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\subsection{\usemenu{slac-pub-7106::context::slac-pub-7106-0-0-3-4}{Factorization Properties}}\label{subsection::slac-pub-7106-0-0-3-4}
Analytic properties of amplitudes are very useful as
consistency checks of the correctness of a calculation, but
they can also sometimes be used to help construct amplitudes.
At tree-level, the principal analytic property is the {\it pole}
behavior as kinematic invariants vanish, due to an almost on-shell
intermediate particle.
As mentioned above, color-ordered amplitudes can only have poles
in channels corresponding to the sum of a sum of {\it cyclically
adjacent} momenta, i.e. 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 $P^\mu_{i,j}$.
We refer to channels formed by three or more adjacent momenta
as multi-particle channels, and the two-particle channels as
collinear channels.
In a multi-particle channel, a true pole can develop as
$P_{1,m}^2 \to 0$,
\be \label{eq:multiparticlepole}
A_n^\tree(1,\ldots,n)\ \sim\
\sum_\lambda 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}),
\ee
where $P_{1,m}$ is the intermediate momentum and
$\lambda$ denotes the helicity of the intermediate state $P$.
Our outgoing-particle helicity convention means that the intermediate
helicity is reversed in going from one product amplitude to the other.
Most multi-parton amplitudes have multi-particle poles, but the MHV
tree amplitudes do not, due to the vanishing of
$A_n^\tree(1^\pm,2^+,\ldots,n^+)$.
When we attempt to factorize an
MHV amplitude on a multi-particle pole, as in \fig{MHVmultipole}(a),
we have only three negative helicities (one from the intermediate
gluon) to distribute among the two product amplitudes. Therefore one
of the two must vanish, so the pole cannot be present. Thus the
vanishing SWI also guarantees the simple structure of the nonvanishing
MHV tree amplitudes: only collinear (two-particle) singularities
of adjacent particles are permitted.
%======== figure ===========================
\figmac{3.5}{factorization}{MHVmultipole}{}
{(a) Factorization of an MHV tree amplitude on a multi-particle pole
--- one of the two product amplitudes always vanishes.
(b) General behavior of a tree-level amplitude in the collinear limit
where $k_a$ is parallel to $k_b$; $S$ stands for the splitting
amplitude $\Split^\tree$.\hfill}
%============================================
An angular momentum obstruction suppresses collinear singularities in
QCD amplitudes. For example, a helicity $+1$ gluon cannot split into
two precisely collinear helicity $\pm1$ gluons and still conserve
angular momentum along the direction of motion. Nor can it split into
a $+\hf$ fermion and $-\hf$ antifermion. The $1/s_{i,i+1}$
from the propagator is cancelled by numerator factors, down to the
square-root of a pole,
${1 \over \sqrt{s_{i,i+1}}} \sim {1 \over \spa{i,}.{i+1}}
\sim {1 \over \spb{i,}.{i+1}}$.
Thus the spinor products, square roots of Lorentz invariants, are ideal
for capturing the collinear behavior in QCD.
The general form of the collinear singularities for tree amplitudes is
shown in \fig{MHVmultipole}(b),
\be \label{eq:treesplit}
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)\ ,
\ee
where $\Split^\tree$ denotes a {\it splitting amplitude},
the intermediate state $P$ has momentum $k_P=k_a+k_b$
and helicity $\lambda$, and $z$ describes the longitudinal momentum
sharing, $k_a \approx zk_P$, $k_b \approx (1-z) k_P$.
Universality of the multi-particle and collinear factorization
limits can be derived in field theory,\cite{20} or
perhaps more elegantly in string theory,\cite{3}
which lumps all the field theory diagrams on each side of the
pole into one string diagram.
An easy way to extract the splitting amplitudes $\Split^\tree$
in \eqn{treesplit} is from the collinear limits of five-point
amplitudes.
For example, the limit of $A_5^\tree(1^-,2^-,3^+,4^+,5^+)$
as $k_4$ and $k_5$ become parallel determines the gluon splitting
amplitude $\Split^\tree_-(a^+,b^+)$:
\bea
A_5^\tree(1^-,2^-,3^+,4^+,5^+) &=&
i\, { {\spa1.2}^4 \over \spa1.2\spa2.3\spa3.4\spa4.5\spa5.1 }
\nonumber \\
&{ {4 \parallel 5} \atop \mathop{\longrightarrow} } &
{1 \over \sqrt{z(1-z)} \spa4.5} \times
\ i\, { {\spa1.2}^4 \over \spa1.2\spa2.3\spa3.{P}\spa{P}.1 }
\nonumber \\
&=& \Split^\tree_-(4^+,5^+) \times A_4^\tree(1^-,2^-,3^+,P^+).
