# 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:

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 .
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 = IntegralLibrary('easy/easy_pylink.so')
# integrate
_, _, result = easy()
# 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.

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

```
$ python generate_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`

:

```
eps^-2: -0.142868356275422825 - 1.63596224151119965e-6*I +/- ( 0.00118022544307414272 + 0.000210769456586696187*I )
eps^-1: 0.639405625715768089 + 1.34277036689902802e-6*I +/- ( 0.00650722394065588166 + 0.000971496627153705891*I )
eps^0 : -0.425514350373418893 + 1.86892487760861536*I +/- ( 0.00706834403694714484 + 0.0186497890361357298*I )
```

The file `generate_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 `generate_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 generate_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
box1L = 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 `generate_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`

.
To change the accuracy settings of the integration, the most important parameters are `epsrel`

, `epsabs`

and `maxeval`

, which
can be added to the integrator argument list:

```
# choose integrator
box.use_Vegas(flags=2,epsrel=0.01, epsabs=1e-07, maxeval=1000000)
```

```
# integrate
str_integral_without_prefactor, str_prefactor, str_integral_with_prefactor = box1L(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('Numerical Result')
print('eps^-2:', integral_with_prefactor.coeff('eps',-2).coeff('value'), '+/- (', integral_with_prefactor_err.coeff('eps',-2).coeff('error'), ')')
print('eps^-1:', integral_with_prefactor.coeff('eps',-1).coeff('value'), '+/- (', integral_with_prefactor_err.coeff('eps',-1).coeff('error'), ')')
print('eps^0 :', 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 built 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` |

triangle3L: |
a 2-loop, 3-point, 7-propagator integral, demonstrates that the symmetry finder can significantly reduce the number of sectors |

elliptic2L_euclidean: |
an integral known to contain elliptic functions, evaluated at a Euclidean phase-space point |

elliptic2L_physical: |
an integral known to contain elliptic functions, evaluated at a physical phase-space point |

triangle2L_split: |
a 2-loop, 3-point, 6-propagator integral without a Euclidean region due to special kinematics |

hypergeo5F4: |
a general dimensionally regulated parameter integral |

4photon1L_amplitude: |
calculation 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 |