RAMBO¶

Overview¶

In hazma, the rambo module can be used to perform various tasks. We include a generic functions for generating phase-space points called generate_phase_space_point and generate_phase_space, which will compute a single or many phase-space points. The generate_phase_space function additionally allows for the user to specify a matrix element. We include a function called integrate_over_phase_space which will perform the integral:

\[\int \left(\prod_{i=1}^{N}\dfrac{d^{3}\vec{p}_{i}}{(2\pi)^3}\dfrac{1}{2E_{i}}\right)(2\pi)^4\delta^{4}\left(P-\sum_{i=1}^{N}p_{i}\right)\left|\mathcal{M}\right|^2\]

where \(P^{\mu}\) is the total four-momentum, \(p_{i}^{\mu}\) are the individual four-momenta for each of the \(N\) final-state particles and \(\mathcal{M}\) is the matrix element. rambo.py also contains functions for computing cross-sections (decay widths) for \(2\to N\) (\(1\to N\)) processes called compute_annihilation_cross_section (compute_decay_width). For example, if we would like to compute the partial decay width of \(\mu\to e\nu\nu\), one could use the following. First, we declare a function for the matrix element. The function for the matrix element must take in a list of the four-momenta and return the matrix element:

# Import the fermi constant
from hazma.parameters import GF
# Import a helper function for scalar products of four-vectors
from hazma.field_theory_helper_functions.common_functions import \
    minkowski_dot as MDot
# Declare the matrix element
def msqrd_mu_to_enunu(momenta):
    pe = momenta[0] # electron four-momentum
    pve = momenta[1] # electron-neutrino four-momentum
    pvmu = momenta[2] # muon-neutrino four-momentum
    pmu = sum(momenta) # muon four-momentum
    # Return matrix element
    return 64. * GF**2 * MDot(pe, pvmu) * MDot(pmu, pve)

Then, the partial decay width can be computed using:

# Import function to compute decay width
from hazma.rambo import compute_decay_width
# import masses of muon and electron
from hazma.parameters import muon_mass as mmu
from hazma.parameters import electron_mass as me
# Specify the masses of the electron and neutrinos
fsp_masses = np.array([me, 0., 0.])
# compute the partial width
partial_width = compute_decay_width(fsp_masses, mmu, num_ps_pts=50000,
                                        mat_elem_sqrd=msqrd_mu_to_enunu)

Using 50000 phase-space points, we are able to within \(5\%\) of the analytical result. In addition, rambo includes a function for performing partial integrations over all variables except the energy of one of the final-state particles called generate_energy_histogram. This function returns a multi-dimensional array with the first index labeling the final-state particles and zeroth component of the second index given the energies and the 1 component of the second index giving the probability that the final-state particle has the particular energy. This function can be used via:

from hazma.rambo import generate_energy_histogram
import numpy as np
num_ps_pts = 100000 #number of phase-space points to use
# masses of final-state particles
masses = np.array([100., 100., 0.0, 0.0])
cme = 1000. # center-of-mass energy
num_bins = 100 # number of energy bins to use
# computing energy histograms
eng_hist = generate_energy_histogram(num_ps_pts, masses, cme,
                                     num_bins=num_bins)
# plot the results
import matplotlib as plt
for i in range(len(masses)):
    # pts[i, 0] are the energies of particle i
    # pts[i, 1] are the probabilities
    plt.loglog(pts[i, 0], pts[i, 1])

RAMBO¶

Overview¶

Functions¶