2. Getting Started¶

After installation, you should have a folder examples in your main pySecDec directory. Here we describe a few of the examples available in the examples directory. A full list of examples is given in List of Examples.

2.1. A Simple Example¶

We first show how to compute a simple dimensionally regulated integral:

$\int_0^1 \mathrm{d} x \int_0^1 \mathrm{d} y \ (x+y)^{-2+\epsilon}.$

To run the example change to the easy directory and run the commands:

$ python generate_easy.py
$ make -C easy
$ python integrate_easy.py

This will evaluate and print the result of the integral:

Numerical Result: + (1.00015897181235158e+00 +/- 4.03392522752491021e-03)*eps^-1 + (3.06903035514056399e-01 +/- 2.82319349818329918e-03) + O(eps)
Analytic Result: + (1.000000)*eps^-1 + (0.306853) + O(eps)

The file generate_easy.py defines the integral and calls pySecDec to perform the sector decomposition. When run it produces the directory easy which contains the code required to numerically evaluate the integral. The make command builds this code and produces a library. The file integrate_easy.py loads the integral library and evaluates the integral. The user is encouraged to copy and adapt these files to evaluate their own integrals.

Note

If the user is interested in evaluating a loop integral there are many convenience functions that make this much easier. Please see Evaluating a Loop Integral for more details.

In generate_easy.py we first import make_package, a function which can decompose, subtract and expand regulated integrals and write a C++ package to evaluate them. To define our integral we give it a name which will be used as the name of the output directory and C++ namespace. The integration_variables are declared along with a list of the name of the regulators. We must specify a list of the requested_orders to which pySecDec should expand our integral in each regulator. Here we specify requested_orders = [0] which instructs make_package to expand the integral up to and including $\mathcal{O}(\epsilon)$ . Next, we declare the polynomials_to_decompose, here sympy syntax should be used.

from pySecDec import make_package

make_package(

name = 'easy',
integration_variables = ['x','y'],
regulators = ['eps'],

requested_orders = [0],
polynomials_to_decompose = ['(x+y)^(-2+eps)'],

)

Once the C++ library has been written and built we run integrate_easy.py. Here the library is loaded using IntegralLibrary. Calling the instance of IntegralLibrary with easy_integral() numerically evaluates the integral and returns the result.

from pySecDec.integral_interface import IntegralLibrary
from math import log

# load c++ library
easy_integral = IntegralLibrary('easy/easy_pylink.so')

# integrate
_, _, result = easy_integral()

# print result
print('Numerical Result:' + result)
print('Analytic Result:' + ' + (%f)*eps^-1 + (%f) + O(eps)' % (1.0,1.0-log(2.0)))

2.2. Evaluating a Loop Integral¶

A simple example of the evaluation of a loop integral with pySecDec is box1L. This example computes a one-loop box with one off-shell leg (with off-shellness s1) and one internal massive line (with mass squared msq), it is shown in Fig. 2.1.

Fig. 2.1 Diagrammatic representation of box1L

To run the example change to the box1L directory and run the commands:

$ python box1L.py
$ make -C box1L
$ python integrate_box1L.py

This will print the result of the integral evaluated with Mandelstam invariants s=4.0, t=-0.75 and s1=1.25, msq=1.0:

leading pole: -0.142868356275422825 - 1.63596224151119965e-6*I +/- ( 0.00118022544307414272 + 0.000210769456586696187*I )
subleading pole: 0.639405625715768089 + 1.34277036689902802e-6*I +/- ( 0.00650722394065588166 + 0.000971496627153705891*I )
finite part: -0.425514350373418893 + 1.86892487760861536*I +/- ( 0.00706834403694714484 + 0.0186497890361357298*I )

The file box1L.py defines the loop integral and calls pySecDec to perform the sector decomposition. When run it produces the directory box1L which contains the code required to numerically evaluate the integral. The make command builds this code and produces a library. The file integrate_box1L.py loads the integral library and evaluates the integral for a specified numerical point.

The content of the python files is described in detail in the following sections. The user is encouraged to copy and adapt these files to evaluate their own loop integrals.

2.2.1. Defining a Loop Integral¶

To explain the input format, let us look at box1L.py from the one-loop box example. The first two lines read

import pySecDec as psd
from pySecDec.loop_integral import loop_package

They say that the module pySecDec should be imported with the alias psd, and that the function loop_package from the module loop_integral is needed.

