- Version 1.2, February, 2024
- Author(s): Rafael L. Delgado
- Email: rafael.delgado@upm.es
Evaluating total cross sections from differential cross sections for 2->n processes, with n>2, can be challenging. This code is intended for solving such problem for massless initial and final state particles and n=3, 4 and 5. Furthermore, the framework is kept as simple as possible for this single task.
Part of the code is Fortran code that compiles to a Python module via F2PY. For the multidimensional numerical integration, the VEGAS algorithm is used via the class vegas.Integrator from the vegas Python module (G.P.Lepage, J.Comput.Phys.27 (1978) 192 and J.Comput.Phys.439 (2021) 110386).
The code is described on Ref. [1]. Furthermore, another upcoming article will describe it in detail.
This code requires a Fortran compiler and a Python 3 installation. The Python packages vegas, Numpy and F2PY should be available. Optionally, MPI can be used for parallelizing the Monte Carlo integration. For using this option, the mpi4py Python module is also required.
The repository includes a Makefile. The only required command is:
make
The repository contains the following folders:
- scripts: two Python scripts as an example of usage
- amplitude: Fortran source files with the amplitudes coded by the user
- src: the actual source code
The compilation process generates a Python 3 module on the repository directory. Such a module should be loaded into Python via the instruction
from amp import amp
Afterwards, the instructions
amp.only_phase_space=False
amp.indisting_final_state_particles=False
amp.only_integration=False
amp.s=1.e-0
load s=1TeV on the evaluation routines. There are three boolean varibles for special purposes:
- amp.only_phase_space: if set to True, only the phase space is integrated
- amp.indisting_final_state_particles: if set to True, the amplitudes
- will be divided by
$n!$ . This is required if all the final state - particles are indistinguishable.
- amp.only_integration: instructs MaMuPaX to directly integrate the value
- of the amplitude instead of treating it as a quantum field theory amplitude.
- The latter requires to compute the square modulus and divide by
$2s$ .
If the MPI option is used, the Python script should be called with the appropriate mpirun command,
mpirun -N 6 ./phase_space.py
where 6 should be changed to match the number of computing cores. Note that the Python script should be made executable.
The folder scripts, contains the Python script phase_space.py
that computes the phase space for the 2->n processes with n=2,3,4 and 5. This is
intended to be a test of the code as well as an usage example. It also contains
the script COMPUTE_bn.py, for computing the integration of the
The matrix elements should be coded in Fortran by the user inside the following files:
- amplitude/M_2to2.h, for a 2->2 process
- amplitude/M_2to3.h, for a 2->3 process
- amplitude/M_2to4.h, for a 2->4 process
- amplitude/M_2to5.h, for a 2->5 process
In each of these files, the code should return a value with
the matrix element on the M variable. The following variables
describing the outgoing particles 4-momenta can be used (notation: i,j=1,2,3,4,5):
-
fi :
$\lVert\vec p_i\rVert/\sqrt{s}$ , three-momentum fractions -
zi :
$1-\cos\theta_i$ , angular functions -
zij :
$1-\cos\theta_{ij}$ , where$\theta_{ij}$ is the angle between the$i$ -th and$j$ -th outgoing particle
The following variables can also be used, although were originally intended for MaMuPaXS internal use:
-
th_c(i) :
$\cos\theta_i$ , where$\theta_i$ is the azimuthal angle of the$i$ -th outgoing particle -
th_s(i) :
$\sin\theta_i$ -
phi(i) :
$\phi_i$ , polar angle ($i\leq 3$ ) -
phi_e(k):
$\phi_i$ , polar angle for$k=i-4$ , where$i$ refers to the$i$ -th outgoing particle ($i\geq 5$ ).
Note that the code assumes, without loss of generality, that
Furthermore, the user can declare parameters and temporary variables in these two files:
- amplitude/params.h: parameters that can be configured inside the Python code and accesed by the user's Fortran code that encodes the amplitude.
- amplitude/tmp_vars.h: temporary variables used by the user's Fotran code.
In this version of the program, the sections that were generated by Maple software have been re-generated with SageMath [2] and SymPy [3]. Since SageMath and SymPy are available under the GPL and BSD licenses, this allows for the distribution of this version of MaMuPaXS under a GPL license (see detailed text in the LICENSE.md file).
Please, note that the old version 1.0 of MaMuPaXS was not available under the GPL license due to part of the analytical expressions being generated by Maple software, hence making it unusable for commercial purposes according to: https://www.maplesoft.com/documentation_center/Maplesoft_EULA.pdf
We ask that if you use this code for work which results in a publication that you cite the following paper:
- [1] Rafael L. Delgado, Raquel Gómez-Ambrosio, Javier Martínez-Martín, Alexandre Salas-Bernárdez, Juan J. Sanz-Cillero, ``SMEFT vs HEFT: multi-Higgs phenomenology'', arXiv:2311.04280 [hep-ph]
For the generation of several sections of the code that compute analytical expressions we have used SageMath and the fcode function of Sympy. See our publication [1] for details and the following referentes for SageMath and Sympy:
-
[2] SageMath, the Sage Mathematics Software System (Version 10.1), The Sage Developers, 2023, https://www.sagemath.org.
-
[3] Meurer A, Smith CP, Paprocki M, Čertík O, Kirpichev SB, Rocklin M, Kumar A, Ivanov S, Moore JK, Singh S, Rathnayake T, Vig S, Granger BE, Muller RP, Bonazzi F, Gupta H, Vats S, Johansson F, Pedregosa F, Curry MJ, Terrel AR, Roučka Š, Saboo A, Fernando I, Kulal S, Cimrman R, Scopatz A. (2017) SymPy: symbolic computing in Python. PeerJ Computer Science 3:e103 https://doi.org/10.7717/peerj-cs.103