CartesianPairContract is like CartesianPair, but with (local) contraction properties. The function fully supports the BMHV algebra and will not expand momenta inside scalar products.
CartesianPairContract is an auxiliary function used in higher level FeynCalc functions that require fast contractions between multiple expressions, where Contract would be too slow.
Overview, CartesianPair, Contract.
CartesianPair[CartesianIndex[i], CartesianMomentum[p]] CartesianPair[CartesianIndex[i], CartesianMomentum[q]]
% /. CartesianPair -> CartesianPairContract
% /. CartesianPairContract -> CartesianPair\overline{p}^i \overline{q}^i
\text{CartesianPairContract}\left(\overline{p},\overline{q}\right)
\overline{p}\cdot \overline{q}
CartesianPair[CartesianIndex[i], CartesianMomentum[p]] CartesianPair[CartesianIndex[j], CartesianMomentum[q]] CartesianPair[CartesianIndex[i], CartesianIndex[j]]
% /. CartesianPair -> CartesianPairContract
% /. CartesianPairContract -> CartesianPair\overline{p}^i \overline{q}^j \bar{\delta }^{ij}
\text{CartesianPairContract}\left(\overline{p},\overline{q}\right)
\overline{p}\cdot \overline{q}
CartesianPair[CartesianIndex[i], CartesianMomentum[p + q]] CartesianPair[CartesianIndex[i], CartesianMomentum[r + s]]
% /. CartesianPair -> CartesianPairContract
% /. CartesianPairContract -> CartesianPair
\left(\overline{p}+\overline{q}\right)^i \left(\overline{r}+\overline{s}\right)^i
\text{CartesianPairContract}\left(\overline{p}+\overline{q},\overline{r}+\overline{s}\right)
(\overline{p}+\overline{q})\cdot (\overline{r}+\overline{s})