2. Getting Started

pySecDec is a set of tools for analysing and numerically computing dimensionally regulated parameter integrals. The standard pySecDec procedure for computing an integral or amplitude consists of three steps:

  1. Write a generate_*.py python file which defines the integral or amplitude and its parameters, running it will generate a C/C++ library,

  2. Compile the C/C++ library,

  3. Write an integrate_*.py python file which chooses the integrator and accuracy goal, running it will perform the numerical integration and return the result.

Currently, pySecDec offers two different integration interfaces:

  1. disteval: a distributed evaluation interface which provides access to the fastest integrator,

  2. intlib: the legacy integration interface, which is more flexible and is maintained for backward compatability.

We recommend the use of the disteval interface, which we describe below.

The best way to use and learn pySecDec is to start from the existing examples. 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:

$ python3 generate_easy.py
$ make -C easy disteval
$ python3 integrate_easy_disteval.py

This will evaluate and print the result of the integral:

Numerical Result: [
    (
        +eps^-1*(+1.0000000000000000e+00+0.0000000000000000e+00j)
        +eps^-1*(+1.2871309125036819e-16+0.0000000000000000e+00j)*plusminus
        +eps^0*(+3.0685281944005316e-01+0.0000000000000000e+00j)
        +eps^0*(+5.1476960702358603e-15+0.0000000000000000e+00j)*plusminus
    )
]
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. The file integrate_easy_disteval.py loads 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/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.

#!/usr/bin/env python3

from pySecDec import make_package

if __name__ == "__main__":

    make_package(
        name = 'easy',
        integration_variables = ['x','y'],
        regulators = ['eps'],
        requested_orders = [0],
        polynomials_to_decompose = ['(x+y)^(-2+eps)']
    )

Once the C/C++ code has been written and built we run integrate_easy_disteval.py. Here the library is loaded using DistevalLibrary. Calling the instance of DistevalLibrary with easy() numerically evaluates the integral and returns the result.

#!/usr/bin/env python3

from pySecDec.integral_interface import DistevalLibrary
from math import log

# load c++ library
easy = DistevalLibrary('easy/disteval/easy.json')

# integrate
result = easy(epsrel=1e-5)

# 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.

Diagrammatic representation of `box1L`

Fig. 2.1 Diagrammatic representation of box1L

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

$ python3 generate_box1L.py
$ make -C box1L disteval
$ python3 integrate_box1L_disteval.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: (-0.14285714285714285+9.18304274527515e-18j) +/- ( (3.4540581316548235e-17+1.4274508808126073e-17j) )
eps^-1: (0.6384337089386416+3.742173166122177e-12j) +/- ( (5.434117151282641e-11+4.5162916148108245e-11j) )
eps^0 : (-0.42635395638872+1.8665025934170225j) +/- ( (6.222832697532139e-07+6.344051364916319e-07j) )

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. The file integrate_box1L_disteval.py loads and evalutes 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 line reads

import pySecDec as psd

This line specifies that the module pySecDec should be imported with the alias psd.

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

if __name__ == "__main__":

    li = psd.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]]],
        # List the names of the external momenta and the labels
        # of the vertecies they are attached to.
        external_lines = [['p1',1], ['p2',2], ['p3',3], ['p4',4]],
        # Define the kinematics and the names of the kinematic
        # invariants.
        replacement_rules = [
            ('p4', '-p1-p2-p3'),
            ('p1*p1', 's1'),
            ('p2*p2', 0),
            ('p3*p3', 0),
            ('p1*p2', 's/2-s1/2'),
            ('p1*p3', '-s/2-t/2'),
            ('p2*p3', 't/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, see e.g. the example elliptic2L_euclidean.

The symbols for the kinematic invariants and the masses also need to be given as an ordered list. The ordering is important when using the intlib interface as the values assigned to these list elements must match the ordering of the values passed in real_parameters at the numerical evaluation stage.

