {\bf 238} (1984) 529.

\end{thebibliography}
%%%%%%%%% end of included bib file

\end{document}
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% add words to TeX's hyphenation exception list
\hyphenation{author another created financial paper re-commend-ed}

% declarations for front matter
\title{Elsevier instructions for the preparation of a
       2-column format camera-ready paper in \LaTeX}

\author{P. de Groot\address{Mathematics and Computer Science Division, 
        Elsevier Science Publishers B.V., \\ 
        P.O. Box 103, 1000 AC Amsterdam, The Netherlands}%
        \thanks{Footnotes should appear on the first page only to
                indicate your present address (if different from your
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                These are obtained with the {\tt\ttbs thanks} command.}
        and 
        X.-Y. Wang\address{Economics Department, University of Winchester, \\
        2 Finch Road, Winchester, Hampshire P3L T19, United Kingdom}}
       
\begin{document}

\begin{abstract}
These pages provide you with an example of the layout and style for
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your paper. This is the output from the \LaTeX{} document style you
requested.
\end{abstract}

% typeset front matter (including abstract)
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\begin{table*}[hbt]
% space before first and after last column: 1.5pc
% space between columns: 3.0pc (twice the above)
\setlength{\tabcolsep}{1.5pc}
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% adapted from TeX book, p. 241
\newlength{\digitwidth} \settowidth{\digitwidth}{\rm 0}
\catcode`?=\active \def?{\kern\digitwidth}
% -----------------------------------------------------
\caption{Biologically treated effluents (mg/l)}
\label{tab:effluents}
\begin{tabular*}{\textwidth}{@{}l@{\extracolsep{\fill}}rrrr}
\hline
                 & \multicolumn{2}{l}{Pilot plant} 
                 & \multicolumn{2}{l}{Full scale plant} \\
\cline{2-3} \cline{4-5}
                 & \multicolumn{1}{r}{Influent} 
                 & \multicolumn{1}{r}{Effluent} 
                 & \multicolumn{1}{r}{Influent} 
                 & \multicolumn{1}{r}{Effluent}         \\
\hline
Total cyanide    & $ 6.5$ & $0.35$ & $  2.0$ & $  0.30$ \\
Method-C cyanide & $ 4.1$ & $0.05$ &         & $  0.02$ \\
Thiocyanide      & $60.0$ & $1.0?$ & $ 50.0$ & $ <0.10$ \\
Ammonia          & $ 6.0$ & $0.50$ &         & $  0.10$ \\
Copper           & $ 1.0$ & $0.04$ & $  1.0$ & $  0.05$ \\
Suspended solids &        &        &         & $<10.0?$ \\
\hline
\multicolumn{5}{@{}p{120mm}}{Reprinted from: G.M. Ritcey,
                             Tailings Management,
                             Elsevier, Amsterdam, 1989, p. 635.}
\end{tabular*}
\end{table*}

\section{TABLES AND ILLUSTRATIONS}

Tables should be made with \LaTeX; illustrations should be originals or
sharp prints. They should be arranged throughout the text and
preferably be included {\em on the same page as they are first
discussed}. They should have a self-contained caption and be positioned
in flush-left alignment with the text margin within the column. If they
do not fit into one column they may be placed across both columns
(using \verb-\begin{table*}- or \verb-\begin{figure*}- so that they
appear at the top of a page).

\subsection{Tables}

Tables should be presented in the form shown in
Table~\ref{tab:effluents}.  Their layout should be consistent
throughout.

Horizontal lines should be placed above and below table headings, above
the subheadings and at the end of the table above any notes. Vertical
lines should be avoided.

If a table is too long to fit onto one page, the table number and
headings should be repeated above the continuation of the table. For
this you have to reset the table counter with
\verb|\addtocounter{table}{-1}|. Alternatively, the table can be turned
by $90^\circ$ (`landscape mode') and spread over two consecutive pages
(first an even-numbered, then an odd-numbered one) created by means of
\verb|\begin{table}[h]| without a caption. To do this, you prepare the
table as a separate \LaTeX{} document and attach the tables to the
empty pages with a few spots of suitable glue.

\subsection{Line drawings}

Line drawings should be drawn in India ink on tracing paper with the
aid of a stencil or should be glossy prints of the same; computer
prepared drawings are also acceptable. They should be attached to your
manuscript page, correctly aligned, using suitable glue and {\em not
transparent tape}. When placing a figure at the top of a page, the top
of the figure should be at the same level as the bottom of the first
text line.

All notations and lettering should be no less than 2\,mm high. The use
of heavy black, bold lettering should be avoided as this will look
unpleasantly dark when printed.

\subsection{Black and white photographs}

Photographs must always be sharp originals ({\em not screened
versions\/}) and rich in contrast. They will undergo the same reduction
as the text and should be pasted on your page in the same way as line
drawings.

\subsection{Colour photographs}

Sharp originals ({\em not transparencies or slides\/}) should be
submitted close to the size expected in publication. Charges for the
processing and printing of colour will be passed on to the author(s) of
the paper. As costs involved are per page, care should be taken in the
selection of size and shape so that two or more illustrations may be
fitted together on one page. Please contact the Technical Editor in the
Camera-Ready Publications Department at Elsevier for a price quotation
and layout instructions before producing your paper in its final form.

\begin{figure}[htb]
\vspace{9pt}
\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\caption{Good sharp prints should be used and not (distorted) photocopies.}
\label{fig:largenenough}
\end{figure}
%
\begin{figure}[htb]
\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\caption{Remember to keep details clear and large enough.}
\label{fig:toosmall}
\end{figure}

\section{EQUATIONS}

Equations should be flush-left with the text margin; \LaTeX{} ensures
that the equation is preceded and followed by one line of white space. 
\LaTeX{} provides the document-style option {\tt fleqn} to get the
flush-left effect.

\begin{equation}
H_{\alpha\beta}(\omega) = E_\alpha^{(0)}(\omega) \delta_{\alpha\beta} +
                          \langle \alpha | W_\pi | \beta \rangle 
\end{equation}

You need not put in equation numbers, since this is taken care of
automatically. The equation numbers are always consecutive and are
printed in parentheses flush with the right-hand margin of the text and
level with the last line of the equation. For multi-line equations, use
the {\tt eqnarray} environment. For complex mathematics, use the
\AmS-\LaTeX{} package.

\begin{thebibliography}{9}
\bibitem{Scho70} S. Scholes, Discuss. Faraday Soc. No. 50 (1970) 222.
\bibitem{Mazu84} O.V. Mazurin and E.A. Porai-Koshits (eds.),
                 Phase Separation in Glass, North-Holland, Amsterdam, 1984.
\bibitem{Dimi75} Y. Dimitriev and E. Kashchieva, 
                 J. Mater. Sci. 10 (1975) 1419.
\bibitem{Eato75} D.L. Eaton, Porous Glass Support Material,
                 US Patent No. 3 904 422 (1975).
\end{thebibliography}

References should be collected at the end of your paper. Do not begin
them on a new page unless this is absolutely necessary. They should be
prepared according to the sequential numeric system making sure that
all material mentioned is generally available to the reader. Use
\verb+\cite+ to refer to the entries in the bibliography so that your
accumulated list corresponds to the citations made in the text body. 

Above we have listed some references according to the
sequential numeric system \cite{Scho70,Mazu84,Dimi75,Eato75}.
\end{document}

