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%% subsection 2.1 Color management [slac-pub-7106-0-0-2-1 in slac-pub-7106-0-0-2: ^slac-pub-7106-0-0-2 >slac-pub-7106-0-0-2-2]
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\subsection{\usemenu{slac-pub-7106::context::slac-pub-7106-0-0-2-1}{Color management}}\label{subsection::slac-pub-7106-0-0-2-1}
First we describe the color decomposition of
amplitudes,\cite{6,7} and review
some diagrammatic techniques\,\cite{8}
for efficiently carrying out the necessary group theory.
The gauge group for QCD is $SU(3)$, but there is no harm in
generalizing it to $SU(N_c)$; indeed this makes some of the group
theory structure more apparent.
Gluons carry an adjoint color index $a=1,2,\ldots,N_c^2-1$,
while quarks and antiquarks carry an $N_c$ or $\overline{N}_c$ index,
$i,\jb=1,\ldots,N_c$.
The generators of $SU(N_c)$ in the fundamental representation are
traceless hermitian $N_c\times N_c$ matrices, $(T^a)_i^{~\jb}$.
We normalize them according to $\Tr(T^aT^b) = \delta^{ab}$
in order to avoid a proliferation of $\sqrt{2}$'s in partial
amplitudes. (Instead the $\sqrt{2}$'s appear in intermediate steps
such as the color-ordered Feynman rules in \fig{RulesFigure}.)
The color factor for a generic Feynman diagram in QCD contains
a factor of $(T^a)_i^{~\jb}$ for each gluon-quark-quark vertex,
a group theory structure constant $f^{abc}$ --- defined by
$[T^a,T^b]\ =\ i\sqrt{2}\,f^{abc}\,T^c$ --- for each pure gluon
three-vertex, and contracted pairs of structure constants
$f^{abe}f^{cde}$ for each pure gluon four-vertex.
The gluon and quark propagators contract many of the indices
together with $\delta_{ab}$, $\delta^{~\jb}_{i}$ factors.
We want to first identify all the different types of color factors
(or ``color structures'') that can appear in a given amplitude,
and then find rules for constructing the kinematic coefficients
of each color structure, which are called sub-amplitudes or partial
amplitudes.
The general color structure of the amplitudes can be exposed if we
first eliminate the structure constants $f^{abc}$ in favor of the
$T^a$'s, using
\be \label{eq:struct}
f^{abc}\ =\ -{i\over\sqrt2}\Bigl(
\Tr\bigl(T^aT^bT^c\bigr) - \Tr\bigl(T^aT^cT^b\bigr) \Bigr),
\ee
which follows from the definition of the structure constants.
At this stage we have 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,
of the form $(T^{a_1}\ldots T^{a_m})_{i_2}^{~\ib_1}$.
To reduce the number of traces and strings we ``Fierz rearrange''
the contracted $T^a$'s, using
\be \label{eq:colorfierz}
(T^a)_{i_1}^{~\jb_1} \, (T^a)_{i_2}^{~\jb_2}\ =\
\delta_{i_1}^{~\jb_2} \delta_{i_2}^{~\jb_1}
- {1\over N_c} \, \delta_{i_1}^{~\jb_1} \delta_{i_2}^{~\jb_2}\,,
\ee
where the sum over $a$ is implicit.
Equation~\docLink{slac-pub-7106-0-0-2.tcx}[eq:colorfierz]{2} is just the statement that the $SU(N_c)$
generators $T^a$ form the complete set of traceless hermitian
$N_c\times N_c$ matrices. The $-1/N_c$ term implements the
tracelessness condition. (To see this, contract both sides
of \eqn{colorfierz} with $\delta^{~i_1}_{\jb_1}$.)
It is often convenient to consider also
$U(N_c)\ =\ SU(N_c) \times U(1)$ gauge theory. The additional $U(1)$
generator is proportional to the identity matrix,
\be \label{eq:photongenerator}
(T^{a_{U(1)}})_{i}^{~\jb} = {1 \over \sqrt{N_c}}\ \delta_{i}^{~\jb}\ ;
\ee
when this is added back the
$U(N_c)$ generators obey \eqn{colorfierz} without the $-1/N_c$ term.
The auxiliary $U(1)$ gauge field is often called the photon, because
it is colorless
(it commutes with $SU(N_c)$, $f^{a_{U(1)}bc}=0$, for all $b,c$)
and therefore it does not couple directly to gluons;
however, quarks carry charge under it.
(Its coupling strength has to be readjusted from QCD to
QED strength for it to represent a real photon.)
