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=pμp2∫dDf(p,q)p⋅q\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]]

{{X1→p⋅q,X2→p2},X1pμX2}\left\{\left\{\text{X1}\to p\cdot q,\text{X2}\to p^2\right\},\frac{\text{X1} p^{\mu }}{\text{X2}}\right\}

pμ(p⋅q)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(p⋅q3)((p⋅q1)(p⋅q2)−p2(q1⋅q2))(1−D)p4+pνgμρ(p⋅q2)((p⋅q1)(p⋅q3)−p2(q1⋅q3))(1−D)p4+pμgνρ(p⋅q1)((p⋅q2)(p⋅q3)−p2(q2⋅q3))(1−D)p4−pμpνpρ(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))(1−D)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}

(p⋅q1)(p⋅q2)(p⋅q3)\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]

−gμν(l2(p1⋅  p2)2−2(l⋅  p1)(l⋅  p2)(p1⋅  p2))(2−D)(p1⋅  p2)2+p2μ  p2ν(2(l⋅  p1)2(p1⋅  p2)2−D(l⋅  p1)2(p1⋅  p2)2)(2−D)(p1⋅  p2)4+p1μ  p1ν(2(l⋅  p2)2(p1⋅  p2)2−D(l⋅  p2)2(p1⋅  p2)2)(2−D)(p1⋅  p2)4−p1ν  p2μ(D(l⋅  p1)(l⋅  p2)(p1⋅  p2)2−l2(p1⋅  p2)3)(2−D)(p1⋅  p2)4−p1μ  p2ν(D(l⋅  p1)(l⋅  p2)(p1⋅  p2)2−l2(p1⋅  p2)3)(2−D)(p1⋅  p2)4-\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}