FeynCalc manual (development version)

CTdec

CTdec[{{qi, a}, {qj, b}, ...}, {p1, p2, ...}] or CTdec[exp, {{qi, a}, {qj, b}, ...}, {p1, p2, ...}] calculates the tensorial decomposition formulas for Cartesian integrals. The more common ones are saved in TIDL.

See also

Overview, Tdec, TIDL, TID.

Examples

Check that dD1qf(p,q)qi=pip2dD1qf(p,q)pq\int d^{D-1} q \, f(p,q) q^i = \frac{p^i}{p^2} \int d^{D-1} q \, f(p,q) p \cdot q

CTdec[{{q, i}}, {p}]

{{X1pq,X2p2},X1piX2}\left\{\left\{\text{X1}\to p\cdot q,\text{X2}\to p^2\right\},\frac{\text{X1} p^i}{\text{X2}}\right\}

%[[2]] /. %[[1]]

pi(pq)p2\frac{p^i (p\cdot q)}{p^2}

CTdec[{{q, i}}, {p}, List -> False]

pi(pq)p2\frac{p^i (p\cdot q)}{p^2}

This calculates integral transformation for any dD1q1dD1q2dD1q3f(p,q1,q2,q3)q1iq2jq3k\int d^{D-1} q_1 d^{D-1} q_2 d^{D-1} q_3 f (p, q_1, q_2, q_3) q_1^i q_2^j q_3^k.

CTdec[{{Subscript[q, 1], i}, {Subscript[q, 2], j}, {Subscript[q, 3], k}}, {p}, List -> False]

pkδij(pq3)((pq1)(pq2)p2(q1q2))(2D)p4+pjδik(pq2)((pq1)(pq3)p2(q1q3))(2D)p4+piδjk(pq1)((pq2)(pq3)p2(q2q3))(2D)p4pipjpk((D1)(pq1)(pq2)(pq3)+2(pq1)(pq2)(pq3)p2(q1q2)(pq3)p2(q1q3)(pq2)p2(q2q3)(pq1))(2D)p6\frac{p^k \delta ^{ij} \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)}{(2-D) p^4}+\frac{p^j \delta ^{ik} \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)}{(2-D) p^4}+\frac{p^i \delta ^{jk} \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)}{(2-D) p^4}-\frac{p^i p^j p^k \left((D-1) \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)}{(2-D) p^6}

Contract[% CVD[p, i] CVD[p, j] CVD[p, k]] // Factor

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