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%% subsection 5.1 \it General Framework [slac-pub-7111-0-0-5-1 in slac-pub-7111-0-0-5: ^slac-pub-7111-0-0-5 >slac-pub-7111-0-0-5-2]
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\subsection{\usemenu{slac-pub-7111::context::slac-pub-7111-0-0-5-1}{\it General Framework}}\label{subsection::slac-pub-7111-0-0-5-1}
First we review briefly the situation at tree level. Color-ordered
amplitudes can have poles only in channels corresponding to the
sum of {\it cyclicly adjacent} momenta, that is as $P^2_{i,j} \to 0$,
where $P^\mu_{i,j} \equiv (k_i+k_{i+1}+\cdots+k_j)^\mu$. This is
because singularities arise from propagators going on-shell, and
propagators for color-ordered graphs always carry momenta of the form
$P^\mu_{i,j}$. The general form of an $n$-point color-ordered tree
amplitude in the limit that $P^2_{1,m}$ vanishes is
$$
\eqalign{
& A_n^\tree(1,\ldots,n)
\mathop{\hbox{\Large$\longrightarrow$}}^{P^2_{1,m} \rightarrow 0}\ \cr
& \hskip .8 cm
\sum_{\lambda=\pm} A_{m+1}^\tree(1,\ldots,m, P^\lambda)
{i\over P^2_{1,m} }
A_{n-m+1}^\tree(m+1,\ldots,n, P^{-\lambda}) \,, \cr}
\equn\label{TreeFactorization}
$$
where $P_{1,m}$ is the intermediate momentum, $A^\tree_{m+1}$ and
$A^\tree_{n-m+1}$ are lower-point scattering amplitudes, and
$\lambda$ denotes the helicity of the intermediate state $P$. The
intermediate helicity is reversed going from one product amplitude to
the other because of the outgoing-particle helicity convention.
For two-particle channels ($m=2$), \eqn{TreeFactorization} needs to be
modified, because a three-point massless scattering amplitude is not
kinematically possible. As $P^2_{12}\to0$, $k_1$ and $k_2$ become
collinear. QCD amplitudes have an angular-momentum obstruction in
this limit. For example, a gluon of helicity $+1$ cannot split into
two collinear helicity $\pm 1$ gluons and conserve angular momentum.
This transforms the full pole in $P^2_{12} = s_{12}$ into the
square-root of a pole, $1/\sqrt{s_{12}}$, a behavior which is well
captured via the spinor products $\spa1.2,\spb1.2$. It is useful to
lump all terms not associated with $A_{n-1}^\tree$
in~\eqn{TreeFactorization} into a `splitting amplitude'
$\Split^\tree$. In particular, as the momenta of adjacent legs $a$
and $b$ become collinear, we have
$$
A_n^\tree(\ldots,a^{\lambda_a},b^{\lambda_b},\ldots)
\ \mathop{\longrightarrow}^{a \parallel b}\
\sum_{\lambda=\pm}
\Split^\tree_{-\lambda}(z,a^{\lambda_a},b^{\lambda_b}) \,
A_{n-1}^\tree(\ldots,P^\lambda,\ldots) \,,
\equn\label{TreeSplit}
$$
where $P$ is the intermediate state with momentum $k_P=k_a+k_b$,
$\lambda$ denotes the helicity of $P$, and $z$ is the
longitudinal momentum fraction, $k_a \approx zk_P$, $k_b \approx (1-z)
k_P$. The universality of these limits can be derived
diagrammatically, but an elegant way to derive it is from string
theory \cite{3}, because all the field theory diagrams on
each side of the pole are lumped into one string diagram.
Given the general form (\docLink{slac-pub-7111-0-0-5.tcx}[TreeSplit]{51}), one may obtain explicit
expressions for the tree-level $g\to gg$ splitting amplitudes from the
four- and five-gluon amplitudes
\cite{53,3,21}. For example, taking the
collinear limits of~\eqn{gggggmmppptree} for
$A_5^\tree(1^-,2^-,3^+,4^+,5^+)$ and comparing
to~eqs.~(\docLink{slac-pub-7111-0-0-4.tcx}[ggggmmpptree]{33}) and~(\docLink{slac-pub-7111-0-0-5.tcx}[TreeSplit]{51}) shows that
$$
\eqalign{
\Split^\tree_{-}(a^{-},b^{-}) &= 0 \,, \cr
%
\Split^\tree_{-}(a^{+},b^{+})
& = {1\over \sqrt{z (1-z)}\spa{a}.b} \,, \cr
%
\Split^\tree_{+}(a^{+},b^{-})
& = {(1-z)^2\over \sqrt{z (1-z)}\spa{a}.b} \,, \cr
%
\Split^\tree_{+}(a^{-},b^{+})
&= {z^2\over \sqrt{z (1-z)}\spa{a}.b} \,. \cr}
\equn\label{TreeSplittingFunctions}
$$
The remaining helicity configurations are obtained using parity.
The $g\to \qb q$ and $q \to qg$ splitting amplitudes can be
obtained in similar fashion.