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_orders specifies the order in the regulator to which the integral should be expanded. For a complete list of possible options see loop_package.

psd.loop_package(
    name = 'box1L',
    loop_integral = li,
    real_parameters = Mandelstam_symbols + mass_symbols,
    requested_orders = [0],
    decomposition_method = "geometric"
)

Here we have specified the decomposition_method parameter; it selects one of the sector decomposition algoirhtms available in pySecDec: use "geometric" for the geometric decomposition method described in [BHJ+15] (this is the default since version 1.6) , "geometric_ku" for the method of [KU10], and "iterative" for the method of [Hei08]. See Sector Decomposition for more details.

2.2.2. Building the integration Library (disteval)

New in version 1.6.

After running the python script generate_box1L.py the folder box1L is created and should contain the following files and subdirectories

Makefile                box1L.hpp               box1L_integral          integrate_box1L.cpp
Makefile.conf           box1L.pdf               disteval                pylink
README                  box1L_data              integral_names.txt      src

In the folder box1L, typing

$ make disteval

will generate and build the C/C++ code for disteval. The make command can also be run in parallel by using the -j option.

Note that disteval libraries are designed with a focus on optimization for modern processors by making use of AVX2 and FMA instruction sets. For CPUs that support these, best performance is achieved by using the newest compiler available on the system (chosen via the CXX variable), and by enabling the support of AVX2 and FMA (via the CXXFLAGS variable). For example:

$ make disteval CXX="g++-12" CXXFLAGS="-mavx2 -mfma"

To build the libraries with NVidia C Compiler (NVCC) for GPU support, type

$ make disteval SECDEC_WITH_CUDA_FLAGS="-arch=sm_XX"

where sm_XX must be replaced by the target NVidia GPU architechtures; see the arch option of NVCC. The SECDEC_WITH_CUDA_FLAGS variable, which enables GPU code compilation, contains flags which are passed to NVCC during code compilation and linking. Multiple GPU architectures may be specified as described in the NVCC manual, for example SECDEC_WITH_CUDA_FLAGS="-gencode arch=compute_XX,code=sm_XX -gencode arch=compute_YY,code=sm_YY" where XX and YY are the target GPU architectures. The script examples/easy/print-cuda-arch.sh can be used to obtain the compute architecture of your current machine.

After building, the integral can be evaluated numerically using the disteval command-line interface or disteval python interface. Alternatively, a C++ library can be produced by building intlib and used via the C++ Interface.

2.2.3. Command-line interface (disteval)

New in version 1.6.

The disteval library, once built, can be used directly from the command line via the pySecDec.disteval Python module:

$ python3 -m pySecDec.disteval box1L/disteval/box1L.json s=4.0 t=-0.75 s1=1.25 msq=1.0
[...]
[
  (
    +eps^-2*(-1.4285714285714279e-01+9.0159338621354360e-18j)
    +eps^-2*(+3.4562234592930473e-17+1.4290950719747641e-17j)*plusminus
    +eps^-1*(+6.3843370937935406e-01+2.5048341561937569e-10j)
    +eps^-1*(+4.4293092326873179e-10+4.6245608965315405e-10j)*plusminus
    +eps^0*(-4.2634981062934296e-01+1.8664974523210687e+00j)
    +eps^0*(+5.5826851229628189e-06+4.8099553795389634e-06j)*plusminus
  )
]

Note that the output is a list of expressions; here a list of a single item. This is because as we shall see in Evaluating a Weighted Sum of Integrals, a single library can produce multiple resulting expressions.

The general usage of the command-line interface is:

$ python3 -m pySecDec.disteval integrand.json [options] <var>=value ...

The evaluation can be controlled via the provided command-line options:

  • --epsabs=<number>: stop if this absolute precision is reached (default: 1e-10);

  • --epsrel=<number>: stop if this relative precision is reached (default: 1e-4);

  • --timeout=<number>: stop after at most this many seconds (defaul: inf);

  • --points=<number>: use this initial Quasi-Monte-Carlo lattice size (default: 1e4);

  • --presamples=<number>: use this many points for presampling (default: 1e4);

  • --shifts=<number>: use this many lattice shifts per integral (default: 32);

  • --lattice-candidates=<number>: use the median QMC rules construction with this many lattice candidates (default: 0);

  • --coefficients=<path>: use coefficients from this directory;

  • --format=<path>: output the result in this format (sympy, mathematica, or json; default: sympy).

