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%% subsection 4.2 Loop Integral Reduction [slac-pub-7106-0-0-4-2 in slac-pub-7106-0-0-4: slac-pub-7106-0-0-4-3]
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\subsection{\usemenu{slac-pub-7106::context::slac-pub-7106-0-0-4-2}{Loop Integral Reduction}}\label{subsection::slac-pub-7106-0-0-4-2}
Even if one takes advantage of the various techniques already outlined,
loop calculations with many external legs can still be very complex.
Most of the complication arises at the stage of doing the loop
integrals. The general one-loop $m$-point integral in $4-2\e$
dimensions (for vanishing internal particle masses) is
\be \label{eq:MPointLoopIntegral}
I_m\bigl[P(\ell^\mu)\bigr] =
\int{d^{4-2\e}\ell \over (2\pi)^{4-2\e} }
{ P(\ell^\mu) \over
\ell^2 (\ell-k_1)^2 (\ell-k_1 - k_2)^2
\cdots (\ell-k_1-k_2 - \cdots - k_{m-1})^2 }
\ee
where $k_i$, $i=1,\ldots,m$, are the momenta flowing out of the loop at
leg $i$, and $P(\ell^\mu)$ is a polynomial in the loop momentum.
As we'll outline, \eqn{MPointLoopIntegral} can be
reduced recursively to a linear combination of
{\it scalar} integrals $I_m[1]$, where $m=2,3,4$.
The problem is that for large $m$ the
reduction coefficients can depend on many kinematic variables, and
are often unwieldy and contain spurious singularities.
Here we illustrate one reduction procedure
that works well for large $m$.\cite{35}
If $m\geq5$, then for generic kinematics we have at least four
independent momenta, say $p_1=k_1$, $p_2=k_1+k_2$, $p_3=k_1+k_2+k_3$,
$p_4=k_1+k_2+k_3+k_4$. We can define a set of dual momenta $v_i^\mu$,
\bea
\hskip-0.5cm
&& v_1^\mu = \pol(\mu,2,3,4), \quad
v_2^\mu = \pol(1,\mu,3,4), \quad
v_3^\mu = \pol(1,2,\mu,4), \quad
v_4^\mu = \pol(1,2,3,\mu),
\nonumber \\
\label{eq:dualmomenta}
\hskip-0.5cm
&&v_i \cdot p_j = \pol(1,2,3,4)\,\delta_{ij},
\eea
and expand the loop momentum in terms of them,
\bea
\ell^\mu &=& {1\over \pol(1,2,3,4)}
\sum_{i=1}^4 v_i^\mu\ \ell\cdot p_i
\nonumber \\
\label{eq:expandloop}
&=& {1\over 2\pol(1,2,3,4)}
\sum_{i=1}^4 v_i^\mu \bigl[ \ell^2 - (\ell-p_i)^2 + p_i^2 \bigr]\ .
\eea
The first step can be verified by contracting both sides with
$p_j^\mu$. In the second step we rewrite $\ell^\mu$ in terms of
the propagator denominators in \eqn{MPointLoopIntegral},
plus a term independent of the loop momentum.
If we insert \eqn{expandloop} into the
degree $p$ polynomial $P(\ell^\mu)$ in \eqn{MPointLoopIntegral},
the former terms cancel propagator denominators, turning an
$m$-point loop integral into $(m-1)$-point integrals with polynomials
of degree $p-1$, while the latter term remains an $m$-point integral,
also of degree $p-1$.
Iterating this procedure, $m$-point integrals can be reduced
to box integrals ($m=4$) plus scalar $m$-point integrals.
Equation~\docLink{slac-pub-7106-0-0-4.tcx}[eq:expandloop]{101} is only valid for the four-dimensional
components of the loop momentum, so one has to be careful when applying
it to dimensionally-regulated amplitudes. In practice, when
using the helicity formalism the loop momenta usually end up
contracted with four-dimensional external momenta and polarization
vectors, in which case $\ell^\mu$ is already projected into
four-dimensions.
The strategy of rewriting the loop momentum polynomial $P(\ell^\mu)$
(which may be contracted with external momenta) in terms of the
propagator denominators $\ell^2$, $(\ell-k_1)^2$, etc.
is a very general one. In special cases --- such as the $N=4$
supersymmetric example in Section 4.4 ---
the form of the contracted $P(\ell^\mu)$ often allows a rapid
reduction without having to invoke the general formalism, and
without undue algebra. However, in other cases one may not be so
fortunate.
The scalar integrals for $m\geq6$ can be reduced to lower-point scalar
integrals by a similar technique.\cite{36,35}
For $m\geq6$ we have a fifth independent vector,
$p_5=k_1+k_2+k_3+k_4+k_5$. Contracting \eqn{expandloop} with $p_5$,
we get
\be \label{eq:expandscalarloop}
\ell \cdot p_5\ =\ {1\over \pol(1,2,3,4)}
\sum_{i=1}^4 v_i\cdot p_5\ \ell\cdot p_i ,
\ee
which can be rewritten as an equality relating a sum of six
propagator denominators to a term independent of the loop momentum.
Inserting this equality into the scalar integral $I_m[1]$, we get an
expression for $I_m[1]$ as a linear combination of six ``daughter''
integrals $I_{m-1}^{(i)}[1]$, where the index $(i)$ indicates which
of the $m$ propagators has been cancelled.
A similar formula reduces the scalar pentagon to a sum of five
boxes.\cite{36,35,37,38}
To reduce box integrals with loop momenta in the numerator,
one may employ either a standard Passarino-Veltman
reduction,\cite{39} or one using dual vectors
like that discussed above.\cite{40,25}
These approaches share the property of \eqn{expandloop}, that
in each step the degree of the loop-momentum polynomial drops by one.
Thus supersymmetric cancellations of $m$-point 1PI graphs
down to $\ell^{m-2}$ are maintained under integral reduction.
The final results for an amplitude may therefore be described as a
linear combination of various bubble, triangle and box scalar integrals.
The biggest problem is that the reduction coefficients from the above
procedures contain spurious kinematic singularities,
which should cancel at the end of the day,
but which can lead to very large intermediate
expressions if one is not careful. For example, although the
Levi-Civita contraction $\pol(1,2,3,4)$ appears in the denominator
of \eqn{expandloop}, it has an unphysical singularity when the
four momenta $k_i$ become co-planar, so it should not appear
in the final result. Despite this fact, the above approach actually
does a good job of keeping the number of terms small, and the requisite
cancellations of $\pol(1,2,3,4)$ denominator factors are not so hard
to obtain.
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