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)