This list of options can also be obtained from within the command line by running:

$ python3 -m pySecDec.disteval --help

2.2.4. Python interface (disteval)

New in version 1.6.

The disteval library can be used from Python via the DistevalLibrary class. An example of this usage be found in integrate_box1L_disteval.py. The example starts by importing the necessary packages and loading the library:

from pySecDec.integral_interface import DistevalLibrary
import sympy as sp

box1L = DistevalLibrary('box1L/disteval/box1L.json', verbose=False)

Then, calling the box1L library to perform the evaluation at the given parameter values:

result = box1L(parameters={"s": 4.0, "t": -0.75, "s1": 1.25, "msq": 1.0},
            epsrel=1e-3, epsabs=1e-10, format="json")

The values of the invariants s, t, s1 and msq are passed in the dictionary parameters, these values can be changed to evaluate different kinematic points. The parameters epsrel and epsabs set the requested relative and absolute precision, respectively. The format option specifies the format of the output, the json format returns a python dictionary of results. The complete set of available options is documented in the DistevalLibrary class.

And finally, the result is parsed and pretty printed:

values = result["sums"]["box1L"]

print('Numerical Result')
print('eps^-2:', values[(-2,)][0], '+/- (', values[(-2,)][1], ')')
print('eps^-1:', values[(-1,)][0], '+/- (', values[(-1,)][1], ')')
print('eps^0 :', values[( 0,)][0], '+/- (', values[( 0,)][1], ')')

Note

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

The integral library can be called multiple times, with different kinematic points, in a loop.

2.2.5. Building the integration Library (intlib)

In the folder box1L, typing

$ make

will create the static library box1L_integral/libbox1L_integral.a and the shared library box1L_pylink.so which can be linked to external programs. The make command can also be run in parallel by using the -j option.

To build the dynamic library libbox1L.so set dynamic as build target:

$ make dynamic

To build the libraries with NVidia C Compiler (NVCC) for GPU support, type

$ make SECDEC_WITH_CUDA_FLAGS="-arch=sm_XX" CXX="nvcc"

where sm_XX must be replaced by the target NVidia GPU architechtures; see the arch option of NVCC. The SECDEC_WITH_CUDA_FLAGS variable, which enables GPU code compilation, contains flags which are passed to NVCC during code compilation and linking. Multiple GPU architectures may be specified as described in the NVCC manual, for example SECDEC_WITH_CUDA_FLAGS="-gencode arch=compute_XX,code=sm_XX -gencode arch=compute_YY,code=sm_YY" where XX and YY are the target GPU architectures. The script examples/easy/print-cuda-arch.sh can be used to obtain the compute architecture of your current machine.

To evaluate the integral numerically a program can now use one of these libraries, this can be done interactively or via a python script as explained in the section Python Interface. Alternatively, a C++ program can be produced as explained in the section C++ Interface.

2.2.6. Python Interface (intlib)

The use of the intlib Python interface is similar to the disteval python interface. To evaluate the integral for a given numerical point follow the integrate_box1L.py example. First it imports the necessary python packages and loads the C++ library.

from pySecDec.integral_interface import IntegralLibrary
import sympy as sp

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.

box1L.use_Qmc()

If the library has been compilted for GPUs (i.e. using nvcc and SECDEC_WITH_CUDA_FLAGS), as described above, the code will run on available GPUs and CPU cores.

Calling the box1L 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.

result_without_prefactor, result_prefactor, result_with_prefactor = \
    box1L(real_parameters=[4.0, -0.75, 1.25, 1.0],
          epsrel=1e-3, epsabs=1e-10, format="json")

The values of the invariants s, t, s1 and msq are passed (in the correct order) in the list real_parameters, these values can be changed to evaluate different kinematic points. The parameters epsrel and epsabs set the requested relative and absolute precision, respectively. The format option specifies the format of the output, the json format returns a python dictionary of results. The complete set of available options is documented in the IntegralLibrary class.

At this point the string result_with_prefactor contains the full result of the integral and can be manipulated as required. The strings result_without_prefactor and result_prefactor contain the value of the integral (usually equal to result_with_prefactor) and the prefactor (usually equal to 1) separately, we do not recommend their use and they exist primarily for backward compatibility.

