FeynCalc manual (development version)

ToPaVe

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.

See also

Overview, PaVeToABCD, ToPaVe2, A0, A00, B0, B1, B00, B11, C0, D0.

Examples

FAD[{q, m1}] 
 
ToPaVe[%, q]

1q2m12\frac{1}{q^2-\text{m1}^2}

iπ2  A0(m12)i \pi ^2 \;\text{A}_0\left(\text{m1}^2\right)

FAD[{q, m1}, {q + p1, m2}] 
 
ToPaVe[%, q]

1(q2m12).((p1+q)2m22)\frac{1}{\left(q^2-\text{m1}^2\right).\left((\text{p1}+q)^2-\text{m2}^2\right)}

iπ2  B0(p12,m12,m22)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]

1(q2m12).((p1+q)2m22).((p2+q)2m32).((p3+q)2m42).((p4+q)2m52)\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)}

iπ2  E0(p12,p22,2(p2  p3)+p22+p32,2(p3  p4)+p32+p42,2(p1  p4)+p12+p42,2(p1  p2)+p12+p22,p32,2(p2  p4)+p22+p42,2(p1  p3)+p12+p32,p42,m22,m12,m32,m42,m52)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]

iπ2pmu  A0(m12)2p2iπ2(m12+p2)pmu  B0(p2,0,m12)2p2\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

iπ2  A0[m12]  Pair[LorentzIndex[mu,D],Momentum[p,D]]2  Pair[Momentum[p,D],Momentum[p,D]](iπ2  B0[Pair[Momentum[p,D],Momentum[p,D]],0,m12]  Pair[LorentzIndex[mu,D],Momentum[p,D]](m12+Pair[Momentum[p,D],Momentum[p,D]]))/(2  Pair[Momentum[p,D],Momentum[p,D]])\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]

iπ2pmu  A0(m12)2p2iπ2(m12+p2)pmu  B0(p2,0,m12)2p2\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

iπ2  Pair[LorentzIndex[mu,D],Momentum[p,D]]  PaVe[0,{},{m12}]2  Pair[Momentum[p,D],Momentum[p,D]](iπ2  Pair[LorentzIndex[mu,D],Momentum[p,D]](m12+Pair[Momentum[p,D],Momentum[p,D]])  PaVe[0,{Pair[Momentum[p,D],Momentum[p,D]]},{0,m12}])/(2  Pair[Momentum[p,D],Momentum[p,D]])\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]])