FeynCalc manual (development version)

FCLoopBasisIncompleteQ

FCLoopBasisIncompleteQ[int, {q1, q2, ...}] checks whether the loop integral or topology int lacks propagators need to have a linearly independent basis .

The input can also consist of an FCTopology object or a list thereof.

See also

Overview, FCLoopBasisOverdeterminedQ.

Examples

FAD[{q1, m1}] 
 
FCLoopBasisIncompleteQ[%, {q1}]

\frac{1}{\text{q1}^2-\text{m1}^2}

\text{False}

SPD[q1, l] FAD[{q1, m1}, {q1 - l + p, m}] 
 
FCLoopBasisIncompleteQ[%, {q1}]

\frac{l\cdot \;\text{q1}}{\left(\text{q1}^2-\text{m1}^2\right).\left((-l+p+\text{q1})^2-m^2\right)}

\text{False}

FAD[{q1, m1}, {q2, m2}] 
 
FCLoopBasisIncompleteQ[%, {q1, q2}]

\frac{1}{\left(\text{q1}^2-\text{m1}^2\right).\left(\text{q2}^2-\text{m2}^2\right)}

\text{True}

FAD[q1, q2, {q1 - l1, m1}, {q2 - l2, m2}] 
 
FCLoopBasisIncompleteQ[%, {q1, q2}]

\frac{1}{\text{q1}^2.\text{q2}^2.\left((\text{q1}-\text{l1})^2-\text{m1}^2\right).\left((\text{q2}-\text{l2})^2-\text{m2}^2\right)}

\text{True}

CSPD[q1, l] CFAD[{q1, m1}, {q1 - l + p, m}] 
 
FCLoopBasisIncompleteQ[%, {q1}]

\frac{l\cdot \;\text{q1}}{(\text{q1}^2+\text{m1}-i \eta ).((-l+p+\text{q1})^2+m-i \eta )}

\text{False}

SFAD[{q1, m1}, {q2, m2}] 
 
FCLoopBasisIncompleteQ[%, {q1, q2}]

\frac{1}{(\text{q1}^2-\text{m1}+i \eta ).(\text{q2}^2-\text{m2}+i \eta )}

\text{True}

FCLoopBasisIncompleteQ[FCTopology[topo, {FAD[p1], 
    FAD[p2], FAD[p1 - q], FAD[p2 - q]}, {p1, p2}, {q}, {}, {}]]

\text{True}

FCLoopBasisIncompleteQ[{
   FCTopology[topo1, {FAD[p1], FAD[p2], FAD[p1 - q], FAD[p2 - q]}, {p1, p2}, {q}, {}, {}], 
   FCTopology[topo2, {FAD[p1], FAD[p2], FAD[p1 - q], FAD[p2 - p1]}, {p1, p2}, {q}, {}, {}], 
   FCTopology[topo3, {FAD[p1], FAD[p1 - q]}, {p1}, {q}, {}, {}] 
  }]

\{\text{True},\text{True},\text{False}\}