BackgroundGluonVertex[{p, mu, a}, {q, nu, b}, {k, la, c}] yields the 3-gluon vertex in the background field gauge, where the first set of arguments corresponds to the external background field. BackgroundGluonVertex[{p, mu, a}, {q, nu, b}, {k, la, c}, {s, si, d}] yields the 4-gluon vertex, with {p, mu ,a} and {k, la, c} denoting the external background fields.
The gauge, dimension and the name of the coupling constant are determined by the options Gauge, Dimension and CouplingConstant.
The Feynman rules are taken from L. Abbot NPB 185 (1981), 189-203; except that all momenta are incoming. Note that Abbot’s coupling constant convention is consistent with the default setting of GluonVertex.
BackgroundGluonVertex[{p, \[Mu], a}, {q, \[Nu], b}, {k, \[Lambda], c}]g_s f^{abc} \left(g^{\mu \nu } (-k+p-q)^{\lambda }+g^{\lambda \mu } (k-p+q)^{\nu }+g^{\lambda \nu } (q-k)^{\mu }\right)
BackgroundGluonVertex[{p, \[Mu], a}, {q, \[Nu], b}, {k, \[Lambda], c}, {s, \[Sigma], d}]-i g_s^2 \left(f^{ad\text{FCGV}(\text{u19})} f^{bc\text{FCGV}(\text{u19})} \left(g^{\lambda \sigma } g^{\mu \nu }-g^{\lambda \nu } g^{\mu \sigma }-g^{\lambda \mu } g^{\nu \sigma }\right)+f^{ac\text{FCGV}(\text{u19})} f^{bd\text{FCGV}(\text{u19})} \left(g^{\lambda \sigma } g^{\mu \nu }-g^{\lambda \nu } g^{\mu \sigma }\right)+f^{ab\text{FCGV}(\text{u19})} f^{cd\text{FCGV}(\text{u19})} \left(g^{\lambda \sigma } g^{\mu \nu }-g^{\lambda \nu } g^{\mu \sigma }+g^{\lambda \mu } g^{\nu \sigma }\right)\right)
BackgroundGluonVertex[{p, \[Mu], a}, {q, \[Nu], b}, {k, \[Lambda], c},Gauge -> \[Alpha]]g_s f^{abc} \left(g^{\mu \nu } \left(-\frac{k}{\alpha }+p-q\right)^{\lambda }+g^{\lambda \mu } \left(k-p+\frac{q}{\alpha }\right)^{\nu }+g^{\lambda \nu } (q-k)^{\mu }\right)
BackgroundGluonVertex[{p, \[Mu], a}, {q, \[Nu], b}, {k, \[Lambda], c}, {s, \[Sigma], d}, Gauge -> \[Alpha]]-i g_s^2 \left(f^{ad\text{FCGV}(\text{u20})} f^{bc\text{FCGV}(\text{u20})} \left(-\frac{g^{\lambda \nu } g^{\mu \sigma }}{\alpha }+g^{\lambda \sigma } g^{\mu \nu }-g^{\lambda \mu } g^{\nu \sigma }\right)+f^{ab\text{FCGV}(\text{u20})} f^{cd\text{FCGV}(\text{u20})} \left(\frac{g^{\lambda \sigma } g^{\mu \nu }}{\alpha }-g^{\lambda \nu } g^{\mu \sigma }+g^{\lambda \mu } g^{\nu \sigma }\right)+f^{ac\text{FCGV}(\text{u20})} f^{bd\text{FCGV}(\text{u20})} \left(g^{\lambda \sigma } g^{\mu \nu }-g^{\lambda \nu } g^{\mu \sigma }\right)\right)