Lecture 12 – Phase space simulation

How to perform a phase space simulation with Python?

Prepare the notebook by importing numpy, matplotlib.pyplot, and pylorentz and download the data file (CSV format) from Google Drive.

Import Python libraries

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import matplotlib.pyplot as plt
import numpy as np
import phasespace
import tensorflow as tf
from pylorentz import Momentum4

Two-body decay

Let’s start with a simple two body decay at rest: \(B^0\rightarrow K^+\pi^-\).

B0_MASS = 5279.65
PION_MASS = 139.57018
KAON_MASS = 493.677
n_events = 100_000

decay = phasespace.nbody_decay(B0_MASS, [PION_MASS, KAON_MASS])
weights, four_momenta = decay.generate(n_events=n_events)

WARNING: All log messages before absl::InitializeLog() is called are written to STDERR
I0000 00:00:1771522795.176880    3133 device_compiler.h:196] Compiled cluster using XLA!  This line is logged at most once for the lifetime of the process.

The simulation produces a dictionary (four_momenta) of tf.Tensor objects. Each object can be addressed with particles['p_i'], where i is the number of the \(i\)-th generated particle.

four_momenta

{'p_0': <tf.Tensor: shape=(100000, 4), dtype=float64, numpy=
 array([[ -803.67993678,   467.06666165,  2444.06953313,  2618.58901417],
        [-1672.1817886 ,  -918.54150558,  1788.1885131 ,  2618.58901417],
        [  347.48298548,  1948.04287955,  1709.36043718,  2618.58901417],
        ...,
        [  270.40729597,  1131.51605322, -2341.81124465,  2618.58901417],
        [ 1091.26615979,   358.17813685, -2349.12221495,  2618.58901417],
        [ 1374.26290117, -1795.61596184,  1313.27582253,  2618.58901417]],
       shape=(100000, 4))>,
 'p_1': <tf.Tensor: shape=(100000, 4), dtype=float64, numpy=
 array([[  803.67993678,  -467.06666165, -2444.06953313,  2661.06098583],
        [ 1672.1817886 ,   918.54150558, -1788.1885131 ,  2661.06098583],
        [ -347.48298548, -1948.04287955, -1709.36043718,  2661.06098583],
        ...,
        [ -270.40729597, -1131.51605322,  2341.81124465,  2661.06098583],
        [-1091.26615979,  -358.17813685,  2349.12221495,  2661.06098583],
        [-1374.26290117,  1795.61596184, -1313.27582253,  2661.06098583]],
       shape=(100000, 4))>}

Each tf.Tensor can be converted to a NumPy array, which can then be converted to a pylorentz.

def to_lorentz(p: tf.Tensor) -> Momentum4:
    p = p.numpy().T
    return Momentum4(p[3], *p[:3])

pion = to_lorentz(four_momenta["p_0"])
kaon = to_lorentz(four_momenta["p_1"])

These objects can be used to do kinematic computations. Let’s first verify that the invariant mass of the kaon+pion system corresponds to the mass of the mother \(B^0\):

B0 = pion + kaon
np.testing.assert_almost_equal(B0.m.mean(), B0_MASS)

Let’s also plot the momentum components of the two daugther particles.

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fig, ax = plt.subplots(1, 4, tight_layout=True, figsize=(11, 3.5))
ax[0].hist(B0.m, bins=100, color="CornFlowerBlue", range=(5279, 5280))
ax[0].set_xlabel(R"i.m.($\pi$K) [MeV/$c^2$]")
ax[0].set_title("(pion-kaon) i.m. \n")

ax[1].hist(kaon.p_x, bins=70, color="lightcoral", hatch="//")
ax[1].hist(
    pion.p_x,
    bins=70,
    color="springgreen",
    histtype="barstacked",
    hatch="\\",
    alpha=0.5,
)
ax[1].set_xlabel("$p_x$ [MeV/$c$]")
ax[1].set_title("x mom. component \n")

