Tdec
Tdec[{{qi, mu}, {qj, nu}, ...}, {p1, p2, ...}]
calculates the tensorial decomposition formulas for Lorentzian integrals. The more common ones are saved in TIDL
.
The automatic symmetrization of the tensor basis is done using Alexey Pak’s algorithm described in arXiv:1111.0868.
See also
Overview, TID, TIDL, OneLoopSimplify.
Examples
Check that ∫dDf(p,q)qμ=p2pμ∫dDf(p,q)p⋅q
Tdec[{q, \[Mu]}, {p}]
%[[2]] /. %[[1]]
{{X1→p⋅q,X2→p2},X2X1pμ}
p2pμ(p⋅q)
This calculates integral transformation for any ∫dDq1dDq2dDq3 f(p,q1,q2,q3)q1μq2νq3ρ.
Tdec[{{Subscript[q, 1], \[Mu]}, {Subscript[q, 2], \[Nu]}, {Subscript[q, 3], \[Rho]}}, {p}, List -> False]
Contract[% FVD[p, \[Mu]] FVD[p, \[Nu]] FVD[p, \[Rho]]] // Factor
(1−D)p4pρgμν(p⋅q3)((p⋅q1)(p⋅q2)−p2(q1⋅q2))+(1−D)p4pνgμρ(p⋅q2)((p⋅q1)(p⋅q3)−p2(q1⋅q3))+(1−D)p4pμgνρ(p⋅q1)((p⋅q2)(p⋅q3)−p2(q2⋅q3))−(1−D)p6pμpνpρ(D(p⋅q1)(p⋅q2)(p⋅q3)+2(p⋅q1)(p⋅q2)(p⋅q3)−p2(q1⋅q2)(p⋅q3)−p2(q1⋅q3)(p⋅q2)−p2(q2⋅q3)(p⋅q1))
(p⋅q1)(p⋅q2)(p⋅q3)