In the integrate_box1L.py an example is shown how to parse the expression with json output format and access individual orders of the regulator.

values = result_with_prefactor["sums"]["box1L"]

print('Numerical Result')
print('eps^-2:', values[(-2,)][0], '+/- (', values[(-2,)][1], ')')
print('eps^-1:', values[(-1,)][0], '+/- (', values[(-1,)][1], ')')
print('eps^0 :', values[( 0,)][0], '+/- (', values[( 0,)][1], ')')

Note

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

The integral library can be called multiple times, with different kinematic points, in a loop. An example of how to loop over several kinematic points is shown in the example integrate_box1L_multiple_points.py.

2.2.7. C++ Interface (intlib)

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

After the lines parsing the input parameters, an secdecutil::Integrator is constructed and its parameters are set:

// Set up Integrator
secdecutil::integrators::Qmc<
                            box1L::integrand_return_t,
                                box1L::maximal_number_of_integration_variables,
                                integrators::transforms::Korobov<3>::type,
                                box1L::user_integrand_t
                            > integrator;
integrator.verbosity = 1;

The amplitude is constructed via a call to name::make_amplitudes() and packed into a name::handler_t.

// Construct the amplitudes
std::vector<box1L::nested_series_t<box1L::sum_t>> unwrapped_amplitudes =
    box1L::make_amplitudes(real_parameters, complex_parameters, "box1L_coefficients", integrator);

// Pack amplitudes into handler
box1L::handler_t<box1L::amplitudes_t> amplitudes
(
    unwrapped_amplitudes,
    integrator.epsrel, integrator.epsabs
    // further optional arguments: maxeval, mineval, maxincreasefac, min_epsrel, min_epsabs, max_epsrel, max_epsabs
);
amplitudes.verbose = true;

If desired, the contour deformation can be adjusted via additional arguments to name::handler_t.

See also

Section 4.1 and Section 5.9.1 for more detailed information about name::make_amplitudes() and name::handler_t.

To numerically integrate the sum of sectors, the name::handler_t::evaluate() function is called:

// compute the amplitudes
const std::vector<box1L::nested_series_t<secdecutil::UncorrelatedDeviation<box1L::integrand_return_t>>> result = amplitudes.evaluate();

The remaining lines print the result:

// print the result
for (unsigned int amp_idx = 0; amp_idx < box1L::number_of_amplitudes; ++amp_idx)
    std::cout << "amplitude" << amp_idx << " = " << result.at(amp_idx) << std::endl;

The C++ program can be built with the command:

$ make integrate_box1L

A kinematic point must be specified when calling the integrate_box1L executable, the input format is:

$ ./integrate_box1L 4.0 -0.75 1.25 1.0

where the arguments are the real_parameters values for (s, t, s1, msq). For integrals depending on complex_parameters, their value is specified by a space separated pair of numbers representing the real and imaginary part.

If your integral is higher than seven dimensional, changing the integral transform to integrators::transforms::Baker::type may improve the accuracy of the result. For further options of the QMC integrator we refer to Section 4.6.2.

2.3. Evaluating a Weighted Sum of Integrals

New in version 1.5.

Let us examine example easy_sum, which demonstrates how two weighted sums of dimensionally regulated integrals can be evaluated. The example computes the following two weighted sums:

& 2 s\ I_1 + 3 s\ I_2, \\ & \frac{s}{2 \epsilon}\ I_1 + \frac{s \epsilon}{3}\ I_2,

where

I_1 & = \int_0^1 \mathrm{d} x \int_0^1 \mathrm{d} y \ (x+y)^{-2+\epsilon}, \\ I_2 & = \int_0^1 \mathrm{d} x \int_0^1 \mathrm{d} y \ (2x+3y)^{-1+\epsilon}.