ax[2].hist(kaon.p_y, bins=70, color="lightcoral", hatch="//")
ax[2].hist(
    pion.p_y,
    bins=70,
    color="springgreen",
    histtype="barstacked",
    hatch="\\",
    alpha=0.5,
)
ax[2].set_xlabel("$p_y$ [MeV/$c$]")
ax[2].set_title("y mom. component \n")

ax[3].hist(kaon.p_z, bins=70, color="lightcoral", hatch="//")
ax[3].hist(
    pion.p_z,
    bins=70,
    color="springgreen",
    histtype="barstacked",
    hatch="\\",
    alpha=0.5,
)
ax[3].set_xlabel("$p_z$ [MeV/$c$]")
ax[3].set_title("z mom. component \n")
plt.show()

But it’s monochromatic!! of course it is… it’s a decay at rest. The momentum components are uniformly distributed in the available phase space.

Three-body decay

Let’s consider now a three body decay like \(B^0\rightarrow K^+\pi^-\pi^0\) and repeat the plot of the relevant kinematic variables. We can also make Dalitz plots this time.

n_events = 50_000
PION0_MASS = 134.9766
decay = phasespace.nbody_decay(B0_MASS, [PION_MASS, PION0_MASS, KAON_MASS])
weights, four_momenta = decay.generate(n_events=n_events)

pim = to_lorentz(four_momenta["p_0"])
pi0 = to_lorentz(four_momenta["p_1"])
kaon = to_lorentz(four_momenta["p_2"])
s1 = (kaon + pim).m2
s2 = (kaon + pi0).m2
s3 = (pim + pi0).m2

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fig, ax = plt.subplots(1, 3, tight_layout=True, figsize=(12, 3.5))
f0 = ax[0].hist2d(s1, s3, bins=70, cmap="turbo", cmin=1)
fig.colorbar(f0[3], ax=ax[0])
ax[0].set_xlabel(R"i.m.$^2(\pi^-K^+)$ [(MeV/$c^2)^2]$")
ax[0].set_ylabel(R"i.m.$^2(\pi^-\pi^0)$ [(MeV/$c^2)^2]$")

f1 = ax[1].hist2d(s2, s3, bins=70, cmap="turbo", cmin=1)
fig.colorbar(f1[3], ax=ax[1])
ax[1].set_xlabel(R"i.m.$^2(\pi^0K^+)$ [(MeV/$c^2)^2]$")
ax[1].set_ylabel(R"i.m.$^2(\pi^-\pi^0)$ [(MeV/$c^2)^2]$")

f2 = ax[2].hist2d(s1, s2, bins=70, cmap="turbo", cmin=1)
fig.colorbar(f2[3], ax=ax[2])
ax[2].set_xlabel(R"i.m.$^2(\pi^-K^+)$ [(MeV/$c^2)^2]$")
ax[2].set_ylabel(R"i.m.$^2(\pi^0K^+)$ [(MeV/$c^2)^2]$")
plt.show()

Decay chain

The phasespace package allows to treat also multiple decays. Let’s consider the \(B^0\rightarrow K^{\ast 0}\gamma\) decay, followed by \(K^{\ast 0}\rightarrow \pi^-K^+\). It can be simulated using the following procedure:

from phasespace import GenParticle

B0_MASS = 5279.65
K0STAR_MASS = 895.55
PION_MASS = 139.57018
KAON_MASS = 493.677
GAMMA_MASS = 0.0

Kp = GenParticle("K+", KAON_MASS)
pim = GenParticle("pi-", PION_MASS)
Kstar = GenParticle("KStar", K0STAR_MASS).set_children(Kp, pim)
gamma = GenParticle("gamma", GAMMA_MASS)
B0 = GenParticle("B0", B0_MASS).set_children(Kstar, gamma)

weights, four_momenta = B0.generate(n_events=100_000)

