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%% subsection 2.1 \it Color Decomposition [slac-pub-7111-0-0-2-1 in slac-pub-7111-0-0-2: ^slac-pub-7111-0-0-2 >slac-pub-7111-0-0-2-2]
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\subsection{\usemenu{slac-pub-7111::context::slac-pub-7111-0-0-2-1}{\it Color Decomposition}}\label{subsection::slac-pub-7111-0-0-2-1}
\label{ColorSubsection}
Color decompositions have a long history, dating back to Chan-Paton
factors in early formulations of string theory \cite{17}.
They are also related to the ``double-line'' formalism introduced by
`t Hooft in the large-$N_c$ (number of colors) approach to
QCD~\cite{26}, although here we will not make any large-$N_c$
or ``leading-color'' approximations.
The basic idea is to use group theory to break up an amplitude
into gauge-invariant pieces which are composed of Feynman diagrams
with a fixed cyclic ordering of external legs.
These pieces are simpler because poles and cuts can only appear in
kinematic invariants made out of cyclicly adjacent sums of momenta,
of the form $(k_i+k_{i+1}+\cdots+ k_j)^2$.
At the four-point level this is not so important, because only one of
the three Mandelstam variables $s$,$t$,$u$ is thereby excluded;
but as the number of external legs grows, the total number of
invariants grows much faster than the number of cyclicly adjacent
ones.
The following brief review focuses on results needed later,
rather than derivations.
A more complete discussion can be found in
refs.~\cite{17,3,27,14,24,21}.
We first generalize the gauge group of QCD to $SU(N_c)$, with the
quarks transforming in the fundamental representation.
The simplest way to implement the color decomposition in field theory
is by rewriting the group structure constants appearing in Feynman
diagrams in terms of fundamental representation matrices
$$
f^{abc} = -{i\over\sqrt2} \Bigl( \Tr\bigl( T^a T^b T^c \bigr)
- \Tr\bigl( T^b T^a T^c \bigr) \Bigr) \,.
\equn
\label{FundConvert}
$$
After making this substitution in a generic Feynman diagram
we obtain a large number of traces, many sharing
$T^a$'s with contracted indices, of the form
$\Tr\bigl(\ldots T^a\ldots\bigr)\,\Tr\bigl(\ldots T^a\ldots\bigr)
\,\ldots\,\Tr\bigl(\ldots)$.
If external quarks are present, then in addition to the traces there
will be some strings of $T^a$'s terminated by fundamental indices.
To reduce the number of traces and strings to a minimum, we rearrange
the contracted $T^a$'s, using
$$
(T^a)_{i_1}^{~\bar j_1} \, (T^a)_{i_2}^{~\bar j_2}\ =\
\delta_{i_1}^{~\bar j_2} \delta_{i_2}^{~\bar j_1}
- {1\over N_c} \, \delta_{i_1}^{~\bar j_1} \delta_{i_2}^{~\bar j_2}\,,
\equn\label{ColorFierz}
$$
where the sum over $a$ is implicit. (If all lines are in the adjoint
representation the second term drops out by a $U(1)$ decoupling
identity \cite{3,17}, which follows from the lack of a
`photon' self-coupling.) A {\it partial amplitude} is the coefficient
of a given color trace in the resulting color decomposition of the
amplitude.
For example, in the $n$-gluon tree amplitude, application of
\eqn{ColorFierz} reduces all color factors to single traces.
Thus its decomposition is
$$
{\cal A}^\tree_n = g^{n-2} \sum_{\sigma\in S_n/Z_n}
\Tr(\si(1)\ldots\si(n)) A^\tree_n(\si(1),\ldots,\si(n)) \,,
\equn\label{TreeAmplitude}
$$
where $A_n^\tree$ are the partial amplitudes,
$\Tr(1\ldots n) \equiv \Tr(T^{a_1}\ldots T^{a_n})$,
with $a_i$ the color index of the $i$-th external
gluon, and $S_n/Z_n$ is the set of non-cyclic
permutations of $\{1,2,\ldots,n\}$, corresponding to the set of
inequivalent traces.
The labels on the gluon momenta $k_i$ and polarization vectors $\pol_i$,
implicit in \eqn{TreeAmplitude}, are also to be permuted by $\si$.
In the next subsection we will go over to a helicity basis, and
the label $i$ will be replaced by $i^{\lambda_i}$, with $\lambda_i$ the
(outgoing) gluon helicity.
Similarly, tree amplitudes with
a pair of external quarks can be reduced to a sum over single
strings of matrices, $(T^{a_3}\cdots T^{a_n})_{i_2}^{~\jb_1}$,
and so on. For a proof that individual partial amplitudes are
gauge invariant, see ref.~\cite{3}.
At one loop, additional color structures are possible;
in the $n$-gluon amplitude double traces appear as well as single
traces. For example, the color decomposition of the one-loop
five-gluon amplitude is
$$
\eqalign{
{\cal A}_{5}^\oneloop = & g^5 \mu_R^{2\e} \Biggl[
\sum_{\sigma \in S_5/Z_5}
N_c \Tr(\si(1)\ldots\si(5))
A_{5;1} (\si(1),\ldots,\si(5)) \cr
& \hskip-16mm + \hskip-4mm
\sum_{\sigma \in S_5/(S_2\times S_3)} \hskip-6mm
\Tr(\si(1)\si(2)) \Tr(\si(3)\si(4)\si(5))
A_{5;3} (\si(1),\si(2);\si(3),\si(4),\si(5)) \Biggr] ; \cr}
\equn\label{LoopColor}
$$
as in~\eqn{TreeAmplitude} the permutation sums are over all
inequivalent traces.