\nonumber \\
\label{eq:explicitfivetofour} &&
\eea
Using also the $2\parallel3$ and $5\parallel1$ limits, plus parity,
we can infer the full set of $g\to gg$ splitting
amplitudes\,\cite{17,21,15,3}
\bea
\Split^{\rm tree}_{-}(a^{-},b^{-}) &=& 0,
\nonumber \\
\Split^{\rm tree}_{-}(a^{+},b^{+})
&=& {1\over \sqrt{z (1-z)}\spa{a}.b},
\nonumber \\
\Split^{\rm tree}_{+}(a^{+},b^{-})
&=& {(1-z)^2\over \sqrt{z (1-z)}\spa{a}.b},
\nonumber \\
\label{eq:gggtree}
\Split^{\rm tree}_{-}(a^{+},b^{-})
&=& -{z^2\over \sqrt{z (1-z)}\spb{a}.b}.
\eea
The $g\to \qb q$ and $q \to qg$ splitting amplitudes are also easy
to obtain, from the limits of \eqn{qqgggmpmpp}, etc.
Since the collinear limits of QCD amplitudes are responsible for
parton evolution, it is not surprising that the residue of the
collinear pole in the square of a splitting amplitude gives the
(color-stripped) polarized Altarelli-Parisi splitting
probability.\cite{22}
\par\noindent
{\bf Exercise:}
Show that the unpolarized $g\to gg$ splitting probability,
from summing over the terms in \eqn{gggtree}, has the familiar form
\be
P_{gg}(z)\ \propto\ { 1+z^4+(1-z)^4 \over z(1-z) }\ ,
\ee
neglecting the plus prescription and $\delta(1-z)$ term.
QCD amplitudes also have universal behavior in the soft limit, where
all components of a gluon momentum vector $k_s$ go to zero.
At tree level one finds
\be \label{eq:treesoftlimit}
A_n^\tree(\ldots,a,s,b,\ldots)
\ \mathop{\longrightarrow}^{k_s\to0}\
\Soft^\tree(a,s,b) \,
A_{n-1}^\tree(\ldots,a,b,\ldots).
\ee
The soft or ``eikonal'' factor,
\be
\Soft^\tree(a,s,b)\ =\ { \spa{a}.{b} \over \spa{a}.{s} \spa{s}.{b} }
\ ,
\ee
depends on both color-ordered neighbors of the soft gluon $s$,
because the sets of graphs where $s$ is radiated from legs
$a$ and $b$ are both singular in the soft limit. On the other hand,
the soft behavior is independent of both the identity
(gluon {\it vs.} quark) and the helicity of partons $a$ and $b$,
reflecting the classical origin of soft radiation.
(See George Sterman's lectures in this volume for a deeper and more
general discussion.\cite{23})
\par\noindent
{\bf Exercise:} Verify the soft behavior, \eqn{treesoftlimit},
for any of the above multiparton tree amplitudes.
As Zoltan Kunszt will explain in more detail,\cite{2}
the universal soft and collinear behavior of tree
amplitudes, and therefore of tree-level cross-sections,
makes possible general procedures for isolating the
infrared divergences in the real, bremsstrahlung contribution to
an arbitrary NLO cross-section, and cancelling these divergences
against corresponding ones in one-loop amplitudes.
But the factorization limits also strongly constrain
the form of tree and loop amplitudes.
It is quite possible that they uniquely determine a rational
function of the $n$-point variables for $n\geq6$, given the lower-point
amplitudes, but this has not yet been proven.
\par\noindent
{\bf Exercise:} Show that
\be \label{eq:collcounterex}
{ \pol(1,2,3,4) \over \spa1.2\spa2.3\spa3.4\spa4.5\spa5.1 }
\ee
provides a counterexample to the uniqueness assertion at the
five-point level, because it is nonzero, yet has nonsingular
collinear limits in all channels.
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