four_momenta

{'KStar': <tf.Tensor: shape=(100000, 4), dtype=float64, numpy=
 array([[   94.75778934, -1393.8028554 , -2149.83128152,  2715.77793272],
        [-2360.02600944,    46.79944535,  1000.76322076,  2715.77793272],
        [  100.52124732,  -984.24522485,  2365.290002  ,  2715.77793272],
        ...,
        [-1012.95910278,  1606.48478225,  1722.37059829,  2715.77793272],
        [ 1565.82882328, -2012.13627529,  -270.05126032,  2715.77793272],
        [ -840.61741583,  1974.59350643,  1402.77682542,  2715.77793272]],
       shape=(100000, 4))>,
 'gamma': <tf.Tensor: shape=(100000, 4), dtype=float64, numpy=
 array([[  -94.75778934,  1393.8028554 ,  2149.83128152,  2563.87206728],
        [ 2360.02600944,   -46.79944535, -1000.76322076,  2563.87206728],
        [ -100.52124732,   984.24522485, -2365.290002  ,  2563.87206728],
        ...,
        [ 1012.95910278, -1606.48478225, -1722.37059829,  2563.87206728],
        [-1565.82882328,  2012.13627529,   270.05126032,  2563.87206728],
        [  840.61741583, -1974.59350643, -1402.77682542,  2563.87206728]],
       shape=(100000, 4))>,
 'K+': <tf.Tensor: shape=(100000, 4), dtype=float64, numpy=
 array([[  142.93198093, -1416.56649873, -2000.38478234,  2504.46530306],
        [-2143.14065442,    -4.570941  ,   682.10819271,  2302.6205343 ],
        [  309.55113446,  -865.42786578,  1698.34406967,  1993.20767985],
        ...,
        [-1079.12932539,  1395.72886108,  1433.02674139,  2325.91095626],
        [ 1388.10371883, -1977.5008459 ,  -304.02391684,  2484.65069016],
        [ -238.60697106,   680.53305591,   372.56884341,   950.04370964]],
       shape=(100000, 4))>,
 'pi-': <tf.Tensor: shape=(100000, 4), dtype=float64, numpy=
 array([[ -48.17419159,   22.76364333, -149.44649918,  211.31262966],
        [-216.88535503,   51.37038635,  318.65502805,  413.15739842],
        [-209.02988714, -118.81735907,  666.94593233,  722.57025287],
        ...,
        [  66.17022261,  210.75592117,  289.3438569 ,  389.86697647],
        [ 177.72510445,  -34.6354294 ,   33.97265652,  231.12724257],
        [-602.01044477, 1294.06045052, 1030.20798201, 1765.73422308]],
       shape=(100000, 4))>}

gamma = to_lorentz(four_momenta["gamma"])
pion = to_lorentz(four_momenta["pi-"])
kaon = to_lorentz(four_momenta["K+"])
Kstar = to_lorentz(four_momenta["KStar"])

Let’s build the Dalitz plots matching particle pairs. The particles measured in the final state are \(K^-,\; \pi^-\) and \(\gamma\).

s1 = (pion + kaon).m2
s2 = (gamma + kaon).m2
s3 = (gamma + pion).m2

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fig, ax = plt.subplots(1, 3, tight_layout=True, figsize=(12, 3.5))

f0 = ax[0].hist2d(s1, s2, bins=70, cmap="turbo")
fig.colorbar(f0[3], ax=ax[0])
ax[0].set_xlabel(R"i.m.$^2(\pi^-K^+)$ [(MeV/$c^2)^2]$")
ax[0].set_ylabel(R"i.m.$^2(K^+\gamma)$ [(MeV/$c^2)^2]$")

f1 = ax[1].hist2d(s2, s3, bins=70, cmap="turbo")
fig.colorbar(f1[3], ax=ax[1])
ax[1].set_xlabel(R"i.m.$^2(K^+\gamma)$ [(MeV/$c^2)^2]$")
ax[1].set_ylabel(R"i.m.$^2(\pi^-\gamma)$ [(MeV/$c^2)^2]$")

f2 = ax[2].hist2d(s1, s3, bins=70, cmap="turbo")
fig.colorbar(f2[3], ax=ax[2])
ax[2].set_xlabel(R"i.m.$^2(\pi^-K^+)$ [(MeV/$c^2)^2]$")
ax[2].set_ylabel(R"i.m.$^2(\pi^-\gamma)$ [(MeV/$c^2)^2]$")
plt.show()

