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%% subsection 2.3 The measurement function [slac-pub-7073-0-0-2-3 in slac-pub-7073-0-0-2: slac-pub-7073-0-0-2-4]
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\subsection{\usemenu{slac-pub-7073::context::slac-pub-7073-0-0-2-3}{The measurement function}}\label{subsection::slac-pub-7073-0-0-2-3}
Using the jet-finding algorithm of the previous section, we can
explicitly define the measurement functions we need to construct
the jet cross section.
When there are only three partons in the final state, no merging
is possible, and the partons themselves will eventually result
in physical jets. Therefore
\beqn
{\cal S}_3&=&\sum_{\sigma(J)}\delta(j,k,l)
\Th(min(d_j,d_k,d_l)-f\,max(d_j,d_k,d_l))
\nonumber \\*&&\,\,\times\,
\Th(R_{jk}-D^2)\Th(R_{jl}-D^2)\Th(R_{kl}-D^2)\,,
\label{S3def}
\eeqn
where we introduced the shorthand notation for the $\delta$
over four-momenta
\beq
\delta(j,k,l)=\delta(k_j-J_1)\delta(k_k-J_2)\delta(k_l-J_3).
\eeq
The indices $j$, $k$ and $l$ take the values $3,~4,~5$ and are different
from each other. $\sigma(J)$ denotes the permutation over the
jet four-momenta $\Jetlist$.
When four partons are present in the final state, the situation is
somewhat more involved. To get three jets starting from four
partons, only the following possibilities may occur:
$\bullet$ no merging, but one parton is dropped being below the hard
scale (this contribution will be denoted by ${\cal S}_i^{(0)}$);
$\bullet$ one merging, occurring in the first step of the algorithm
(${\cal S}_{ij}^{(1)}$);
$\bullet$ one merging, occurring in the second step of the algorithm
(${\cal S}_i^{(2)}$);
$\bullet$ one merging, occurring in the third step of the algorithm
(${\cal S}_i^{(3)}$).
\noindent
Consistently, we will then write
\beq
{\cal S}_4=\sum_i\left({\cal S}_i^{(0)}+{\cal S}_i^{(2)}
+{\cal S}_i^{(3)}\right)+\sum_{\stackrel{i,j}{i