FeynCalc manual (development version)

FCLoopPakOrder

FCLoopPakOrder[poly, {x1, x2, ...}] determines a canonical ordering of the Feynman parameters x1, x2, ... in the polynomial poly.

The function uses the algorithm of Alexey Pak arXiv:1111.0868. Cf. also the PhD thesis of Jens Hoff 10.5445/IR/1000047447 for the detailed description of a possible implementation.

The current implementation is based on the PolyOrdering function from FIRE 6 arXiv:1901.07808

The function can also directly perform the renaming of the Feynman parameter variables returning the original polynomial in the canonical form. This is done by setting the option Rename to True.

See also

Overview, FCTopology, GLI, FCLoopToPakForm, FCLoopPakOrder.

Examples

Canonicalizing a polynomial

Let us consider the following product of U and F polynomials of some loop integral

poly = (x[1]*x[2] + x[1]*x[3] + x[2]*x[3] + x[2]*x[4] + x[3]*x[4] + x[1]*x[5] + 
     x[2]*x[5] + x[4]*x[5])* (m1^2*x[1]^2*x[2] + m3^2*x[1]*x[2]^2 + m1^2*x[1]^2*x[3] + 
     m1^2*x[1]*x[2]*x[3] + m2^2*x[1]*x[2]*x[3] +   m3^2*x[1]*x[2]*x[3] + 
     m3^2*x[2]^2*x[3] + m2^2*x[1]*x[3]^2 + m2^2*x[2]*x[3]^2 + m1^2*x[1]*x[2]*x[4] - 
      SPD[q, q]*x[1]*x[2]*x[4] + m3^2*x[2]^2*x[4] + m1^2*x[1]*x[3]*x[4] - 
      SPD[q, q]*x[1]*x[3]*x[4] +   m2^2*x[2]*x[3]*x[4] + m3^2*x[2]*x[3]*x[4] - 
      SPD[q, q]*x[2]*x[3]*x[4] + m2^2*x[3]^2*x[4] + m1^2*x[1]^2*x[5] +   
      m1^2*x[1]*x[2]*x[5] + m3^2*x[1]*x[2]*x[5] - SPD[q, q]*x[1]*x[2]*x[5] + 
      m3^2*x[2]^2*x[5] + m2^2*x[1]*x[3]*x[5] -   SPD[q, q]*x[1]*x[3]*x[5] + 
      m2^2*x[2]*x[3]*x[5] - SPD[q, q]*x[2]*x[3]*x[5] + m1^2*x[1]*x[4]*x[5] - 
      SPD[q, q]*x[1]*x[4]*x[5] + m3^2*x[2]*x[4]*x[5] + m2^2*x[3]*x[4]*x[5] - 
      SPD[q, q]*x[3]*x[4]*x[5])

$$(x(1) x(2)+x(3) x(2)+x(4) x(2)+x(5) x(2)+x(1) x(3)+x(3) x(4)+x(1) x(5)+x(4) x(5)) \left(\text{m1}^2 x(1)^2 x(2)+\text{m1}^2 x(1)^2 x(3)+\text{m1}^2 x(1) x(2) x(3)+\text{m1}^2 x(1) x(2) x(4)+\text{m1}^2 x(1) x(3) x(4)+\text{m1}^2 x(1)^2 x(5)+\text{m1}^2 x(1) x(2) x(5)+\text{m1}^2 x(1) x(4) x(5)+\text{m2}^2 x(1) x(3)^2+\text{m2}^2 x(2) x(3)^2+\text{m2}^2 x(1) x(2) x(3)+\text{m2}^2 x(3)^2 x(4)+\text{m2}^2 x(2) x(3) x(4)+\text{m2}^2 x(1) x(3) x(5)+\text{m2}^2 x(2) x(3) x(5)+\text{m2}^2 x(3) x(4) x(5)+\text{m3}^2 x(1) x(2)^2+\text{m3}^2 x(2)^2 x(3)+\text{m3}^2 x(1) x(2) x(3)+\text{m3}^2 x(2)^2 x(4)+\text{m3}^2 x(2) x(3) x(4)+\text{m3}^2 x(2)^2 x(5)+\text{m3}^2 x(1) x(2) x(5)+\text{m3}^2 x(2) x(4) x(5)-q^2 x(1) x(2) x(4)-q^2 x(1) x(3) x(4)-q^2 x(2) x(3) x(4)-q^2 x(1) x(2) x(5)-q^2 x(1) x(3) x(5)-q^2 x(2) x(3) x(5)-q^2 x(1) x(4) x(5)-q^2 x(3) x(4) x(5)\right)$$