The following part contains the definition of the loop integral li:

li = psd.loop_integral.LoopIntegralFromGraph(
# give adjacency list and indicate whether the propagator connecting the numbered vertices is massive or massless in the first entry of each list item.
internal_lines = [['m',[1,2]],[0,[2,3]],[0,[3,4]],[0,[4,1]]],
# contains the names of the external momenta and the label of the vertex they are attached to
external_lines = [['p1',1],['p2',2],['p3',3],['p4',4]],

# define the kinematics and the names for the kinematic invariants
replacement_rules = [
                        ('p1*p1', 's1'),
                        ('p2*p2', 0),
                        ('p3*p3', 0),
                        ('p4*p4', 0),
                        ('p3*p2', 't/2'),
                        ('p1*p2', 's/2-s1/2'),
                        ('p1*p4', 't/2-s1/2'),
                        ('p2*p4', 's1/2-t/2-s/2'),
                        ('p3*p4', 's/2'),
                        ('m**2', 'msq')
                   ]
)

Here the class LoopIntegralFromGraph is used to Feynman parametrize the loop integral given the adjacency list. Alternatively, the class LoopIntegralFromPropagators can be used to construct the Feynman integral given the momentum representation.

The symbols for the kinematic invariants and the masses also need to be given as an ordered list. The ordering is important as the numerical values assigned to these list elements at the numerical evaluation stage should have the same order.

Mandelstam_symbols = ['s','t','s1']
mass_symbols = ['msq']

Next, the function loop_package is called. It will create a folder called box1L. It performs the algebraic sector decomposition steps and writes a package containing the C++ code for the numerical evaluation. The argument requested_order specifies the order in the regulator to which the integral should be expanded. For a complete list of possible options see loop_package.

loop_package(

name = 'box1L',

loop_integral = li,

real_parameters = Mandelstam_symbols + mass_symbols,

# the highest order of the final epsilon expansion --> change this value to whatever you think is appropriate
requested_order = 0,

# the optimization level to use in FORM (can be 0, 1, 2, 3)
form_optimization_level = 2,

# the WorkSpace parameter for FORM
form_work_space = '100M',

# the method to be used for the sector decomposition
# valid values are ``iterative`` or ``geometric`` or ``geometric_ku``
decomposition_method = 'iterative',
# if you choose ``geometric[_ku]`` and 'normaliz' is not in your
# $PATH, you can set the path to the 'normaliz' command-line
# executable here
#normaliz_executable='/path/to/normaliz',

)

2.2.2. Building the C++ Library¶

After running the python script box1L.py the folder box1L is created and should contain the following files and subdirectories

Makefile    Makefile.conf    README    box1L.hpp    codegen    integrate_box1L.cpp    pylink    src

in the folder box1L, typing

$ make

will create the libraries libbox1L.a and box1L_pylink.so which can be linked to an external program calling these integrals. The make command can also be run in parallel by using the -j option.

To evaluate the integral numerically a program can call one of these libraries. How to do this interactively or via a python script is explained in the section Python Interface. Alternatively, a C++ program can be produced as explained in the section C++ Interface.

2.2.3. Python Interface (basic)¶

To evaluate the integral for a given numerical point we can use integrate_box1L.py. First it imports the necessary python packages and loads the C++ library.

from __future__ import print_function
from pySecDec.integral_interface import IntegralLibrary
import sympy as sp

# load c++ library
box = IntegralLibrary('box1L/box1L_pylink.so')

Next, an integrator is configured for the numerical integration. The full list of available integrators and their options is given in integral_interface.

# choose integrator
box.use_Vegas(flags=2) # ``flags=2``: verbose --> see Cuba manual

Calling the box library numerically evaluates the integral. Note that the order of the real parameters must match that specified in box1L.py. A list of possible settings for the library, in particular details of how to set the contour deformation parameters, is given in IntegralLibrary.

# integrate
str_integral_without_prefactor, str_prefactor, str_integral_with_prefactor = box(real_parameters=[4.0, -0.75, 1.25, 1.0])

At this point the string str_integral_with_prefactor contains the full result of the integral and can be manipulated as required. In the integrate_box1L.py an example is shown how to parse the expression with sympy and access individual orders of the regulator.

Note

Instead of parsing the result, it can simply be printed with the line print(str_integral_with_prefactor).