The situation for color-ordered one-loop amplitudes is similar to
tree level. The one-loop analog of \eqn{TreeFactorization} is
schematically depicted in \fig{MultiFactFigure}, and is given by
$$
\hskip -.2 cm
\eqalign{
& A_n^{\rm loop}(1,\ldots,n)
\mathop{\hbox{\Large$\longrightarrow$}}^{P^2_{1,m} \rightarrow 0}\ \cr
& \hskip .2 cm
\sum_{\lambda=\pm} \biggl[
A_{m+1}^{\rm loop} (1,\ldots,m,P^\lambda)
{i\over P^2_{1,m} }
A_{n-m+1}^\tree(m+1,\ldots,n,P^{-\lambda}) \cr
%
& \hskip .4 cm +
A_{m+1}^\tree(1,\ldots,m,P^\lambda)
{i\over P^2_{1,m} }
A_{n-m+1}^{\rm loop}(m+1,\ldots,n,P^{-\lambda}) \cr
%
& \hskip .4 cm +
A_{m+1}^\tree(1,\ldots,m,P^\lambda)
{i \, \Fact_n(1, \!\ldots\!, n)\over P^2_{1,m} }
A_{n-m+1}^\tree(m+1,\ldots,n,P^{-\lambda}) \biggr]\,, \cr }
\equn\label{LoopFact}
$$
where the one-loop {\it factorization function}, $\Fact_n$, is
independent of helicities and does not cancel the pole in $P^2_{1,m}$.
In an infrared divergent theory, such as QCD, amplitudes do not
factorize `naively': $\Fact_n$ may contain logarithms of kinematic
invariants built out of momenta from {\it both\/} sides of the pole in
$P^2_{1,m}$; $\ln(-s_{n, 1})$ is an example of such a logarithm. The
factorization functions are nonetheless universal functions depending
on the infrared divergences present in the amplitudes
\cite{54}.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=MultiFact.eps,width=3.6in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{MultiFactFigure}
A schematic representation of the behavior of one-loop amplitudes as
a kinematic invariant vanishes. }
\end{figure}
%
%
The collinear limits for color-ordered one-loop amplitudes are a
special case and have the form
$$
\eqalign{
A_{n}^{\rm loop}\ \mathop{\longrightarrow}^{a \parallel b}\
\sum_{\lambda=\pm} \biggl(
& \Split^\tree_{-\lambda} (z, a^{\lambda_a},b^{\lambda_b})\,
A_{n-1}^{\rm loop}(\ldots(a+b)^\lambda\ldots) \cr
& + \Split^{\rm loop}_{-\lambda}(z,a^{\lambda_a},b^{\lambda_b})\,
A_{n-1}^\tree(\ldots(a+b)^\lambda\ldots) \biggr) \,,
\cr}
\equn\label{LoopSplit}
$$
which is schematically depicted in \fig{CollinearFigure}. The
splitting amplitudes
$\Split^\tree_{-\lambda}(a^{\lambda_a},b^{\lambda_b})$ and
$\Split^{\rm loop}_{-\lambda}(a^{\lambda_a},b^{\lambda_b})$ are
universal: they depend only on the two momenta becoming collinear, and
not upon the specific amplitude under consideration. The explicit
$\Split^{\rm loop}_{-\lambda}(a^{\lambda_a},b^{\lambda_b})$ were
originally determined from the four- and five-point one-loop
amplitudes \cite{22,24} in much the same way as we
obtained the tree-level splitting amplitudes above. (See appendix~B
of ref.~\cite{14}.) Soft limits --- the behavior as any
particular $k_i \rightarrow 0$ --- are also useful for constraining
the form of one-loop amplitudes, and have a form analogous to
\eqn{LoopSplit}.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=Collinear.eps,width=2.8in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{CollinearFigure}
A schematic representation of the behavior of one-loop amplitudes as
the momenta of two legs become collinear. }
\end{figure}
%
%
In performing explicit calculations, factorization provides an
extremely stringent check since one must obtain the correct limits in
all channels. A sign or labeling error, for example, will invariably
be detected in some limits. In some cases one can also use
factorization to construct ans\"atze for higher-point amplitudes
\cite{12,14}. One writes down a sufficiently general form
for a higher-point amplitude, containing arbitrary coefficients which
are then fixed by imposing the correct behavior as kinematic variables
vanish. A collinear bootstrap of this form would, however, miss
functions that are nonsingular in all collinear limits. For
five-point amplitudes it is possible to write down such a function,
namely
$$
{\varepsilon(1,2,3,4)
\over \spa1.2 \spa2.3 \spa3.4 \spa4.5 \spa5.1}
\, ,
\equn\label{FivePointAmbiguity}
$$
since the contracted antisymmetric tensor $\varepsilon(1,2,3,4) \equiv
4i \varepsilon_{\mu\nu\rho\sigma} k_1^\mu k_2^\nu k_3^\rho k_4^\sigma$
vanishes when any two of the five vectors $k_i$ become collinear
(using $\sum_{i=1}^5 k_i = 0$). However, it is quite possible that the
factorization constraint uniquely specifies the rational functions of
color-ordered $n\ge 6$-point amplitudes, given the lower point
amplitudes. A heuristic explanation of this conjecture is that as the
number of external legs increases, by dimensional analysis the
amplitudes require ever increasing powers of momenta in the
denominators. Thus one expects more kinematic poles from the
denominator than zeros from the numerator. We know of no counter-examples to
this conjecture, but don't have a proof either.