FeynCalc manual (development version)

 

Color algebra

See also

Overview.

Notation

FeynCalc objects relevant for the color algebra are

SUNT[a]

TaT^a

SUNF[a, b, c]

fabcf^{abc}

SUND[a, b, c]

dabcd^{abc}

SUNDelta[a, b]

δab\delta ^{ab}

SUNN

NN

CA

CAC_A

CF

CFC_F

There are two main functions to deal with colored objects: SUNSimplify and SUNTrace

SUNT[a, a]
SUNSimplify[%]

Ta.TaT^a.T^a

CFC_F

SUNT[a, b, a, b]
SUNSimplify[%]

Ta.Tb.Ta.TbT^a.T^b.T^a.T^b

12CF(CA2CF)-\frac{1}{2} C_F \left(C_A-2 C_F\right)

SUNT[b, d, a, b, d]
SUNSimplify[%]

Tb.Td.Ta.Tb.TdT^b.T^d.T^a.T^b.T^d

Ta(CA2+1)4CA2\frac{T^a \left(C_A^2+1\right)}{4 C_A^2}

The color factors CAC_A and CFC_F are reconstructed from NcN_c using heuristics. The reconstruction can be disabled by setting the option SUNNToCACF to False

SUNSimplify[SUNT[b, d, a, b, d], SUNNToCACF -> False]

(N2+1)Ta4N2\frac{\left(N^2+1\right) T^a}{4 N^2}

The color traces are not evaluated by default. The evaluation can be forced either by applying SUNSimplify or setting the option SUNTraceEvaluate to True

SUNTrace[SUNT[a, b]]

tr(Ta.Tb)\text{tr}\left(T^a.T^b\right)

SUNTrace[SUNT[a, b, b, a]]

tr(Ta.Tb.Tb.Ta)\text{tr}\left(T^a.T^b.T^b.T^a\right)

SUNTrace[SUNT[a, b]] // SUNSimplify

δab2\frac{\delta ^{ab}}{2}

SUNTrace[SUNT[a, b, b, a]] // SUNSimplify

CACF2C_A C_F^2

SUNTrace[SUNT[a, b], SUNTraceEvaluate -> True]

δab2\frac{\delta ^{ab}}{2}

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\text{tr}\left(T^b.T^a\right) \left(T^aT^bT^c\right){}_{ij}

12CFTijc\frac{1}{2} C_F T_{ij}^c

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] &

14δad(CA2CF)δbc14δac(CA2CF)δbd+14δab(CA2CF)δcd18ifadFCGV(sun941)dbcFCGV(sun941)+18idadFCGV(sun941)fbcFCGV(sun941)+18dadFCGV(sun941)dbcFCGV(sun941)18dbdFCGV(sun941)dacFCGV(sun941)+18dcdFCGV(sun941)dabFCGV(sun941)\frac{1}{4} \delta ^{ad} \left(C_A-2 C_F\right) \delta ^{bc}-\frac{1}{4} \delta ^{ac} \left(C_A-2 C_F\right) \delta ^{bd}+\frac{1}{4} \delta ^{ab} \left(C_A-2 C_F\right) \delta ^{cd}-\frac{1}{8} i f^{ad\text{FCGV}(\text{sun941})} d^{bc\text{FCGV}(\text{sun941})}+\frac{1}{8} i d^{ad\text{FCGV}(\text{sun941})} f^{bc\text{FCGV}(\text{sun941})}+\frac{1}{8} d^{ad\text{FCGV}(\text{sun941})} d^{bc\text{FCGV}(\text{sun941})}-\frac{1}{8} d^{bd\text{FCGV}(\text{sun941})} d^{ac\text{FCGV}(\text{sun941})}+\frac{1}{8} d^{cd\text{FCGV}(\text{sun941})} d^{ab\text{FCGV}(\text{sun941})}

One can automatically rename dummy indices using the SUNIndexNames option

SUNTrace[SUNT[a, b, c, d]] // SUNSimplify[#, SUNTraceEvaluate -> True, SUNIndexNames -> {j}] &

14δad(CA2CF)δbc14δac(CA2CF)δbd+14δab(CA2CF)δcd18ifadjdbcj+18idadjfbcj+18dadjdbcj18dbdjdacj+18dcdjdabj\frac{1}{4} \delta ^{ad} \left(C_A-2 C_F\right) \delta ^{bc}-\frac{1}{4} \delta ^{ac} \left(C_A-2 C_F\right) \delta ^{bd}+\frac{1}{4} \delta ^{ab} \left(C_A-2 C_F\right) \delta ^{cd}-\frac{1}{8} i f^{adj} d^{bcj}+\frac{1}{8} i d^{adj} f^{bcj}+\frac{1}{8} d^{adj} d^{bcj}-\frac{1}{8} d^{bdj} d^{acj}+\frac{1}{8} d^{cdj} d^{abj}