FeynCalc manual (development version)

Color algebra

See also

Overview.

Notation for colored objects

FeynCalc objects relevant for the color algebra are

SUNT[a]

$$T^a$$

SUNF[a, b, c]

$$f^{abc}$$

SUND[a, b, c]

$$d^{abc}$$

SUNDelta[a, b]

$$\delta ^{ab}$$

SUNN

$$N$$

CA

$$C_A$$

CF

$$C_F$$

There are two main functions to deal with colored objects: SUNSimplify and SUNTrace. In general, SUNSimplify will also simplify color traces when possible

SUNT[a, a]
SUNSimplify[%]

$$T^a.T^a$$

$$C_F$$

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

$$T^a.T^b.T^a.T^b$$

$$-\frac{1}{2} C_F \left(C_A-2 C_F\right)$$

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

$$T^b.T^d.T^a.T^b.T^d$$

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

SUNF[a, r, s] SUNF[b, r, s]
SUNSimplify[%]

$$f^{ars} f^{brs}$$

$$C_A \delta ^{ab}$$

SUNF[a, b, c]  SUNF[a, b, c]
SUNSimplify[%]

$$\left(f^{abc}\right)^2$$

$$2 C_A^2 C_F$$

SUNF[a, b, c] SUND[d, b, c] 
SUNSimplify[%]

$$d^{bcd} f^{abc}$$

$$0$$

SUND[a, b, c] SUND[a, b, c] 
SUNSimplify[%]

$$\left(d^{abc}\right)^2$$

$$-2 \left(4-C_A^2\right) C_F$$

The color factors $C_A$ and $C_F$ are reconstructed from $N_c$ using heuristics. The reconstruction can be disabled by setting the option SUNNToCACF to False

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

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

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]]

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

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

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

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

$$\frac{\delta ^{ab}}{2}$$

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

$$C_A C_F^2$$

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

$$\frac{\delta ^{ab}}{2}$$

Use SUNTF to get color matrices with explicit fundamental indices

SUNTF[{a, b, c}, i, j] SUNTrace[SUNT[b, a]]
% // SUNSimplify

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

$$\frac{1}{2} C_F T_{ij}^c$$

SUNDelta[a, b] SUNTF[{a, b}, i, j] SUNTF[{c, d}, j, i]
SUNSimplify[%]

$$\delta ^{ab} \left(T^aT^b\right){}_{ij} \left(T^cT^d\right){}_{ji}$$

$$\frac{1}{2} C_F \delta ^{cd}$$

Color traces with more than 3 distinct 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

$$\text{tr}\left(T^a.T^b.T^c.T^d\right)$$

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

$$\frac{1}{4} \delta ^{ad} \left(C_A-2 C_F\right) \delta ^{bc}-\frac{1}{4} \delta ^{ac} \left(C_A-2 C_F\right) \delta ^{bd}+\frac{1}{4} \delta ^{ab} \left(C_A-2 C_F\right) \delta ^{cd}-\frac{1}{8} i f^{ad\text{FCGV}(\text{sun1521})} d^{bc\text{FCGV}(\text{sun1521})}+\frac{1}{8} i d^{ad\text{FCGV}(\text{sun1521})} f^{bc\text{FCGV}(\text{sun1521})}+\frac{1}{8} d^{ad\text{FCGV}(\text{sun1521})} d^{bc\text{FCGV}(\text{sun1521})}-\frac{1}{8} d^{bd\text{FCGV}(\text{sun1521})} d^{ac\text{FCGV}(\text{sun1521})}+\frac{1}{8} d^{cd\text{FCGV}(\text{sun1521})} d^{ab\text{FCGV}(\text{sun1521})}$$

One can automatically rename dummy indices using the SUNIndexNames option

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

$$\frac{1}{4} \delta ^{ad} \left(C_A-2 C_F\right) \delta ^{bc}-\frac{1}{4} \delta ^{ac} \left(C_A-2 C_F\right) \delta ^{bd}+\frac{1}{4} \delta ^{ab} \left(C_A-2 C_F\right) \delta ^{cd}-\frac{1}{8} i f^{adj} d^{bcj}+\frac{1}{8} i d^{adj} f^{bcj}+\frac{1}{8} d^{adj} d^{bcj}-\frac{1}{8} d^{bdj} d^{acj}+\frac{1}{8} d^{cdj} d^{abj}$$