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%% subsection 3.2 Recursive Techniques [slac-pub-7106-0-0-3-2 in slac-pub-7106-0-0-3: slac-pub-7106-0-0-3-3]
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\subsection{\usemenu{slac-pub-7106::context::slac-pub-7106-0-0-3-2}{Recursive Techniques}}\label{subsection::slac-pub-7106-0-0-3-2}
By now you can see that color-ordering, plus the spinor helicity
formalism, can vastly reduce the number of diagrams, and terms
per diagram, that have to be evaluated. However, with more external
legs the results still get more complex and difficult to carry out by
hand. Fortunately, a technique is available for generating tree
amplitudes recursively in the number of
legs.\cite{15} Even if one cannot
simplify analytically the expressions obtained in this way,
the recursive approach lends itself to efficient numerical evaluation.
In order to get a tree-level recursion relation, we need to
construct an auxiliary quantity with one leg off-shell.
For the construction of pure-glue amplitudes,
we define the {\it off-shell current} $J^\mu(1,2,\ldots,n)$
to be the sum of color-ordered ($n+1)$-point Feynman graphs,
where legs $1,2,\ldots,n$ are on-shell gluons, and leg ``$\mu$''
is off-shell, as shown in \fig{GluonCurrentFigure}.
The uncontracted vector index on the off-shell leg is also denoted by
$\mu$; the off-shell propagator is defined to be included in $J^\mu$.
Since $J^\mu$ is an off-shell quantity, it is gauge-dependent.
For example, $J^\mu$ depends on the reference momenta for the
on-shell gluons, which must therefore be kept fixed until
after one has extracted an on-shell result. One can also construct
amplitudes with external quarks recursively, by introducing an off-shell
quark current\,\cite{15} as well as the gluon current
$J^\mu$, but we will not do so here.
%======== figure ===========================
\figmac{1.5}{gluoncurrent}{GluonCurrentFigure}{}
{The off-shell gluon current $J^\mu(1,2,\ldots,n)$. Leg ``$\mu$'' is
the only off-shell leg.\hfill}
%============================================
It is easy to write down a recursion relation for $J^\mu$, by
following the off-shell line back into the diagram.
One first encounters either a three-gluon vertex or a four-gluon
vertex. Each of the off-shell lines branching
out from this vertex attaches to a smaller number of on-shell
gluons, thus we have the recursion relation\cite{15}
depicted in \fig{GluonRecursionFigure},
\bea
\hskip0mm
J^\mu(1,\ldots,n) \hskip-1.5mm &=& \hskip-1.5mm
{-i\over P^2_{1,n}} \Biggl[
\sum_{i=1}^{n-1} V_3^{\mu\nu\rho}(P_{1,i},P_{i+1,n})
\ J_\nu(1,\ldots,i)\ J_\rho(i+1,\ldots,n)
\nonumber \\
&& \hskip-5mm
+ \sum_{j=i+1}^{n-1} \sum_{i=1}^{n-2} V_4^{\mu\nu\rho\sigma}
\ J_\nu(1,\ldots,i)\ J_\rho(i+1,\ldots,j)\ J_\sigma(j+1,\ldots,n)
\Biggr] ,
\nonumber \\
\label{eq:Jrecursion}
&&
\eea
where the $V_i$ are just the color-ordered gluon self-interactions,
\bea
V_3^{\mu\nu\rho}(P,Q) &=& {i\over\sqrt{2}}
\left( \eta^{\nu\rho}(P-Q)^\mu + 2\eta^{\rho\mu}Q^\nu
- 2\eta^{\mu\nu}P^\rho \right),
\nonumber \\
\label{eq:videfn}
V_4^{\mu\nu\rho\sigma} &=& {i\over2}
\left( 2\eta^{\mu\rho}\eta^{\nu\sigma} - \eta^{\mu\nu}\eta^{\rho\sigma}
- \eta^{\mu\sigma}\eta^{\nu\rho} \right),
\eea
and
\be \label{eq:pdef}
P_{i,j}\ \equiv\ k_i+k_{i+1}+\cdots+k_j .
