FeynHelpers manual (development version)

pySecDec usage examples

The main idea behind the FeynHelpers interface to pySecDec is to facilitate the generation of pySecDec scripts for integrals written in the FeynCalc notation (i.e. as GLIs with the corresponding lists of FCTopology symbols).

The main high-level function of this interface is called PSDCreatePythonScripts. In the simplest case we need two provide following arguments and options

• the 1st argument is some GLI
• the 2nd argument is the FCTopology to which this GLI belongs
• the 3rd argument is where to put the directory with pySecDec scripts. For quick tests one can simply use NotebookDirectory[]
• the option PSDRequestedOrder specifies the order in \varepsilon to which the integral should be evaluated (default is 0)
• the option PSDRealParameterRules is a list of rules for replacing kinematic invariants with numerical values which are real numbers. For complex numbers you need to use PSDComplexParameterRules
• if the script directory already exists, the function will by default refuse to overwrite it. Setting the option OverwriteTarget to True you can tell the code that you do not care about that

Here is a simple 1-loop example that incorporates all of the above

int = GLI[prop1L, {1, 1}]
topo = FCTopology[prop1L, {FAD[{p1, m1}], FAD[{p1 + q, m2}]}, {p1}, {q}, {Hold[SPD][q] -> qq}, {}]
files = PSDCreatePythonScripts[int, topo, NotebookDirectory[], 
  PSDRealParameterRules -> {qq -> 1., m1 -> 2., m2 -> 3.}, OverwriteTarget -> True]

The output is a list containing two elements which are full paths to the two pySecDec script files generate_int.py and integrate_int.py. You can now switch to the terminal, enter the corresponding directory and perform the integral evaluation by first running

python generate_int.py

Here is a sample output of this script

running "sum_package" for loopint
running "make_package" for "loopint_integral"
computing Jacobian determinant for primary sector 0
total number sectors before symmetry finding: 2
total number sectors after symmetry finding (iterative): 2
total number sectors after symmetry finding (light Pak): 2
total number sectors after symmetry finding (full Pak): 2
writing FORM files for sector 1
writing FORM files for sector 2
expanding the prefactor exp(EulerGamma*eps)*gamma(eps) (regulators: [eps] , orders: [0] )
 + (1)*eps**-1 + (0)
"loopint_integral" done

Now you need to compile the generated library files. This can be done via

make -j8 -C loopint

where 8 stands for the number threads to be run simultaneously. It depends on how powerful the CPU in your machine is.

Finally, entering

python integrate_int.py

will perform the actual numerical evaluation and save the obtained results to numres_*_psd.txt, numres_*_mma.m and numres_*_maple.mpl. Here * stands for the numerical values of kinematic invariants present in the integral. You can modify those values without the need to recompile the libraries by simply editing the arrays num_params_real and num_params_complex in integrate_int.py.

For Mathematica users the file numres_*_mma.m is probably the most useful one. You can load the content of this file into your Mathematica session using the function PSDLoadNumericalResults. To that aim you just need to give it the output of PSDCreatePythonScripts and set the options PSDRealParameterRules and PSDComplexParameterRules to the same values that were used when invoking PSDCreatePythonScripts

PSDLoadNumericalResults[files, PSDRealParameterRules -> {qq -> 1., m1 -> 2., m2 -> 3.}]