FCMultiLoopTID[amp, {q1, q2, ...}] does a multi-loop
tensor integral decomposition, transforming the Lorentz indices away
from the loop momenta q1, q2, ... The decomposition is
applied only to the loop integrals where loop momenta are contracted
with Dirac matrices or epsilon tensors.
Overview, FCLoopFindTensorBasis, TID.
FCI[FVD[q1, \[Mu]] FVD[q2, \[Nu]] FAD[q1, q2, {q1 - p1}, {q2 - p1}, {q1 - q2}]]
FCMultiLoopTID[%, {q1, q2}]\frac{\text{q1}^{\mu } \;\text{q2}^{\nu }}{\text{q1}^2.\text{q2}^2.(\text{q1}-\text{p1})^2.(\text{q2}-\text{p1})^2.(\text{q1}-\text{q2})^2}
\frac{\text{p1}^{\mu } \;\text{p1}^{\nu }-\text{p1}^2 g^{\mu \nu }}{(1-D) \;\text{p1}^2 \;\text{q1}^2.\text{q2}^2.(\text{q1}-\text{p1})^2.(\text{q1}-\text{q2})^2}-\frac{\text{p1}^{\mu } \;\text{p1}^{\nu }-\text{p1}^2 g^{\mu \nu }}{2 (1-D) \;\text{p1}^2 \;\text{q1}^2.\text{q2}^2.(\text{q1}-\text{p1})^2.(\text{q2}-\text{p1})^2}-\frac{D \;\text{p1}^{\mu } \;\text{p1}^{\nu }-\text{p1}^2 g^{\mu \nu }}{4 (1-D) \;\text{q1}^2.\text{q2}^2.(\text{q1}-\text{p1})^2.(\text{q1}-\text{q2})^2.(\text{q2}-\text{p1})^2}+\frac{D \;\text{p1}^{\mu } \;\text{p1}^{\nu }-\text{p1}^2 g^{\mu \nu }}{2 (1-D) \;\text{p1}^4 \;\text{q1}^2.(\text{q1}-\text{q2})^2.(\text{q2}-\text{p1})^2}
In the case of vanishing Gram determinants one can apply the same
procedure as in the case of TID or FCLoopTensorReduce: one uses
FCLoopFindTensorBasis to find a linear independent basis of
external momenta and then supplies this basis to the function.
FCClearScalarProducts[]
SPD[p1] = m1^2;
SPD[p2] = m2^2;
SPD[p1, p2] = m1 m2;FCMultiLoopTID[FVD[q1, mu] FAD[{q1, m}, {q1 + p1}, {q1 + p2}], {q1}]
\text{\$Aborted}
FCLoopFindTensorBasis[{p1, p2}, {}, n]\left( \begin{array}{c} \;\text{p1} \\ \;\text{p2} \\ \;\text{p2}\to \;\text{p1} \;\text{FCGV}(\text{Prefactor})\left(\frac{\text{m2}}{\text{m1}}\right) \\ \end{array} \right)
FCMultiLoopTID[FVD[q1, mu] FAD[{q1, m}, {q1 + p1}, {q1 + p2}], {q1},
TensorReductionBasisChange -> {{p1, p2} -> {p1}}]\frac{\text{p1}^{\text{mu}}}{2 \;\text{m1}^2 \;\text{q1}^2.\left((\text{q1}-\text{p2})^2-m^2\right)}-\frac{\left(m^2+\text{m1}^2\right) \;\text{p1}^{\text{mu}}}{2 \;\text{m1}^2 \left(\text{q1}^2-m^2\right).(\text{q1}-\text{p1})^2.(\text{q1}-\text{p2})^2}-\frac{\text{p1}^{\text{mu}}}{2 \;\text{m1}^2 \;\text{q1}^2.(-\text{p1}+\text{p2}+\text{q1})^2}