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%% subsection 4.3 Unitarity constraints [slac-pub-7106-0-0-4-3 in slac-pub-7106-0-0-4: slac-pub-7106-0-0-4-4]
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\subsection{\usemenu{slac-pub-7106::context::slac-pub-7106-0-0-4-3}{Unitarity constraints}}\label{subsection::slac-pub-7106-0-0-4-3}
In Section 3.4 we discussed the analytic behavior of tree amplitudes,
namely their pole structure. At the loop level, amplitudes have cuts
as well as poles. I won't elaborate on the
factorization (pole) structure of one-loop amplitudes, but they do
exhibit the same kind of universality as tree amplitudes,
which leads to strong constraints and consistency
checks on calculations.\cite{41,9,42}
Unitarity of the $S$-matrix, $S^\dagger S=1$, implies that the
scattering $T$ matrix, defined by $S=1+iT$, obeys
$(T-T^\dagger)/i = T^\dagger T$.
One can expand this equation perturbatively in $g$, and recognize
the matrix sum on the right-hand side as including an integration
over momenta of intermediate states.
Thus the imaginary or absorptive parts of loop amplitudes ---
which contain the branch-cut information --- can be
determined from phase-space integrals of products of lower-order
amplitudes.\cite{43}
For one-loop multi-parton amplitudes, there are several reasons why
this calculation of the cuts is much easier than a direct loop
calculation:
\par\noindent
$\bullet$ One can simplify the tree amplitudes {\it before} feeding
them into the cut calculation.
\par\noindent
$\bullet$ The tree amplitudes are usually quite simple, because they
possess ``effective'' supersymmetry, even if the full loop amplitudes
do not.
\par\noindent
$\bullet$ One can further use on-shell conditions for the intermediate
legs in evaluating the cuts.
The catch is that it is not always possible to reconstruct the
full loop amplitude from its cuts. In general there can be an
additive ``polynomial ambiguity'' --- besides the usual
logarithms and dilogarithms of loop amplitudes, one may add
polynomial terms (actually rational functions) in the kinematic
variables, which cannot be detected by the cuts.
This ambiguity turns out to be absent in one-loop massless
supersymmetric amplitudes, due to the loop-momentum
cancellations discussed in Section 4.1.\cite{9,33}
For example, in the five-gluon amplitude, \eqn{gggggmmppploop},
all the polynomial terms in both $A^{N=4}$ and $A^{N=1}$ are
intimately linked to the logarithms, while in $A^{\rm scalar}$
they are not linked.
The polynomial terms in non-supersymmetric one-loop amplitudes
cannot generally be reconstructed from unitarity cuts evaluated
in four-dimensions.
It is possible to use dimensional analysis to extract the
${\cal O}(\e^0)$ polynomial terms if one has evaluated the cuts
to ${\cal O}(\e)$ in dimensional regularization,\cite{44}
but this task is significantly harder than evaluation to
${\cal O}(\e^0)$.
In practice, polynomial ambiguities can often be fixed, recursively in
the number of external legs, by requiring consistent collinear
factorization of an amplitude in all
channels.\cite{41,42}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%======== figure ===========================
\figmac{3.8}{samesidecut}{SameSideCutFigure}{}
{The possible intermediate helicities for a cut of a MHV amplitude,
when both negative helicity gluons lie on the same side of the cut.
\hfill}
%============================================