First, we import the necessary python packages and open the if __name__ == "__main__" guard, as required by multiprocessing.

from pySecDec import MakePackage
from pySecDec import sum_package

if __name__ == "__main__":

First, the coefficients of the integrals for each weighted sum are specified. Each coefficient is specified as a string with an arbitrary arithmetic (i.e. rational) expression. Coefficients can also be specified as instances of the Coefficient class. Coefficients can depend on the regulators, the sum_package function will automatically determine the correct orders to which the coefficients and integrals should be expanded in order to obtain the requested_orders.

coefficients = {
    "sum1": [
        '2*s',   # easy1
        '3*s'    # easy2
    ],
    "sum2": [
        's/(2*eps)', # easy1
        's*eps/3'  # easy2
    ]
}

The integrals are specified using the MakePackage wrapper function (which has the same arguments as make_package), for loop integrals the LoopPackage wrapper may be used (it has the same arguments as loop_package).

integrals = [
    MakePackage('easy1',
        integration_variables = ['x','y'],
        polynomials_to_decompose = ['(x+y)^(-2+eps)'],
        ),
    MakePackage('easy2',
        integration_variables = ['x','y'],
        polynomials_to_decompose = ['(2*x+3*y)^(-1+eps)'],
        )
]

Finally, the list of integrals and coefficients are passed to sum_package. This will generate a C++ library which efficiently evaluates both weighted sums of integrals, sharing the results of the integrals between the different sums.

sum_package('easy_sum', integrals,
    coefficients = coefficients, real_parameters=['s'],
    regulators=['eps'], requested_orders=[0])

The generated C/C++ code can be compiled and then called via the disteval CLI or disteval python interface. Alternatively, a library can be generated and called via the python or C++ interface.

2.4. Using Expansion By Regions (Generic Integral)

New in version 1.5.

The example make_regions_ebr provides a simple introduction to the expansion by regions functionality within pySecDec. For a more detailed discussion of expansion by regions see our paper [PSD21].

The necessary packages are loaded and the if __name__ == "__main__" guard is opened.

from pySecDec import sum_package, make_regions

if __name__ == "__main__":

Expansion by regions is applied to a generic integral using the make_regions function.

regions_generators = make_regions(
    name = 'make_regions_ebr',
    integration_variables = ['x'],
    regulators = ['delta'],
    requested_orders = [0],
    smallness_parameter = 't',
    polynomials_to_decompose = ['(x)**(delta)','(t + x + x**2)**(-1)'],
    expansion_by_regions_order = 0,
    real_parameters = ['t'],
    complex_parameters = [],
    decomposition_method = 'geometric_infinity_no_primary',
    polytope_from_sum_of=[1]
)

In order to obtain a result for the expanded integral, we must sum the all of the relevant regions. The output of make_regions can be passed to sum_package in order to generate a C++ library suitable for evaluating the expanded integral.

sum_package(
    'make_regions_ebr',
    regions_generators,
    regulators = ['delta'],
    requested_orders = [0],
    real_parameters = ['t']
)

The generated C++ code can be compiled and then called via the disteval CLI or disteval python interface. Alternatively, a library can be generated and called via the python or C++ interface.

2.5. Using Expansion By Regions (Loop Integral)

New in version 1.5.

The example generate_box1L_ebr demonstrates how expansion by regions can be applied to loop integrals within pySecDec by applying it to the 1-loop box integral as described in Section 4.2 of [Mis18]. For a more detailed discussion of expansion by regions see our paper [PSD21].

First, the necessary packages are loaded and the if __name__ == "__main__" guard is opened.

from pySecDec import sum_package, loop_regions
import pySecDec as psd

# This example is the one-loop box example in Go Mishima's paper arXiv:1812.04373

if __name__ == "__main__":

The loop integral can be constructed via the convenience functions in loop_integral, here we use LoopintegralFromGraph. Note that powerlist=["1+n1","1+n1/2","1+n1/3","1+n1/5"], here n1 is an extra regulator required to regulate the singularities which appear when expanding this loop integral. We use the “trick” of introducing only a single regulator divided by different prime numbers for each power, rather than unique regulators for each propagator (though this is also supported by pySecDec). Poles in the extra regulator n1 may appear in individual regions but are expected to cancel when all regions are summed.

