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, FCLoopFindTensorBasis, TID.

Examples

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

q1μ  q2νq12.q22.(q1p1)2.(q2p1)2.(q1q2)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  p1νp12gμν(1D)  p12  q12.q22.(q1p1)2.(q1q2)2p1μ  p1νp12gμν2(1D)  p12  q12.q22.(q1p1)2.(q2p1)2D  p1μ  p1νp12gμν4(1D)  q12.q22.(q1p1)2.(q1q2)2.(q2p1)2+D  p1μ  p1νp12gμν2(1D)  p14  q12.(q1q2)2.(q2p1)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}

In the case of vanishing Gram determinants one can apply the same procedure as in the case of TID or FCLoopTensorReduce: one uses FCLoopFindTensorBasis to find a linear independent basis of external momenta and then supplies this basis to the function.

FCClearScalarProducts[]
SPD[p1] = m1^2;
SPD[p2] = m2^2;
SPD[p1, p2] = m1 m2;
FCMultiLoopTID[FVD[q1, mu] FAD[{q1, m}, {q1 + p1}, {q1 + p2}], {q1}]

0h2gltw65l2pe

$Aborted\text{\$Aborted}

FCLoopFindTensorBasis[{p1, p2}, {}, n]

(  p1  p2  p2  p1  FCGV(Prefactor)(m2m1))\left( \begin{array}{c} \;\text{p1} \\ \;\text{p2} \\ \;\text{p2}\to \;\text{p1} \;\text{FCGV}(\text{Prefactor})\left(\frac{\text{m2}}{\text{m1}}\right) \\ \end{array} \right)

FCMultiLoopTID[FVD[q1, mu] FAD[{q1, m}, {q1 + p1}, {q1 + p2}], {q1}, 
  TensorReductionBasisChange -> {{p1, p2} -> {p1}}]

p1mu2  m12  q12.((q1p2)2m2)(m2+m12)  p1mu2  m12(q12m2).(q1p1)2.(q1p2)2p1mu2  m12  q12.(p1+p2+q1)2\frac{\text{p1}^{\text{mu}}}{2 \;\text{m1}^2 \;\text{q1}^2.\left((\text{q1}-\text{p2})^2-m^2\right)}-\frac{\left(m^2+\text{m1}^2\right) \;\text{p1}^{\text{mu}}}{2 \;\text{m1}^2 \left(\text{q1}^2-m^2\right).(\text{q1}-\text{p1})^2.(\text{q1}-\text{p2})^2}-\frac{\text{p1}^{\text{mu}}}{2 \;\text{m1}^2 \;\text{q1}^2.(-\text{p1}+\text{p2}+\text{q1})^2}