Using FCLoopPakOrder we can obtain a canonical ordering for this polynomial

sigma = FCLoopPakOrder[poly, x]

$$\left( \begin{array}{ccccc} 1 & 3 & 2 & 5 & 4 \\ \end{array} \right)$$

This output implies that the polynomial will become canonically ordered upon renaming the Feynman parameter variables as follows

fpVars = Table[x[i], {i, 1, 5}]

$$\{x(1),x(2),x(3),x(4),x(5)\}$$

repRule = Thread[Rule[Extract[fpVars, List /@ First[sigma]], fpVars]]

$$\{x(1)\to x(1),x(3)\to x(2),x(2)\to x(3),x(5)\to x(4),x(4)\to x(5)\}$$

This way we obtain the canonical form of our polynomial poly

poly /. repRule

$$(x(1) x(2)+x(3) x(2)+x(5) x(2)+x(1) x(3)+x(1) x(4)+x(3) x(4)+x(3) x(5)+x(4) x(5)) \left(\text{m1}^2 x(1)^2 x(2)+\text{m1}^2 x(1)^2 x(3)+\text{m1}^2 x(1) x(2) x(3)+\text{m1}^2 x(1)^2 x(4)+\text{m1}^2 x(1) x(3) x(4)+\text{m1}^2 x(1) x(2) x(5)+\text{m1}^2 x(1) x(3) x(5)+\text{m1}^2 x(1) x(4) x(5)+\text{m2}^2 x(1) x(2)^2+\text{m2}^2 x(2)^2 x(3)+\text{m2}^2 x(1) x(2) x(3)+\text{m2}^2 x(1) x(2) x(4)+\text{m2}^2 x(2) x(3) x(4)+\text{m2}^2 x(2)^2 x(5)+\text{m2}^2 x(2) x(3) x(5)+\text{m2}^2 x(2) x(4) x(5)+\text{m3}^2 x(1) x(3)^2+\text{m3}^2 x(2) x(3)^2+\text{m3}^2 x(1) x(2) x(3)+\text{m3}^2 x(3)^2 x(4)+\text{m3}^2 x(1) x(3) x(4)+\text{m3}^2 x(3)^2 x(5)+\text{m3}^2 x(2) x(3) x(5)+\text{m3}^2 x(3) x(4) x(5)-q^2 x(1) x(2) x(4)-q^2 x(1) x(3) x(4)-q^2 x(2) x(3) x(4)-q^2 x(1) x(2) x(5)-q^2 x(1) x(3) x(5)-q^2 x(2) x(3) x(5)-q^2 x(1) x(4) x(5)-q^2 x(2) x(4) x(5)\right)$$

Checking equivalence

Let us consider the following two polynomials

poly1 = -1/4*(x[2]^2*x[3]) - (x[1]^2*x[4])/4 - 
   (x[1]^2*x[5])/4 + (x[1]*x[2]*x[5])/2 - (x[2]^2*x[5])/4 + x[3]*x[4]*x[5]

$$-\frac{1}{4} x(4) x(1)^2-\frac{1}{4} x(5) x(1)^2+\frac{1}{2} x(2) x(5) x(1)-\frac{1}{4} x(2)^2 x(3)-\frac{1}{4} x(2)^2 x(5)+x(3) x(4) x(5)$$

poly2 = -1/4*(x[1]^2*x[2]) - (x[1]^2*x[3])/4 + 
   x[2]*x[3]*x[4] + (x[1]*x[3]*x[5])/2 - (x[3]*x[5]^2)/4 - (x[4]*x[5]^2)/4

$$-\frac{1}{4} x(2) x(1)^2-\frac{1}{4} x(3) x(1)^2+\frac{1}{2} x(3) x(5) x(1)-\frac{1}{4} x(3) x(5)^2-\frac{1}{4} x(4) x(5)^2+x(2) x(3) x(4)$$

These polynomials are not identical

poly1 === poly2

$$\text{False}$$

However, one can easily recognize that they are actually the same upon renaming Feynman parameters x[i] in a suitable way. FCLoopPakOrder can do such renamings automatically

canoPoly1 = FCLoopPakOrder[poly1, x, Rename -> True] 
 
canoPoly2 = FCLoopPakOrder[poly2, x, Rename -> True]

$$-\frac{1}{4} x(3) x(1)^2-\frac{1}{4} x(5) x(1)^2+\frac{1}{2} x(2) x(3) x(1)-\frac{1}{4} x(2)^2 x(3)-\frac{1}{4} x(2)^2 x(4)+x(3) x(4) x(5)$$

$$-\frac{1}{4} x(3) x(1)^2-\frac{1}{4} x(5) x(1)^2+\frac{1}{2} x(2) x(3) x(1)-\frac{1}{4} x(2)^2 x(3)-\frac{1}{4} x(2)^2 x(4)+x(3) x(4) x(5)$$

When comparing the canonicalized versions of both polynomials we see that they are indeed identical

canoPoly1 === canoPoly2

$$\text{True}$$