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.
Overview, TID, TIDL, OneLoopSimplify.
Check that \int d^D f(p,q) q^{\mu}= \frac{p^{\mu}}{p^2} \int d^D f(p,q) p \cdot q
Tdec[{q, \[Mu]}, {p}]
%[[2]] /. %[[1]]\left\{\left\{\text{X1}\to p\cdot q,\text{X2}\to p^2\right\},\frac{\text{X1} p^{\mu }}{\text{X2}}\right\}
\frac{p^{\mu } (p\cdot q)}{p^2}
This calculates integral transformation for any \int d^D q_1 d^D q_2 d^D q_3 f(p,q_1,q_2,q_3) q_1^{\mu} q_2^{\nu}q_3^{\rho}.
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\frac{p^{\rho } g^{\mu \nu } \left(p\cdot q_3\right) \left(\left(p\cdot q_1\right) \left(p\cdot q_2\right)-p^2 \left(q_1\cdot q_2\right)\right)}{(1-D) p^4}+\frac{p^{\nu } g^{\mu \rho } \left(p\cdot q_2\right) \left(\left(p\cdot q_1\right) \left(p\cdot q_3\right)-p^2 \left(q_1\cdot q_3\right)\right)}{(1-D) p^4}+\frac{p^{\mu } g^{\nu \rho } \left(p\cdot q_1\right) \left(\left(p\cdot q_2\right) \left(p\cdot q_3\right)-p^2 \left(q_2\cdot q_3\right)\right)}{(1-D) p^4}-\frac{p^{\mu } p^{\nu } p^{\rho } \left(D \left(p\cdot q_1\right) \left(p\cdot q_2\right) \left(p\cdot q_3\right)+2 \left(p\cdot q_1\right) \left(p\cdot q_2\right) \left(p\cdot q_3\right)-p^2 \left(q_1\cdot q_2\right) \left(p\cdot q_3\right)-p^2 \left(q_1\cdot q_3\right) \left(p\cdot q_2\right)-p^2 \left(q_2\cdot q_3\right) \left(p\cdot q_1\right)\right)}{(1-D) p^6}
\left(p\cdot q_1\right) \left(p\cdot q_2\right) \left(p\cdot q_3\right)
To calculate a tensor decomposition with specific kinematic
constraints, use the option FinalSubstitutions. Notice that
kinematic configurations involving zero Gram determinants are not
supported.
Tdec[{{l, \[Mu]}, {l, \[Nu]}}, {p1, p2},
FinalSubstitutions -> {SPD[p1] -> 0, SPD[p2] -> 0}, FCE -> True, List -> False]-\frac{g^{\mu \nu } \left(l^2 (\text{p1}\cdot \;\text{p2})^2-2 (l\cdot \;\text{p1}) (l\cdot \;\text{p2}) (\text{p1}\cdot \;\text{p2})\right)}{(2-D) (\text{p1}\cdot \;\text{p2})^2}+\frac{\text{p2}^{\mu } \;\text{p2}^{\nu } \left(2 (l\cdot \;\text{p1})^2 (\text{p1}\cdot \;\text{p2})^2-D (l\cdot \;\text{p1})^2 (\text{p1}\cdot \;\text{p2})^2\right)}{(2-D) (\text{p1}\cdot \;\text{p2})^4}+\frac{\text{p1}^{\mu } \;\text{p1}^{\nu } \left(2 (l\cdot \;\text{p2})^2 (\text{p1}\cdot \;\text{p2})^2-D (l\cdot \;\text{p2})^2 (\text{p1}\cdot \;\text{p2})^2\right)}{(2-D) (\text{p1}\cdot \;\text{p2})^4}-\frac{\text{p1}^{\nu } \;\text{p2}^{\mu } \left(D (l\cdot \;\text{p1}) (l\cdot \;\text{p2}) (\text{p1}\cdot \;\text{p2})^2-l^2 (\text{p1}\cdot \;\text{p2})^3\right)}{(2-D) (\text{p1}\cdot \;\text{p2})^4}-\frac{\text{p1}^{\mu } \;\text{p2}^{\nu } \left(D (l\cdot \;\text{p1}) (l\cdot \;\text{p2}) (\text{p1}\cdot \;\text{p2})^2-l^2 (\text{p1}\cdot \;\text{p2})^3\right)}{(2-D) (\text{p1}\cdot \;\text{p2})^4}