The color algebra can easily be carried out
diagrammatically.\cite{8}
Starting with any given Feynman diagram, one interprets it
as just the color factor for the full diagram, and then
makes the two substitutions, \eqns{struct}{colorfierz},
which are represented diagrammatically in \fig{FabcdabFigure}.
In \fig{TreeExampleFigure} we use these steps to
simplify a sample diagram for five-gluon scattering at tree level.
The final line is the diagrammatic representation of a single trace,
$\Tr\bigl(T^{a_1}T^{a_2}T^{a_3}T^{a_4}T^{a_5}\bigr)$, plus all possible
permutations.
Notice that the $-1/N_c$ terms in \eqn{colorfierz} do not
contribute here, because the photon does not couple to gluons.
%======== figure ===========================
\figmac{3.0}{fabc_dab}{FabcdabFigure}{}
{Diagrammatic equations for simplifying $SU(N_c)$ color algebra.
Curly lines (``gluon propagators'') represent adjoint indices,
oriented solid lines (``quark propagators'') represent fundamental
indices, and ``quark-gluon vertices'' represent the generator matrices
$(T^a)_i^{~\jb}$.\hfill}
%===========================================
It is easy to see that any tree diagram for $n$-gluon scattering can
be reduced to a sum of ``single trace'' terms. This observation leads
to the {\it color decomposition} of the the $n$-gluon
tree amplitude,\cite{6}
\be \label{eq:treegluecolor}
{\cal A}^\tree_n \LP \{k_i,\lambda_i,a_i\}\RP
= g^{n-2} \hskip-1.3mm \sum_{\sigma \in S_n/Z_n} \hskip-1.3mm
\Tr\LP T^{a_{\sigma(1)}}\cdots T^{a_{\sigma(n)}}\RP\
A_n^\tree(\sigma(1^{\lambda_1}),\ldots,\sigma(n^{\lambda_n})).
\ee
Here $g$ is the gauge coupling (${g^2\over4\pi}=\as$),
$k_i, \lambda_i$ are the gluon momenta and helicities,
and $A_n^\tree(1^{\lambda_1},\ldots,n^{\lambda_n})$ are the
{\it partial amplitudes}, which contain all the kinematic information.
$S_n$ is the set of all permutations of $n$ objects, while $Z_n$ is
the subset of cyclic permutations, which preserves the trace;
one sums over the set $S_n/Z_n$ in order to sweep out all distinct
cyclic orderings in the trace.
The real work is still to come, in calculating the independent
partial amplitudes $A_n^\tree$. However, the partial amplitudes
are simpler than the full amplitude because they are
{\it color-ordered}: they only receive contributions from
diagrams with a particular cyclic ordering of the gluons.
Because of this, the singularities of the partial amplitudes, poles
and (in the loop case) cuts, can only occur in a limited
set of momentum channels, those made out of sums of cyclically adjacent
momenta. For example, the five-point partial amplitudes
$A_5^\tree(1^{\lambda_1},2^{\lambda_2},3^{\lambda_3},
4^{\lambda_4},5^{\lambda_5})$ can only have poles in
$s_{12}$, $s_{23}$, $s_{34}$, $s_{45}$, and $s_{51}$,
and not in $s_{13}$, $s_{24}$, $s_{35}$, $s_{41}$, or $s_{52}$,
where $s_{ij} \equiv (k_i+k_j)^2$.
Similarly, tree amplitudes $\qb qgg\cdots g$ with two external
quarks can be reduced to single strings of $T^a$ matrices,
\be \label{eq:treequarkgluecolor}
{\cal A}^\tree_n
= g^{n-2} \hskip-1.3mm \sum_{\sigma \in S_{n-2}} \hskip-1.3mm
\LP T^{a_{\sigma(3)}}\cdots T^{a_{\sigma(n)}}\RP_{i_2}^{~\jb_1}
A_n^\tree(1_\qb^{\lambda_1},2_q^{\lambda_2},
\sigma(3^{\lambda_3}),\ldots,\sigma(n^{\lambda_n})),
\ee
where numbers without subscripts refer to gluons.
\par\noindent
{\bf Exercise:} Write down the color decomposition for the tree
amplitude $\qb q\Qb Qg$.
%======== figure ===========================
\figmac{3.5}{tree_example}{TreeExampleFigure}{}
{A sample diagram for tree-level five-gluon scattering, reduced
to a single trace.\hfill}
%===============================================
Color decompositions at loop level are equally straightforward.
In \fig{LoopExampleFigure} we simplify a sample diagram for four-gluon
scattering at one loop.