# convert complex numbers from c++ to sympy notation
str_integral_with_prefactor = str_integral_with_prefactor.replace(',','+I*')
str_prefactor = str_prefactor.replace(',','+I*')
str_integral_without_prefactor = str_integral_without_prefactor.replace(',','+I*')

# convert result to sympy expressions
integral_with_prefactor = sp.sympify(str_integral_with_prefactor.replace('+/-','*value+error*'))
integral_with_prefactor_err = sp.sympify(str_integral_with_prefactor.replace('+/-','*value+error*'))
prefactor = sp.sympify(str_prefactor)
integral_without_prefactor = sp.sympify(str_integral_without_prefactor.replace('+/-','*value+error*'))
integral_without_prefactor_err = sp.sympify(str_integral_without_prefactor.replace('+/-','*value+error*'))

# examples how to access individual orders
print('leading pole:', integral_with_prefactor.coeff('eps',-2).coeff('value'), '+/- (', integral_with_prefactor_err.coeff('eps',-2).coeff('error'), ')')
print('subleading pole:', integral_with_prefactor.coeff('eps',-1).coeff('value'), '+/- (', integral_with_prefactor_err.coeff('eps',-1).coeff('error'), ')')
print('finite part:', integral_with_prefactor.coeff('eps',0).coeff('value'), '+/- (', integral_with_prefactor_err.coeff('eps',0).coeff('error'), ')')

An example of how to loop over several kinematic points is shown in the example multiple_kinematic_points.py.

2.2.4. C++ Interface (advanced)¶

Usually it is easier to obtain a numerical result using the Python Interface. However, the library can also be used directly from C++. Inside the generated box1L folder the file integrate_box1L.cpp demonstrates this.

The function print_integral_info shows how to access the important variables of the integral library.

In the main function a kinematic point must be specified by setting the real_parameters variable, for example:

int main()
{
//  User Specified Phase-space point
    const std::vector<box1L::real_t> real_parameters = {4.0, -0.75, 1.25, 1.0}; // EDIT: kinematic point specified here
    const std::vector<box1L::complex_t> complex_parameters = {  };

The name::make_integrands() function returns an secdecutil::IntegrandContainer for each sector and regulator order:

//  Generate the integrands (optimization of the contour if applicable)
    const std::vector<box1L::nested_series_t<box1L::integrand_t>> sector_integrands = box1L::make_integrands(real_parameters, complex_parameters);

The sectors can be added before integration:

//  Add integrands of sectors (together flag)
    const box1L::nested_series_t<box1L::integrand_t> all_sectors = std::accumulate(++sector_integrands.begin(), sector_integrands.end(), *sector_integrands.begin() );

An secdecutil::Integrator is constructed and its parameters are set:

//  Integrate
    secdecutil::cuba::Vegas<box1L::integrand_return_t> integrator;
    integrator.flags = 2; // verbose output --> see cuba manual

To numerically integrate the functions the secdecutil::Integrator::integrate() function is applied to each secdecutil::IntegrandContainer using secdecutil::deep_apply():

const box1L::nested_series_t<secdecutil::UncorrelatedDeviation<box1L::integrand_return_t>> result_all = secdecutil::deep_apply( all_sectors, integrator.integrate );

The remaining lines print the result:

    std::cout << "------------" << std::endl << std::endl;

    std::cout << "-- integral info -- " << std::endl;
    print_integral_info();
    std::cout << std::endl;

    std::cout << "-- integral without prefactor -- " << std::endl;
    std::cout << result_all << std::endl << std::endl;

    std::cout << "-- prefactor -- " << std::endl;
    const box1L::nested_series_t<box1L::integrand_return_t> prefactor = box1L::prefactor(real_parameters, complex_parameters);
    std::cout << prefactor << std::endl << std::endl;

    std::cout << "-- full result (prefactor*integral) -- " << std::endl;
    std::cout << prefactor*result_all << std::endl;
    return 0;
}

After editing the real_parameters as described above the C++ program can be build and executed with the commands

$ make integrate_box1L
$ ./integrate_box1L

2.3. List of Examples¶

Here we list the available examples. For more details regarding each example see [PSD17].

easy:	a simple parametric integral, described in Section 2.1
box1L:	a simple 1-loop, 4-point, 4-propagator integral, described in Section 2.2
triangle2L:	a 2-loop, 3-point, 6-propagator diagram, also known as P126
box2L_numerator:	a massless planar on-shell 2-loop, 4-point, 7-propagator box with a numerator, either defined as an inverse propagator `box2L_invprop.py` or in terms of contracted Lorentz vectors `box2L_contracted_tensor.py`
formfactor3L:	a 2-loop, 3-point, 7-propagator integral, demonstrates that the symmetry finder can significantly reduce the number of sectors
elliptic2L:	an integral known to contain elliptic functions
Zbb_vertex_correction:	a 2-loop, 3-point, 6-propagator integral without a Euclidean region due to special kinematics
Hypergeo5F4:	a general dimensionally regulated parameter integral
4photon1L:	calcuation of the 4-photon amplitude, showing how to use pySecDec as an integral library in a larger context
two_regulators:	an integral involving poles in two different regulators.
userdefined_cpp:	a collection of examples demonstrating how to combine polynomials to be decomposed with other user-defined functions

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