FeynCalc manual (development version)

GluonPropagator

GluonPropagator[p, {mu, a}, {nu, b}] or GluonPropagator[p, mu, a, nu, b] yields the gluon propagator.

GluonPropagator[p, {mu}, {nu}] or GluonPropagator[p, mu, nu] omits the SUNDelta.

GP can be used as an abbreviation of GluonPropagator.

The gauge and the dimension are determined by the options Gauge and Dimension. The following settings of Gauge are possible:

See also

Overview, GluonSelfEnergy, GluonVertex, GluonGhostVertex, GhostPropagator, GluonGhostVertex.

Examples

GluonPropagator[p, \[Mu], a, \[Nu], b] 
 
Explicit[%]

Πabμν(p)\Pi _{ab}^{\mu \nu }(p)

iδabgμνp2-\frac{i \delta ^{ab} g^{\mu \nu }}{p^2}

GP[p, \[Mu], a, \[Nu], b, Gauge -> \[Alpha]] 
 
Explicit[%]

Πabμν(p)\Pi _{ab}^{\mu \nu }(p)

iδab((1α)pμpνp2gμν)p2\frac{i \delta ^{ab} \left(\frac{(1-\alpha ) p^{\mu } p^{\nu }}{p^2}-g^{\mu \nu }\right)}{p^2}

GluonPropagator[p, \[Mu], a, \[Nu], b, Gauge -> {Momentum[n], 1}, Explicit -> True]

iδab(pμnν+pνnμ(np+iη)n2pμpνp2nμnν(np+iη)21gμν)p2\frac{i \delta ^{ab} \left(\frac{p^{\mu } \overline{n}^{\nu }+p^{\nu } \overline{n}^{\mu }}{(\overline{n}\cdot \overline{p}+i \eta )}-\frac{\overline{n}^2 p^{\mu } p^{\nu }-p^2 \overline{n}^{\mu } \overline{n}^{\nu }}{(\overline{n}\cdot \overline{p}+i \eta )^{21}}-g^{\mu \nu }\right)}{p^2}

GP[p, \[Mu], \[Nu]]

Πgμν(p)\Pi _g^{\mu \nu }(p)

Explicit[%]

igμνp2-\frac{i g^{\mu \nu }}{p^2}

GluonPropagator[p, \[Mu], a, \[Nu], b, CounterTerm -> 1] // Explicit

iCAgs2Snδab(11pμpν3196p2gμν)ε-\frac{i C_A g_s^2 S_n \delta ^{ab} \left(\frac{11 p^{\mu } p^{\nu }}{3}-\frac{19}{6} p^2 g^{\mu \nu }\right)}{\varepsilon }

GluonPropagator[p, \[Mu], a, \[Nu], b, CounterTerm -> 2] // Explicit

iCAgs2Snδab(16p2gμν13pμpν)ε-\frac{i C_A g_s^2 S_n \delta ^{ab} \left(-\frac{1}{6} p^2 g^{\mu \nu }-\frac{1}{3} p^{\mu } p^{\nu }\right)}{\varepsilon }

GluonPropagator[p, \[Mu], a, \[Nu], b, CounterTerm -> 3] // Explicit

2iTfgs2Snδab(43p2gμν4pμpν3)ε-\frac{2 i T_f g_s^2 S_n \delta ^{ab} \left(\frac{4}{3} p^2 g^{\mu \nu }-\frac{4 p^{\mu } p^{\nu }}{3}\right)}{\varepsilon }

GluonPropagator[p, \[Mu], a, \[Nu], b, CounterTerm -> 4] // Explicit

iCAgs2Snδab(10pμpν3103p2gμν)ε-\frac{i C_A g_s^2 S_n \delta ^{ab} \left(\frac{10 p^{\mu } p^{\nu }}{3}-\frac{10}{3} p^2 g^{\mu \nu }\right)}{\varepsilon }

GluonPropagator[p, \[Mu], a, \[Nu], b, CounterTerm -> 5] // Explicit

iCAgs2Snδab(10pμpν3103p2gμν)ε+iTfgs2Snδab(43p2gμν4pμpν3)ε\frac{i C_A g_s^2 S_n \delta ^{ab} \left(\frac{10 p^{\mu } p^{\nu }}{3}-\frac{10}{3} p^2 g^{\mu \nu }\right)}{\varepsilon }+\frac{i T_f g_s^2 S_n \delta ^{ab} \left(\frac{4}{3} p^2 g^{\mu \nu }-\frac{4 p^{\mu } p^{\nu }}{3}\right)}{\varepsilon }