Color algebra
See also
Overview.
Notation
FeynCalc objects relevant for the color algebra are
Ta
fabc
dabc
δab
N
CA
CF
There are two main functions to deal with colored objects: SUNSimplify
and SUNTrace
SUNT[a, a]
SUNSimplify[%]
Ta.Ta
CF
SUNT[a, b, a, b]
SUNSimplify[%]
Ta.Tb.Ta.Tb
−21CF(CA−2CF)
SUNT[b, d, a, b, d]
SUNSimplify[%]
Tb.Td.Ta.Tb.Td
4CA2Ta(CA2+1)
The color factors CA and CF are reconstructed from Nc using heuristics. The reconstruction can be disabled by setting the option SUNNToCACF
to False
SUNSimplify[SUNT[b, d, a, b, d], SUNNToCACF -> False]
4N2(N2+1)Ta
The color traces are not evaluated by default. The evaluation can be forced either by applying SUNSimplify
or setting the option SUNTraceEvaluate
to True
tr(Ta.Tb)
SUNTrace[SUNT[a, b, b, a]]
tr(Ta.Tb.Tb.Ta)
SUNTrace[SUNT[a, b]] // SUNSimplify
2δab
SUNTrace[SUNT[a, b, b, a]] // SUNSimplify
CACF2
SUNTrace[SUNT[a, b], SUNTraceEvaluate -> True]
2δab
Use SUNTF
to get color matrices with explicit fundamental indices
SUNTF[{a, b, c}, i, j] SUNTrace[SUNT[b, a]]
% // SUNSimplify
tr(Tb.Ta)(TaTbTc)ij
21CFTijc
Color traces with more than 3 matrices are not evaluated by default (assuming that no other simplifications are possible). The evaluation can be forced using the option SUNTraceEvaluate
set to True
SUNTrace[SUNT[a, b, c, d]] // SUNSimplify[#, SUNTraceEvaluate -> True] &
41δad(CA−2CF)δbc−41δac(CA−2CF)δbd+41δab(CA−2CF)δcd−81ifadFCGV(sun941)dbcFCGV(sun941)+81idadFCGV(sun941)fbcFCGV(sun941)+81dadFCGV(sun941)dbcFCGV(sun941)−81dbdFCGV(sun941)dacFCGV(sun941)+81dcdFCGV(sun941)dabFCGV(sun941)
One can automatically rename dummy indices using the SUNIndexNames
option
SUNTrace[SUNT[a, b, c, d]] // SUNSimplify[#, SUNTraceEvaluate -> True, SUNIndexNames -> {j}] &
41δad(CA−2CF)δbc−41δac(CA−2CF)δbd+41δab(CA−2CF)δcd−81ifadjdbcj+81idadjfbcj+81dadjdbcj−81dbdjdacj+81dcdjdabj