ToPaVe[exp, q] converts all scalar 1-loop integrals in
exp that depend on the momentum q to scalar
Passarino Veltman functions A0, B0,
C0, D0 etc.
Overview, PaVeToABCD, ToPaVe2, A0, A00, B0, B1, B00, B11, C0, D0.
FAD[{q, m1}]
ToPaVe[%, q]\frac{1}{q^2-\text{m1}^2}
i \pi ^2 \;\text{A}_0\left(\text{m1}^2\right)
FAD[{q, m1}, {q + p1, m2}]
ToPaVe[%, q]\frac{1}{\left(q^2-\text{m1}^2\right).\left((\text{p1}+q)^2-\text{m2}^2\right)}
i \pi ^2 \;\text{B}_0\left(\text{p1}^2,\text{m1}^2,\text{m2}^2\right)
% // StandardForm
(*I \[Pi]^2 B0[Pair[Momentum[p1, D], Momentum[p1, D]], m1^2, m2^2]*)FAD[{q, m1}, {q + p1, m2}, {q + p2, m3}, {q + p3, m4}, {q + p4, m5}]
ToPaVe[%, q]\frac{1}{\left(q^2-\text{m1}^2\right).\left((\text{p1}+q)^2-\text{m2}^2\right).\left((\text{p2}+q)^2-\text{m3}^2\right).\left((\text{p3}+q)^2-\text{m4}^2\right).\left((\text{p4}+q)^2-\text{m5}^2\right)}
i \pi ^2 \;\text{E}_0\left(\text{p1}^2,\text{p2}^2,-2 (\text{p2}\cdot \;\text{p3})+\text{p2}^2+\text{p3}^2,-2 (\text{p3}\cdot \;\text{p4})+\text{p3}^2+\text{p4}^2,-2 (\text{p1}\cdot \;\text{p4})+\text{p1}^2+\text{p4}^2,-2 (\text{p1}\cdot \;\text{p2})+\text{p1}^2+\text{p2}^2,\text{p3}^2,-2 (\text{p2}\cdot \;\text{p4})+\text{p2}^2+\text{p4}^2,-2 (\text{p1}\cdot \;\text{p3})+\text{p1}^2+\text{p3}^2,\text{p4}^2,\text{m2}^2,\text{m1}^2,\text{m3}^2,\text{m4}^2,\text{m5}^2\right)
By default, ToPaVe has the option
PaVeToABCD set to True. This means that some
of the PaVe functions are automatically converted to direct
Passarino-Veltman functions (A0, A00,
B0, B1, B00, B11,
C0, D0). This also has consequences for
TID
TID[FVD[q, mu] FAD[{q, m1}, {q + p}], q, ToPaVe -> True]\frac{i \pi ^2 p^{\text{mu}} \;\text{A}_0\left(\text{m1}^2\right)}{2 p^2}-\frac{i \pi ^2 \left(\text{m1}^2+p^2\right) p^{\text{mu}} \;\text{B}_0\left(p^2,0,\text{m1}^2\right)}{2 p^2}
% // StandardForm\frac{i \pi ^2 \;\text{A0}\left[\text{m1}^2\right] \;\text{Pair}[\text{LorentzIndex}[\text{mu},D],\text{Momentum}[p,D]]}{2 \;\text{Pair}[\text{Momentum}[p,D],\text{Momentum}[p,D]]}-\left.\left(i \pi ^2 \;\text{B0}\left[\text{Pair}[\text{Momentum}[p,D],\text{Momentum}[p,D]],0,\text{m1}^2\right] \;\text{Pair}[\text{LorentzIndex}[\text{mu},D],\text{Momentum}[p,D]] \left(\text{m1}^2+\text{Pair}[\text{Momentum}[p,D],\text{Momentum}[p,D]]\right)\right)\right/(2 \;\text{Pair}[\text{Momentum}[p,D],\text{Momentum}[p,D]])
If you want to avoid direct functions in the output of
TID and other functions that employ ToPaVe,
you need to set the option PaVeToABCD to False
globally.
SetOptions[ToPaVe, PaVeToABCD -> False];TID[FVD[q, mu] FAD[{q, m1}, {q + p}], q, ToPaVe -> True]\frac{i \pi ^2 p^{\text{mu}} \;\text{A}_0\left(\text{m1}^2\right)}{2 p^2}-\frac{i \pi ^2 \left(\text{m1}^2+p^2\right) p^{\text{mu}} \;\text{B}_0\left(p^2,0,\text{m1}^2\right)}{2 p^2}
% // StandardForm\frac{i \pi ^2 \;\text{Pair}[\text{LorentzIndex}[\text{mu},D],\text{Momentum}[p,D]] \;\text{PaVe}\left[0,\{\},\left\{\text{m1}^2\right\}\right]}{2 \;\text{Pair}[\text{Momentum}[p,D],\text{Momentum}[p,D]]}-\left.\left(i \pi ^2 \;\text{Pair}[\text{LorentzIndex}[\text{mu},D],\text{Momentum}[p,D]] \left(\text{m1}^2+\text{Pair}[\text{Momentum}[p,D],\text{Momentum}[p,D]]\right) \;\text{PaVe}\left[0,\{\text{Pair}[\text{Momentum}[p,D],\text{Momentum}[p,D]]\},\left\{0,\text{m1}^2\right\}\right]\right)\right/(2 \;\text{Pair}[\text{Momentum}[p,D],\text{Momentum}[p,D]])