FeynCalc manual (development version)

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=pμp2dDf(p,q)pq\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]]

{{X1pq,X2p2},X1pμX2}\left\{\left\{\text{X1}\to p\cdot q,\text{X2}\to p^2\right\},\frac{\text{X1} p^{\mu }}{\text{X2}}\right\}

pμ(pq)p2\frac{p^{\mu } (p\cdot q)}{p^2}

This calculates integral transformation for any dDq1dDq2dDq3\int d^D q_1 d^D q_2 d^D q_3 f(p,q1,q2,q3)q1μq2νq3ρf(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

pρgμν(pq3)((pq1)(pq2)p2(q1q2))(1D)p4+pνgμρ(pq2)((pq1)(pq3)p2(q1q3))(1D)p4+pμgνρ(pq1)((pq2)(pq3)p2(q2q3))(1D)p4pμpνpρ(D(pq1)(pq2)(pq3)+2(pq1)(pq2)(pq3)p2(q1q2)(pq3)p2(q1q3)(pq2)p2(q2q3)(pq1))(1D)p6\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}

(pq1)(pq2)(pq3)\left(p\cdot q_1\right) \left(p\cdot q_2\right) \left(p\cdot q_3\right)