For gauge group $U(N_c)$, the partial amplitudes $A_{5;2}$ multiplying
traces of the form $\Tr(1)\Tr(2345)$ would also have to be included,
but for $SU(N_c)$ the trace of a single generator vanishes.
The decomposition of the $n$-gluon amplitude into single-trace
($A_{n;1}$) and double-trace ($A_{n;j>2}$) components is entirely
analogous. Were one to consider the large-$N_c$ limit, the single-trace
terms would give rise to the leading contributions, and we will refer
to the corresponding partial amplitudes as leading-color partial amplitudes;
the double-trace terms have subleading-color partial amplitudes as
coefficients.
The rules for constructing leading-color partial amplitudes
such as $A_n^\tree$ and $A_{n;1}$ are {\it color-ordered} Feynman
rules, which are depicted in \fig{FeynmanColorFigure} for the standard
Lorentz-Feynman gauge.
These rules are obtained from
ordinary Feynman rules by restricting attention to a given ordering of
color matrices. Applying \eqn{FundConvert} to \eqn{FeynmanVertex}
and extracting the coefficient of $\Tr(T^a T^b T^c)$ gives the
color-ordered three-vertex in \fig{FeynmanColorFigure};
and similarly for the color-ordered four-vertex,
the coefficient of $\Tr(T^a T^b T^c T^d)$.
The only diagrams to be computed are those that can be drawn
in a planar fashion with the external legs following the
ordering of the color trace under consideration.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=FeynmanColor.eps,width=2.5in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{FeynmanColorFigure}
Color-ordered Feynman rules in Lorentz-Feynman gauge. Curly lines
represent gluons and lines with arrows fermions.}
\end{figure}
%
The immediate advantage of rewriting Feynman rules in this way is that
fewer diagrams contribute to a given partial amplitude, and its
analytic structure is simpler. As a simple example, with conventional
Feynman diagrams one would have a total of four conventional Feynman
diagrams, depicted in \fig{FourPointTreeFigure} for the four-point
tree amplitude. With color-ordered Feynman rules one would compute
the partial amplitude $A_4(1,2,3,4)$ associated with the color trace
$\Tr(T^{a_1} T^{a_2} T^{a_3} T^{a_4})$, omitting
diagram~\docLink{slac-pub-7111-0-0-2.tcx}[FourPointTreeFigure]{5}c since the ordering of the legs do
not follow the ordering of the color trace. Thus $A_4(1,2,3,4)$ has
no pole in $(k_1+k_3)^2$. The other partial amplitudes can be
obtained by permuting the arguments of $A_4(1,2,3,4)$. For the
five-gluon amplitude, there are 10 color-ordered diagrams as opposed
to 40 total. Obviously the simplifications obtained using partial
amplitudes increase rapidly with the number of external legs.
%FIGURE
%
\begin{figure}
\begin{center}
\epsfig{file=FourPointTree.eps,width=1.7in,clip=}
\end{center}
\vskip -.7 cm \caption[]{
\label{FourPointTreeFigure}
The four-point Feynman diagrams. Color-ordered Feynman rules do not
include diagram (c) for $A_4(1,2,3,4)$.}
\end{figure}
%
At one loop, one also has to compute subleading-color partial
amplitudes, such as the double-trace coefficients $A_{5;3}$ in
\eqn{LoopColor}, which cannot be obtained directly from color-ordered
rules. Fortunately there exist general formulas relating such
quantities to permutation sums of color-ordered objects
\cite{27,14,24}. For example, the gluon-loop
contribution to the four-gluon amplitude can be found from the
relation
$$
\eqalign{
A_{4;3}(1,2;3,4) &=
A_{4;1}(1,2,3,4) + A_{4;1}(1,3,2,4) + A_{4;1}(2,1,3,4) \cr
& \hskip -1mm
+ A_{4;1}(2,3,1,4) + A_{4;1}(3,1,2,4) + A_{4;1}(3,2,1,4)\,. \cr}
\equn\label{FourPointPerms}
$$
Such formul\ae\ can be derived from string theory
\cite{28,14}, although the most straightforward way to prove
them is using color flow diagrams in field theory \cite{24}. To
understand formula (\docLink{slac-pub-7111-0-0-2.tcx}[FourPointPerms]{7}) heuristically, it is useful
to focus on the box diagram. Using ordinary Feynman rules and
expanding out the structure constants using eqs.~(\docLink{slac-pub-7111-0-0-2.tcx}[FundConvert]{3})
and (\docLink{slac-pub-7111-0-0-2.tcx}[ColorFierz]{4}) it is straightforward to check that the box
diagrams contribute to $A_{4;1}$ and $A_{4;3}$ in such a way
that~\eqn{FourPointPerms} is satisfied. Roughly speaking, gauge
invariance then requires the remaining diagrams to tag along properly
with the box diagram.
Thus we can restrict our discussion henceforth to amplitudes with
a fixed ordering of external legs, which we call
{\it primitive amplitudes}.
In the $n$-gluon cases discussed above, the set of primitive
amplitudes coincides with the leading-color partial amplitudes
$A_n^\tree$ and $A_{n;1}$, but this is not always the case.
For example, one-loop amplitudes with external fermions have
leading-color (as well as subleading-color) partial amplitudes
that are sums of several primitive amplitudes \cite{24}.
%%%%%%%%%%%%%%%%%%%%%%%%