Again the $-1/N_c$ terms in \eqn{colorfierz} are not
present, but now both single and double trace structures are generated,
leading to the one-loop color decomposition,\cite{7}
\bea
&& \hskip-8mm{\cal A}^\oneloop_n \LP \{k_i,\lambda_i,a_i\}\RP
\nonumber \\
&& \hskip 6mm
= g^n\Biggl[
\sum_{\sigma \in S_n/Z_n}
N_c\,\Tr\LP T^{a_{\sigma(1)}}\cdots T^{a_{\sigma(n)}}\RP\
A_{n;1}(\sigma(1^{\lambda_1}),\ldots,\sigma(n^{\lambda_n}))
\nonumber \\
&& \hskip 15mm
+\ \sum_{c=2}^{\lfloor{n/2}\rfloor+1}
\sum_{\sigma \in S_n/S_{n;c}}
\Tr\LP T^{a_{\sigma(1)}}\cdots T^{a_{\sigma(c-1)}}\RP\
\Tr\LP T^{a_{\sigma(c)}}\cdots T^{a_{\sigma(n)}}\RP\
\nonumber \\
\label{eq:loopgluecolor}
&& \hskip 53mm
\times\ A_{n;c}(\sigma(1^{\lambda_1}),\ldots,\sigma(n^{\lambda_n}))
\Biggr]\, ,
\eea
where $A_{n;c}$ are the partial amplitudes,
$Z_n$ and $S_{n;c}$ are the subsets of $S_n$
that leave the corresponding single and double trace structures
invariant, and $\lfloor x \rfloor$ is the greatest integer less than or
equal to $x$.
%======== figure ===========================
\figmac{3.5}{loop_example}{LoopExampleFigure}{}
{A diagram for one-loop four-gluon scattering, reduced to single
and double traces.\hfill}
%===============================================
The $A_{n;1}$ are the more basic objects in \eqn{loopgluecolor},
and are called {\it primitive amplitudes}, because:
\par\noindent
{\it a}. Like the tree partial amplitudes $A_n^\tree$ in
\eqn{treegluecolor}, they are color-ordered.
\par\noindent
{\it b}. It turns out that the remaining $A_{n;c>1}$ can be
generated\,\cite{7,9} as sums of permutations
of the $A_{n;1}$. (For amplitudes with external quarks as well as gluons,
the primitive amplitudes are not a subset of the partial amplitudes;
new color-ordered objects have to be defined.\cite{10})
One might worry that the color and helicity decompositions
will lead to a huge proliferation in the number of primitive/partial
amplitudes that have to be computed. Actually it is not too bad,
thanks to symmetries such as parity --- which allows one to
simultaneously reverse all helicities in an amplitude ---
and charge conjugation --- which allows one to exchange a quark and
anti-quark, or equivalently flip the helicity on a quark line.
For example, using parity and cyclic ($Z_5$) symmetry,
the five-gluon amplitude has only four independent tree-level partial
amplitudes:
\bea
&& A_5^\tree(1^+,2^+,3^+,4^+,5^+),\qquad\qquad
A_5^\tree(1^-,2^+,3^+,4^+,5^+),
\nonumber \\
\label{eq:treefiveg}
&& A_5^\tree(1^-,2^-,3^+,4^+,5^+),\qquad\qquad
A_5^\tree(1^-,2^+,3^-,4^+,5^+).
\eea
In fact, we'll see that the first two tree partial amplitudes
vanish, and there is a group theory relation between the last two,
so there is only one independent nonvanishing object to calculate.
At one-loop there are four independent objects --- \eqn{treefiveg} with
$A_5^\tree$ replaced by $A_{5;1}$ --- but only the last two
contribute to the NLO cross-section, due to the tree-level vanishings.
The group theory relation just mentioned derives from the fact that
the tree color decomposition, \eqn{treegluecolor}, is equally valid
for gauge group $U(N_c)$ as $SU(N_c)$, but any amplitude
containing the extra $U(1)$ photon must vanish. Hence if we
substitute the $U(1)$ generator --- the identity matrix --- into
the right-hand-side of \eqn{treegluecolor}, and collect
the terms with the same remaining color structure, that linear
combination of partial amplitudes must vanish. We get
\bea
0 &=& A_n^\tree(1,2,3,\ldots,n) + A_n^\tree(2,1,3,\ldots,n)
+ A_n^\tree(2,3,1,\ldots,n)
\nonumber \\
\label{eq:treephotondecouple}
&& \hskip 10mm
+ \cdots + A_n^\tree(2,3,\ldots,1,n),
\eea
often called a ``photon decoupling equation''\cite{7}
or ``dual Ward identity''\cite{3}
(because \eqn{treephotondecouple} can be derived from string theory,
a.k.a. dual theory).