\ee
The $J^\mu$ satisfy the photon decoupling relation,
\be \label{eq:Jdecouple}
J^\mu(1,2,3,\ldots,n) + J^\mu(2,1,3,\ldots,n)
+ \cdots + J^\mu(2,3,\ldots,n,1)\ =\ 0,
\ee
the reflection identity
\be \label{eq:Jreflectionid}
J^\mu(1,2,3,\ldots,n) =\ (-1)^{n+1}\,J^\mu(n,\ldots,3,2,1),
\ee
and current conservation,
\be \label{eq:Jconserve}
P^\mu_{1,n}\ J_\mu(1,2,\ldots,n)\ =\ 0.
\ee
%======== figure ===========================
\figmac{4.5}{gluonrecursion}{GluonRecursionFigure}{}
{The recursion relation for the off-shell gluon current
$J^\mu(1,2,\ldots,n)$.\hfill}
%============================================
In some cases, the recursion relations can be solved in closed
form.\cite{15,16}
The simplest case is (as expected) when all on-shell gluons have the
same helicity, for which we choose the common reference momentum $q$,
and then
\be \label{eq:Jallplus}
J^\mu(1^+,2^+,\ldots,n^+)\ =\
{ \langle q^-|\gamma^\mu \Psl_{1,n}|q^+\rangle
\over\sqrt{2} \spa{q}.1\spa1.2\cdots\spa{n-1,}.{n}\spa{n}.{q} }
\ .
\ee
Let's verify that this expression solves~\eqn{Jrecursion}.
Note first that the $V_4$ term does not contribute at all, nor the
first term in $V_3$, because after
Fierzing we get a factor of $\spa{q}.{q} = 0$.
Thus the right-hand side of~\eqn{Jrecursion} becomes
(using $\spa{q}.{q} = 0$ to commute and rearrange terms)
\bea
&&{ 1 \over \sqrt{2} P^2_{1,n}
\spa{q}.1\spa1.2\cdots\spa{n-1,}.{n}\spa{n}.{q} }
\sum_{i=1}^{n-1}
{ \spa{i,}.{i+1} \over \spa{i}.{q} \spa{q,}.{i+1} }
\nonumber \\
&& \times \Bigl( \langle q^-|\gamma^\mu \Psl_{i+1,n}|q^+\rangle
\langle q^-|\Psl_{i+1,n}\Psl_{1,i}|q^+\rangle
\nonumber \\
&&\qquad
- \langle q^-|\gamma^\mu \Psl_{1,i}|q^+\rangle
\langle q^-|\Psl_{1,i}\Psl_{i+1,n}|q^+\rangle \Bigr)
\nonumber \\
&=& { \langle q^-|\gamma^\mu \Psl_{1,n}|q^+\rangle
\over\sqrt{2} P^2_{1,n}
\spa{q}.1\spa1.2\cdots\spa{n-1,}.{n}\spa{n}.{q} }
\nonumber \\
\label{eq:checkJrecursion}
&& \times \left[ \sum_{i=1}^{n-1}
{ \spa{i,}.{i+1} \over \spa{i}.{q} \spa{q,}.{i+1} }
\langle q^-| \Psl_{i+1,n} \right] \Psl_{1,n} | q^+\rangle\ .
\eea
Using the identity
\be \label{eq:secondident}
\sum_{i=1}^{n-1}
{ \spa{i,}.{i+1} \over \spa{i}.{q} \spa{q,}.{i+1} }
\langle q^-| \Psl_{i+1,n}\ =\
{ \langle 1^-| \Psl_{1,n} \over \spa1.{q} }\ ,
\ee
we get the desired result, \eqn{Jallplus}.
\par\noindent
{\bf Exercise:} Prove the identity, \eqn{secondident}, by first
proving the identity
\be \label{eq:firstident}
\sum_{i=j}^{k-1}
{ \spa{i,}.{i+1} \over \spa{i}.{q} \spa{q,}.{i+1} }
\ =\ { \spa{j}.{k} \over \spa{j}.{q} \spa{q}.{k} }\ .
