PairContract3 is like Pair, but with local contraction properties among PairContract3s. The function fully supports the BMHV algebra and, unlike PairContract or PairContract2 will always expand momenta inside scalar products.
PairContract3 is an auxiliary function used in higher level FeynCalc functions that require fast contractions between multiple expressions, where Contract would be too slow.
Overview, Contract, PairContract, PairContract2.
Pair[LorentzIndex[\[Mu]], Momentum[p]] Pair[LorentzIndex[\[Mu]], Momentum[q]]
% /. Pair -> PairContract3\overline{p}^{\mu } \overline{q}^{\mu }
\overline{p}\cdot \overline{q}
Pair[LorentzIndex[\[Mu]], Momentum[p]] Pair[LorentzIndex[\[Nu]], Momentum[q]] Pair[LorentzIndex[\[Mu]], LorentzIndex[\[Nu]]]
% /. Pair -> PairContract3\overline{p}^{\mu } \overline{q}^{\nu } \bar{g}^{\mu \nu }
\overline{p}\cdot \overline{q}
Pair[LorentzIndex[\[Mu]], Momentum[p + q]] Pair[LorentzIndex[\[Mu]], Momentum[r + s]]
% /. Pair -> PairContract3
\left(\overline{p}+\overline{q}\right)^{\mu } \left(\overline{r}+\overline{s}\right)^{\mu }
\overline{p}\cdot \overline{r}+\overline{p}\cdot \overline{s}+\overline{q}\cdot \overline{r}+\overline{q}\cdot \overline{s}