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%% subsection 4.2 \it Cut Constructibility [slac-pub-7111-0-0-4-2 in slac-pub-7111-0-0-4: slac-pub-7111-0-0-4-3]
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\subsection{\usemenu{slac-pub-7111::context::slac-pub-7111-0-0-4-2}{\it Cut Constructibility}}\label{subsection::slac-pub-7111-0-0-4-2}
One-loop amplitudes satisfying a certain power-counting
criterion (for example supersymmetric amplitudes)
can be obtained directly from four-dimensional tree
amplitudes via the Cutkosky rules. That is, when the criterion is
satisfied, one may fix all rational functions appearing in the
amplitudes directly from terms (through $\Ord(\eps^0)$) in the
amplitudes which contain cuts. We refer to such amplitudes as
`cut-constructible'. (Amplitudes not satisfying the criterion can
still be obtained from cuts, but one must evaluate the cuts to
higher order in $\eps$, which is more work.)
In the decomposition of $A_{5;1}$ given in \eqn{gggggmmppploop},
the $N=4$ and $N=1$ supersymmetric components are cut-constructible,
while the scalar component is not. Correspondingly, rational
functions in the first two components (i.e., ${\pi^2\over6}$ and $2$)
are intimately linked to the logarithms, while the last three
rational terms in $A_{5;1}^{\rm scalar}$ are not so linked.
In a one-loop calculation one encounters integrals of the form
$$
\eqalign{
{\cal I}_m[P(p^\mu)] & \equiv \int
{d^{4-2\e}p\over (2\pi)^{4-2\e}} \;
{P(p^\mu)\over p^2 (p-K_1)^2 (p-K_1-K_2)^2 \cdots (p+K_{m})^2}\, \cr
& \equiv {i(-1)^m\over(4\pi)^{2-\e}} I_m[P(p^\mu)]\,, }
\equn\label{GeneralMPoint}
$$
where $m$ is the number of propagators in the loop, $K_i$ are sums of
external momenta $k_i$, and $P(p^\mu)$ is the loop-momentum
polynomial. A {\it cut-constructible} amplitude is one for which one
can arrange that all the $P(p^\mu)$ have degree at most $m-2$, except
for $m=2$ when $P$ should be at most linear. Any amplitude satisfying
this power-counting criterion can be fully reconstructed from its cuts
(through $\Ord(\eps^0)$)~\cite{15}. The basic idea behind the
proof is that only a restricted set of analytic functions appear in a
cut-constructible amplitude. The standard Passarino-Veltman method
\cite{49} reduces the generic tensor integral $I_m[P(p^\mu)]$ to a
linear combination of basic integrals with from 2 to $m$ external
legs. (The kinematics of the lower-point integrals are obtained by
cancelling denominator factors in the original integral. In a
diagrammatic representation of the integrals, this corresponds to
pinching together adjacent external legs.) A key feature of
Passarino-Veltman reduction is that integrals obeying the
power-counting criterion can be reduced entirely to scalar integrals
(integrals with $P=1$). The proof of cut-constructibility is then
based on showing that the cuts provide sufficient information to fix
the coefficients of all the scalar integrals. As we shall exemplify,
amplitudes not satisfying the power-counting criterion contain
additional rational functions, which spoil the argument.
As an illustration, any cut-constructible massless four-point
amplitude must be given by a linear combination of the five scalar
integrals depicted in \fig{FourBasisFigure}. (The triangle integral
with legs $3$ and $4$ pinched is equal to the integral with legs $1$
and $2$ pinched in \fig{FourBasisFigure}b and is therefore not
included in the figure; similarly, the one with $2$ and $3$ pinched is
equal to the one in \fig{FourBasisFigure}c.) All these integrals
can be generated by Passarino-Veltman reduction of a box Feynman
diagram; the triangle and bubble integrals can also be generated from
other Feynman diagrams. (Bubbles on external legs vanish in
dimensional regularization, and are therefore not included.) The
coefficients of the integrals are fixed by the cuts because each
integral contains logarithms unique to it: the box contains the
product $\ln(-s)\ln(-t)$, the triangles $\ln(-s)^2$ or $\ln(-t)^2$,
and the two bubbles contain $\ln(-s)$ or $\ln(-t)$. Consequently no
linear combination of these integrals with rational coefficients can
be formed which is cut-free.
The proof for an arbitrary number of external legs is similar,
although more complicated. By systematically inspecting all scalar
integrals that enter into an $n$-point amplitude, one may
show that the cuts fix the coefficients of all integrals uniquely
\cite{15}. One may also show that the errors induced by ignoring
the difference between using $D=4-2\eps$ and $D=4$ momenta on the cut
legs do not affect the cuts through $\Ord(\eps^0)$.
This observation is of considerable
practical use because $D=4$ tree amplitudes are simpler
than those with legs in $D=4-2\eps$.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=FourBasis.eps,width=1.8in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{FourBasisFigure}
The independent scalar integrals that may appear in a massless
four-point calculation.}
\end{figure}
%
%
The proof breaks down for amplitudes that do not satisfy the
power-counting criterion. For example, the scalar bubble with
momentum $K$,
$$
I_2[1](K) = {\rg\over\e(1-2\e)}(-K^2)^{-\e}
= {1\over\e} + \ln(-K^2 ) + 2 + \Ord(\eps),
\equn\label{ScalarBubbleExample}
$$
obeys the criterion. It contains a rational function, `2', but the latter is
always accompanied by $\ln(-K^2)$.
On the other hand, the linear combination
$$
\Bigl( {K^{\mu}K^{\nu} \over 3} -{\eta^{\mu\nu} K^2 \over 12 } \Bigr)
I_2[1](K) -I_2[p^{\mu} p^{\nu}](K)
= -{ 1 \over 18} ( {K^{\mu}K^{\nu} }-\eta^{\mu\nu} K^2 ) + \Ord(\eps)
\equn\label{NoCutsExample}
$$
does not obey the criterion, because $I_2[p^{\mu} p^{\nu}](K)$ is
quadratic in the loop momentum. The combination~(\docLink{slac-pub-7111-0-0-4.tcx}[NoCutsExample]{28})
is free of cuts through $\Ord(\eps^0)$; there is no logarithm attached to
it at this order.
The presence of such a combination within an amplitude cannot be detected
using the $\Ord(\eps^0)$ cuts.
In general, the power counting associated with a given amplitude
depends on the specific gauge choice and diagrammatic organization.
However, it suffices to find one organization of the diagrams satisfying
the power-counting criterion. The string-inspired method discussed in
section~\docLink{slac-pub-7111-0-0-3.tcx}[StringInspiredSection]{3} provides such an organization; it
can satisfy the power-counting criterion even when the corresponding
diagrams in conventional Feynman gauge do not.
%
An important class of cut-constructible amplitudes are those in supersymmetric
gauge theory. In section~\docLink{slac-pub-7111-0-0-3.tcx}[SusyDecompositionSubsection]{3.2} we showed
that for $n$-gluon amplitudes the leading two powers of loop momentum
cancel in a supermultiplet contribution; the same result holds for
amplitudes with external fermions~\cite{15}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%