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%% subsection 4.1 \it Cutkosky Rules [slac-pub-7111-0-0-4-1 in slac-pub-7111-0-0-4: ^slac-pub-7111-0-0-4 >slac-pub-7111-0-0-4-2]
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\subsection{\usemenu{slac-pub-7111::context::slac-pub-7111-0-0-4-1}{\it Cutkosky Rules}}\label{subsection::slac-pub-7111-0-0-4-1}
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=FourPtCut.eps,width=3.in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{FourPtCutFigure}
The $s$- and $t$-channel cuts of a one-loop four-gluon amplitude.
The cut lines can be gluons, fermions, or scalars.}
\end{figure}
%
%
Consider the $s$-channel cut of the four-point amplitude
represented pictorially in \fig{FourPtCutFigure}a. The Mandelstam
variables are as usual $s=(k_1+k_2)^2$ and $t=(k_2+k_3)^2$.
According to the Cutkosky rules, the $s$-channel cut (with $s>0$ and
$t<0$) of this amplitude is
$$
\eqalign{
-i\, {\rm Disc} \;
A_{4;1}(1,2,3,4)\Bigr|_{\scut} & =
\int {d^{4-2\eps} p \over (2\pi)^{4-2\eps}} \;
2\pi \delta^{\tiny(\!\tiny+\!)}(\ell_1^2)\, 2\pi\delta^{\tiny(\!\tiny+\!)}
(\ell_2^2) \cr
& \hskip .3 cm \times
A_4^\tree (-\ell_1, 1,2,\ell_2) \,
A_4^\tree (-\ell_2 ,3,4 ,\ell_1) \, , \cr}
\equn
\label{TreeProduct}
$$
where $\ell_1=p$ and $\ell_2 = p - k_1 - k_2$,
$\delta^{\tiny(+)}$ is
the positive-energy branch of the delta-function
and `Disc' means the discontinuity across the branch cut.
Color-ordering requires us to maintain the clockwise
ordering of the legs in sewing the tree amplitudes.
Suppose the amplitude had the form $A_{4;1} = c \ln(-s) + \cdots
= c (\ln|s| - i \pi) + \cdots$,
where the coefficient $c$ is a rational function.
Then the phase space integral~(\docLink{slac-pub-7111-0-0-4.tcx}[TreeProduct]{24})
would generate the $ i \pi$ term but drop the
$\ln|s|$ term. Since we wish to obtain both types of terms, real and
imaginary, we replace the phase-space integral by the cut of an
unrestricted loop momentum integral \cite{14};
that is, we replace the $\delta$-functions with Feynman propagators,
$$
\eqalign{
A_{4;1}&(1, 2, 3, 4)\Bigr|_{\scut} = \cr
& \hskip -3mm\left.
\left[
\int\! {d^{4-2\eps}p\over (2\pi)^{4-2\eps}} \;
{i\over \ell_1^2 } \,
A_4^\tree (-\ell_1, 1,2,\ell_2) \,{i\over \ell_2^2} \,
A_4^\tree (-\ell_2 ,3,4,\ell_1) \right]\right|_{\scut} \, .\cr}
\equn
\label{TreeProductDef}
$$
While \eqn{TreeProduct} includes only imaginary parts,
\eqn{TreeProductDef} contains both real and imaginary parts. As
indicated, \eqn{TreeProductDef} is valid only for those terms with an
$s$-channel branch cut; terms without an $s$-channel cut may not be
correct. A very useful property of this formula is that one may
continue to use on-shell conditions for the cut intermediate legs
inside the tree amplitudes without affecting the result. Only terms
containing no cut in this channel would change. A similar equation
holds for the $t$-channel cut depicted in \fig{FourPtCutFigure}b.
Combining the two cuts into a single function, one obtains the full
amplitude, up to possible ambiguities in rational functions.
This procedure generalizes to an arbitrary number of external legs.
Isolate the cut in a single momentum channel by taking exactly one
of the momentum invariants to be above threshold, and the rest of
the cyclicly adjacent ones to be negative (space-like).
To construct all terms with cuts in an amplitude, combine the
contributions from the various channels into a single function with
the correct cuts in all channels.
Below we describe how to link the rational functions appearing in
amplitudes to terms with cuts, so that complete
amplitudes can be obtained from Cutkosky rules.