FeynCalc manual (development version)

LightPak

LightPak is an option for FCLoopPakOrder and other functions for finding equivalent topologies or integrals using Pak algorithm. When set to True, instead of using the full Pak algorithm (which can be slow for complicated integrals) we use only a lightweight version that is not guaranteed to find all mappings but requires significantly less time.

The light Pak algorithm is described in the pySecDec manual. Essentially, it means that in the step 5 of the full Pak algorithm we keep only the first matrix in the vector, so that the next iteration step generates significantly less matrices than in the full version.

See also

Overview, FCLoopIsolate.

Examples

Canonicalizing this characteristic Polynomial of a loop integral with 11 propagators requires almost half a minute using the full Pak algorithm. The light Pak finishes here almost immediately.

poly = (x[1]*x[2]*x[3]*x[4]*x[5]*x[6]*x[7]*x[8]*(x[9]*x[10] + x[9]*x[11] + 
       x[10]*x[11]) - x[1]*x[2]*x[3]*x[4]*x[5]*x[6]*x[7]*x[8]*(m1^2*x[1] + m1^2*x[2] + 
       m1^2*x[3] + m1^2*x[4] + m1^2*x[5] + m1^2*x[6] + m1^2*x[7] + m1^2*x[8] + m1^2*x[9] + 
       m1^2*x[10] + m3^2*x[11])*(x[9]*x[10] + x[9]*x[11] + x[10]*x[11]))

x(1) x(2) x(3) x(4) x(5) x(6) x(7) x(8) (x(9) x(10)+x(11) x(10)+x(9) x(11))-x(1) x(2) x(3) x(4) x(5) x(6) x(7) x(8) (x(9) x(10)+x(11) x(10)+x(9) x(11)) \left(\text{m1}^2 x(1)+\text{m1}^2 x(2)+\text{m1}^2 x(3)+\text{m1}^2 x(4)+\text{m1}^2 x(5)+\text{m1}^2 x(6)+\text{m1}^2 x(7)+\text{m1}^2 x(8)+\text{m1}^2 x(9)+\text{m1}^2 x(10)+\text{m3}^2 x(11)\right)

FCLoopPakOrder[poly, {x[1], x[2], x[3], x[4], x[5], x[6], x[7], x[8], x[9], x[10], x[11]}, 
  LightPak -> True]

\left( \begin{array}{ccccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \end{array} \right)

canoPoly1 = FCLoopPakOrder[poly, {x[1], x[2], x[3], x[4], x[5], x[6], x[7], x[8], x[9], x[10],x[11]}, 
    LightPak -> True, Rename -> True];

canoPoly2 = FCLoopPakOrder[poly /. {x[3] -> x[11], x[11] -> x[3]}, 
    {x[1], x[2], x[3], x[4], x[5], x[6], x[7], x[8], x[9], x[10], x[11]}, LightPak -> True, Rename -> True];

Obviously, we can still find equivalence relations with using this algorithm

canoPoly1 - canoPoly2

0