FeynCalc manual (development version)

FCMultiLoopTID

FCMultiLoopTID[amp, {q1, q2, ...}] does a multi-loop tensor integral decomposition, transforming the Lorentz indices away from the loop momenta q1, q2, ... The decomposition is applied only to the loop integrals where loop momenta are contracted with Dirac matrices or epsilon tensors.

See also

Overview, TID.

Examples

FCI[FVD[q1, \[Mu]] FVD[q2, \[Nu]] FAD[q1, q2, {q1 - p1}, {q2 - p1}, {q1 - q2}]] 
 
FCMultiLoopTID[%, {q1, q2}]

q1μ  q2νq12.q22.(q1−p1)2.(q2−p1)2.(q1−q2)2\frac{\text{q1}^{\mu } \;\text{q2}^{\nu }}{\text{q1}^2.\text{q2}^2.(\text{q1}-\text{p1})^2.(\text{q2}-\text{p1})^2.(\text{q1}-\text{q2})^2}

p1μ  p1ν−p12gμν(1−D)  p12  q12.q22.(q1−p1)2.(q1−q2)2−p1μ  p1ν−p12gμν2(1−D)  p12  q12.q22.(q1−p1)2.(q2−p1)2−D  p1μ  p1ν−p12gμν4(1−D)  q12.q22.(q1−p1)2.(q1−q2)2.(q2−p1)2+D  p1μ  p1ν−p12gμν2(1−D)  p14  q12.(q1−q2)2.(q2−p1)2\frac{\text{p1}^{\mu } \;\text{p1}^{\nu }-\text{p1}^2 g^{\mu \nu }}{(1-D) \;\text{p1}^2 \;\text{q1}^2.\text{q2}^2.(\text{q1}-\text{p1})^2.(\text{q1}-\text{q2})^2}-\frac{\text{p1}^{\mu } \;\text{p1}^{\nu }-\text{p1}^2 g^{\mu \nu }}{2 (1-D) \;\text{p1}^2 \;\text{q1}^2.\text{q2}^2.(\text{q1}-\text{p1})^2.(\text{q2}-\text{p1})^2}-\frac{D \;\text{p1}^{\mu } \;\text{p1}^{\nu }-\text{p1}^2 g^{\mu \nu }}{4 (1-D) \;\text{q1}^2.\text{q2}^2.(\text{q1}-\text{p1})^2.(\text{q1}-\text{q2})^2.(\text{q2}-\text{p1})^2}+\frac{D \;\text{p1}^{\mu } \;\text{p1}^{\nu }-\text{p1}^2 g^{\mu \nu }}{2 (1-D) \;\text{p1}^4 \;\text{q1}^2.(\text{q1}-\text{q2})^2.(\text{q2}-\text{p1})^2}