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]
q2−m121
iπ2A0(m12)
FAD[{q, m1}, {q + p1, m2}]
ToPaVe[%, q]
(q2−m12).((p1+q)2−m22)1
iπ2B0(p12,m12,m22)
% // 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]
(q2−m12).((p1+q)2−m22).((p2+q)2−m32).((p3+q)2−m42).((p4+q)2−m52)1
iπ2E0(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)
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]
2p2iπ2pmuA0(m12)−2p2iπ2(m12+p2)pmuB0(p2,0,m12)
2Pair[Momentum[p,D],Momentum[p,D]]iπ2A0[m12]Pair[LorentzIndex[mu,D],Momentum[p,D]]−(iπ2B0[Pair[Momentum[p,D],Momentum[p,D]],0,m12]Pair[LorentzIndex[mu,D],Momentum[p,D]](m12+Pair[Momentum[p,D],Momentum[p,D]]))/(2Pair[Momentum[p,D],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]
2p2iπ2pmuA0(m12)−2p2iπ2(m12+p2)pmuB0(p2,0,m12)
2Pair[Momentum[p,D],Momentum[p,D]]iπ2Pair[LorentzIndex[mu,D],Momentum[p,D]]PaVe[0,{},{m12}]−(iπ2Pair[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}])/(2Pair[Momentum[p,D],Momentum[p,D]])