Width distribution

These distributions aren’t so interesting, because the masses of each particle are one fixed value. So let’s simulate a more realistic \(K^\ast\) particle; not monochromatic, but with a width of 47 MeV.[1] The mass is extracted from a Gaussian distribution centered at the B0_MASS value and with \(\sigma = 47/2.36 \sim 20\) MeV. See more info on how to do this with the phasespace package here.

import tensorflow as tf
import tensorflow_probability as tfp

K0STAR_WIDTH = 47 / 2.36


def kstar_mass(min_mass, max_mass, n_events):
    min_mass = tf.cast(min_mass, tf.float64)
    max_mass = tf.cast(max_mass, tf.float64)
    kstar_mass_cast = tf.cast(K0STAR_MASS, dtype=tf.float64)
    tf.cast(K0STAR_WIDTH, tf.float64)
    tf.broadcast_to(kstar_mass_cast, shape=(n_events,))
    return tfp.distributions.TruncatedNormal(
        loc=K0STAR_MASS,
        scale=K0STAR_WIDTH,
        low=min_mass,
        high=max_mass,
    ).sample()

K = GenParticle("K+", KAON_MASS)
pion = GenParticle("pi-", PION_MASS)
Kstar = GenParticle("KStar", kstar_mass).set_children(K, pion)
gamma = GenParticle("gamma", GAMMA_MASS)
B0 = GenParticle("B0", B0_MASS).set_children(Kstar, gamma)
weights, four_momenta = B0.generate(n_events=100_000)

gamma = to_lorentz(four_momenta["gamma"])
pion = to_lorentz(four_momenta["pi-"])
kaon = to_lorentz(four_momenta["K+"])
Kstar = to_lorentz(four_momenta["KStar"])

Now you have all the 4-vectors to plot the invariant mass distributions for the different steps of the decay chains.

s1 = (pion + kaon).m2
s2 = (gamma + kaon).m2
s3 = (gamma + pion).m2

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fig, ax = plt.subplots(1, 3, tight_layout=True, figsize=(12, 3.5))
f0 = ax[0].hist2d(s1, s2, bins=70, cmap="rainbow")
fig.colorbar(f0[3], ax=ax[0])
ax[0].set_xlabel(R"i.m.$^2(\pi^-K^+)$ [(MeV/$c^2)^2]$")
ax[0].set_ylabel(R"i.m.$^2(K^+\gamma)$ [(MeV/$c^2)^2]$")

f1 = ax[1].hist2d(s2, s3, bins=70, cmap="rainbow")
fig.colorbar(f1[3], ax=ax[1])
ax[1].set_xlabel(R"i.m.$^2(K^+\gamma)$ [(MeV/$c^2)^2]$")
ax[1].set_ylabel(R"i.m.$^2(\pi^-\gamma)$ [(MeV/$c^2)^2]$")

f2 = ax[2].hist2d(s1, s3, bins=70, cmap="rainbow")
fig.colorbar(f2[3], ax=ax[2])
ax[2].set_xlabel(R"i.m.$^2(\pi^-K^+)$ [(MeV/$c^2)^2]$")
ax[2].set_ylabel(R"i.m.$^2(\pi^-\gamma)$ [(MeV/$c^2)^2]$")
plt.show()

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[1]

Do you remember the difference between a BW width and a Lorentzian FWHM? The widths you get out of the PDG booklet are FWHM’s.