In the five-point case, we can use \eqn{treephotondecouple} to get
\bea
A_5^\tree(1^-,2^+,3^-,4^+,5^+) &=& - A_5^\tree(1^-,3^-,2^+,4^+,5^+)
\nonumber \\
&& - A_5^\tree(1^-,3^-,4^+,2^+,5^+)
\nonumber \\
\label{eq:fivephotondecouple}
&& - A_5^\tree(1^-,3^-,4^+,5^+,2^+).
\eea
The partial amplitude where the two negative helicities
are not adjacent has been expressed in terms of the partial
amplitude where they are adjacent, as desired.
Since color is confined and unobservable,
the QCD-improved parton model cross-sections
of interest to us are averaged over initial colors and summed over
final colors. These color sums can be performed very easily
using the diagrammatic techniques. For example,
\fig{ColorSumFigure} illustrates the evaluation of the color sums
needed for the tree-level four-gluon cross-section.
In this case we can use the much simpler $U(N_c)$ color algebra,
omitting the $-1/N_c$ term in~\eqn{colorfierz},
because the $U(1)$ contribution vanishes. (This shortcut is not
valid for general loop amplitudes, or if external quarks are present.)
Using also the reflection identity discussed below, \eqn{reflectionid},
the total color sum becomes
\bea
\sum_{{\rm colors}} [\A_4^{{\rm tree}\,*} \A_4^\tree]
&=& 2\,g^4\, A_4^{{\rm tree}\,*}(1,2,3,4)
\times \biggl[ A_4^\tree(1,2,3,4) ( N_c^4 + N_c^2 )
\nonumber \\
&&\quad
+ \Bigl( A_4^\tree(2,1,3,4) + A_4^\tree(2,3,1,4) \Bigr)
( N_c^2 + N_c^2 ) \biggr]
\nonumber \\
&&\qquad\qquad +\ \hbox{2 more permutations}
\nonumber \\
&&
\nonumber \\
\label{eq:fourgluesum}
&=& g^4 \, N_c^2 (N_c^2-1) \sum_{\si\in S_3}
|A_4^\tree(\si(1),\si(2),\si(3),4)|^2\ ,
\eea
where we have used the decoupling identity, \eqn{treephotondecouple},
in the last step.
%======== figure ===========================
\figmac{2.0}{colorsum}{ColorSumFigure}{}
{Diagrammatic evaluation of color sums for the tree-level
four-gluon cross-section.\hfill}
%===========================================
Because we have stripped all the color factors out of the partial
amplitudes, the {\it color-ordered Feynman rules} for constructing
these objects are purely kinematic (no $T^a$'s or $f^{abc}$'s are
left). The rules are given in \fig{RulesFigure}, for quantization
in Lorentz-Feynman gauge. (Later we will discuss alternate gauges.)
To compute a tree partial amplitude, or a {\it color-ordered} loop
partial amplitude such as $A_{n;1}$,
\par\noindent
1. Draw all {\it color-ordered graphs}, i.e. all planar graphs where
the cyclic ordering of the
external legs matches the ordering of the $T^{a_i}$ matrices
in the corresponding color structure,
\par\noindent
2. Evaluate each graph using the color-ordered vertices of
\fig{RulesFigure}.
\par\noindent
Starting with the standard Feynman rules in terms of $f^{abc}$, etc.,
you can check that this prescription works because:
\par\noindent
1) of all possible graphs, only the color-ordered graphs can
contribute to the desired color structure, and
\par\noindent
2) the color-ordered vertices are obtained by inserting
\eqn{struct} into the standard Feynman rules and extracting a single
ordering of the $T^a$'s; hence they keep only the portion of a
color-ordered graph which does contribute to the correct color
structure.
\par\noindent
Many partial amplitudes are {\it not} color-ordered --- for example
the $A_{n;c}$ for $c>1$ in \eqn{loopgluecolor} --- and so the above
rules do not apply. However, as mentioned above one can usually
express such quantities as sums over permutations of color-ordered
``primitive amplitudes'' --- for example the $A_{n;1}$ --- to which
the rules do apply.
%======== figure ===========================
\figmac{3.2}{rules}{RulesFigure}{}
{Color-ordered Feynman rules, in Lorentz-Feynman gauge, omitting
ghosts. Straight lines represent fermions, wavy lines gluons.
All momenta are taken outgoing.\hfill}
%===========================================
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%