li = psd.loop_integral.LoopIntegralFromGraph(
internal_lines = [['mt',[3,1]],['mt',[1,2]],['mt',[2,4]],['mt',[4,3]]],
external_lines = [['p1',1],['p2',2],['p3',3],['p4',4]],
powerlist=["1+n1","1+n1/2","1+n1/3","1+n1/5"],
regulators=["n1","eps"],
Feynman_parameters=["x%i" % i for i in range(1,5)], # renames the parameters to get the same polynomials as in 1812.04373
replacement_rules = [
                        # note that in those relations all momenta are incoming
                        # general relations:
                        ('p4', '-p1-p2-p3'),
                        ('p1*p1', 'm1sq'),
                        ('p2*p2', 'm2sq'),
                        ('p3*p3', 'm3sq'),
                        ('p1*p2', 's/2-(m1sq+m2sq)/2'),
                        ('p1*p3', 't/2-(m1sq+m3sq)/2'),
                        ('p2*p3', 'u/2-(m2sq+m3sq)/2'),
                        ('u', '(m1sq+m2sq+m3sq+m4sq)-s-t'),
                        # relations for our specific case:
                        ('mt**2', 'mtsq'),
                        ('m1sq',0),
                        ('m2sq',0),
                        ('m3sq','mHsq'),
                        ('m4sq','mHsq'),
                        ('mHsq', 0),
                    ])

Expansion by regions is applied to a loop integral using the loop_regions function. We expand around a small mass mtsq.

generators_args = loop_regions(
    name = "box1L_ebr",
    loop_integral=li,
    smallness_parameter = "mtsq",
    expansion_by_regions_order=0)

The output of loop_regions can be passed to sum_package in order to generate a C++ library suitable for evaluating the expanded integral.

sum_package("box1L_ebr",
            generators_args,
            li.regulators,
            requested_orders = [0,0],
            real_parameters = ['s','t','mtsq'],
            complex_parameters = [])

The generated C++ code can be compiled and then called via the disteval CLI or disteval python interface. Alternatively, a library can be generated and called via the python or C++ interface.

2.6. List of Examples

Here we list the available examples. For more details regarding each example see [PSD17], [PSD18], [PSD21] and [PSD23].

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

pentabox_fin:

a 2-loop, 5-point, 8-propagator diagram, evaluated in 6-2 \epsilon dimensions where it is finite

triangle3L:

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

formfactor4L:

a single-scale 4-loop 3-point integral in 6-2 \epsilon dimensions

bubble6L:

a single-scale 6-loop 2-point integral, evaluated at a Euclidean phase-space point

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

banana_3mass:

a 3-loop 2-point integral with three different internal masses known to contain hyperelliptic functions, evaluated at a physical phase-space point

hyperelliptic:

a 2-loop 4-point nonplanar integral known to contain hyperelliptic 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

Nbox2L_split:

three 2-loop, 4-point, 5-propagator integrals that need split=True due to special kinematics

hypergeo5F4:

a general dimensionally regulated parameter integral

hz2L_nonplanar:

a 2-loop, 4-point, 7-propagator integral with internal and external masses

box1L_ebr:

uses expansion by regions to expand a 1-loop box with a small internal mass, this integral is also considered in Section 4.2 of [Mis18], demonstrates the use of an additional regulators as described in [PSD21]

bubble1L_ebr:

uses expansion by regions to expand a 1-loop, 2-point integral in various limits,

bubble1L_dotted_ebr:

uses expansion by regions to expand a 1-loop, 2-point integral, demonstrates the t and z methods described in [PSD21]

bubble2L_largem_ebr:

uses expansion by regions to expand a 1-loop, 2-point integral with a large mass

bubble2L_smallm_ebr:

uses expansion by regions to expand a 1-loop, 2-point integral with a small mass

formfactor1L_ebr:

uses expansion by regions to compute various 1-loop, 3-point form factor integrals from the literature, demonstrates the use of add_monomial_regulator_power to introduce an additional regulator as described in [PSD21]

triangle2L_ebr:

uses expansion by regions to compute a 2-loop, 3-point integral with a large mass

make_regions_ebr:

uses expansion by regions to compute a simple generic integral with a small parameter

easy_sum:

calculates the sum of two integrals with different coefficients, demonstrates the use of sum_package

yyyy1L:

calculates a 1-loop 4-photon helicity amplitude, demonstrates the use of sum_package

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

regions:

prints a list of the regions obtained by applying expansion by regions to formfactor1L_massless

region_tools:

demonstrates the standalone usage of suggested_extra_regulator_exponent, extra_regulator_constraints and find_regions