\ee
The ``eikonal'' identity, \eqn{firstident}, also plays a role in
understanding the structure of the soft singularities of QED
amplitudes, when these are obtained from QCD partial amplitudes by
the replacement $T^a \to 1$ (see Sections 3.4 and 3.5).
The current where the first on-shell gluon has negative helicity
can be obtained similarly,
\be \label{eq:Joneminus}
J^\mu(1^-,2^+,\ldots,n^+)\ =\
{ \langle 1^-|\gamma^\mu \Psl_{2,n}|1^+\rangle
\over\sqrt{2} \spa1.2\cdots\spa{n}.{1} }
\sum_{m=3}^n { \langle 1^-|\ksl_m \Psl_{1,m}|1^+\rangle
\over P_{1,m-1}^2 P_{1,m}^2 }\ ,
\ee
where the reference momentum choice is
$q_1=k_2$, $q_2= \cdots = q_n = k_1$.
\par\noindent
{\bf Exercise:} Show this.
\par\noindent
Amplitudes with $(n+1)$ legs are obtained from the currents
$J^\mu(1,2,\ldots,n)$ by amputating the off-shell propagator
(multiplying by $i \, P^2_{1,n}$), contracting the $\mu$
index with the appropriate on-shell polarization vector
$\pol_{n+1}^\mu$, and taking $P^2_{1,n} = k_{n+1}^2 \to 0$.
In the case of $J^\mu(1^+,2^+,\ldots,n^+)$, there is no $P^2_{1,n}$
pole in the current, so the amplitude must vanish for both
helicities of gluon $(n+1)$, in accord with \eqn{treevanishing}.
In the case of $J^\mu(1^-,2^+,\ldots,n^+)$, the pole term requirement
picks out the term $m=n$ in~\eqn{Joneminus}. Using reference momentum
$q_{n+1} = k_n$ for $\pol_{n+1}^-$, we obtain
(replacing $\Psl_{1,n} \to -\ksl_{n+1}$, etc.),
\bea
&& \hskip-15mm A_{n+1}^\tree(1^-,2^+,\ldots,n^+,(n+1)^-)
\nonumber \\
&=& -i \,
{ \langle n^+|\gamma_\mu | (n+1)^+\rangle
\over\sqrt{2}\spb{n,}.{n+1} }
{ \langle 1^-|\gamma^\mu \Psl_{1,n}|1^+\rangle
\over\sqrt{2} \spa1.2\cdots\spa{n}.{1} }
{ \langle 1^-|\ksl_n \Psl_{1,n}|1^+\rangle
\over P^2_{1,n-1} }
\nonumber \\
\label{eq:mhvinterm}
&=& -i \, { \spa{1,}.{n+1} \over \spa1.2\cdots\spa{n}.{1} }
{ \spa{n+1,}.1\spa1.{n}\spb{n,}.{n+1}\spa{n+1,}.1
\over s_{n,n+1} }\ ,
\eea
or
\be \label{eq:mhvadjacent}
A_n^\tree(1^-,2^-,3^+,4^+,\ldots,n^+)\ =\
i\, { {\spa1.2}^4 \over \spa1.2\cdots\spa{n}.{1} }\ .
\ee
Applying the decoupling identity, \eqn{treephotondecouple}, and the
spinor identity, \eqn{firstident}, it is easy to obtain the remaining
{\it maximally helicity violating} (MHV) or
Parke-Taylor\,\cite{17} helicity amplitudes,
\be \label{eq:mhvall}
A_{jk}^\treemhv\ \equiv\
A_n^\tree(1^+,\ldots,j^-,\ldots,k^-,\ldots,n^+)\ =\
i\, { {\spa{j}.{k}}^4 \over \spa1.2\cdots\spa{n}.{1} }\ .
\ee
These remarkably simple amplitudes were first conjectured by Parke and
Taylor\,\cite{17} on the basis of their collinear limits (see
below) and photon decoupling relations, and were rigorously proven
correct by Berends and Giele\,\cite{15} using the above
recursive approach. The other nonvanishing helicity configurations
(beginning at $n=6$) are typically more complicated.
The MHV amplitudes can be used as the basis of approximation schemes,
however.\cite{18}
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