import numpy as np
[docs]
def Kallen(x, y, z):
r"""
The Källén triangle function.
Parameters
----------
x, y, z : array_like
A complex number or sequence of complex numbers.
Returns
-------
λ : array_like
The broadcasted shape from `x`, `y`, `z`.
Notes
-----
The function is defined as :cite:`Kallen:1964lxa`
.. math:: \lambda(x, y, z) = x^2 + y^2 + z^2 - 2 (xy + yz + zx) .
An alternate form can be derived for squared inputs
.. math:: \lambda(q^2, M_1^2, M_2^2) = (q^2 - (M_1 + M_2)^2)(q^2 - (M_1 - M_2)^2) .
"""
return x**2 + y**2 + z**2 - 2 * (x * y + y * z + z * x)
[docs]
def momentum_cms(s, M1, M2):
r"""
The center-of-mass momentum of a two-body system.
Parameters
----------
s : array_like
Four-momentum squared of the two-body system.
M1 : float, complex
Mass of the first particle.
M2 : float, complex
Mass of the second particle.
Returns
-------
q : array_like
The same shape as input `s`.
Notes
-----
The CMS momentum is defined as
.. math:: q(s, M_1, M_2) = \frac{1}{2}\sqrt{\frac{\lambda(s, M_1^2, M_2^2)}{s}},
where :math:`\lambda(s, M_1^2, M_2^2)` is the Källén triangle function.
Examples
-------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from dispersionrelations.kinematics import momentum_cms
>>> M1 = 0.3
>>> M2 = 0.5
>>> E_plot = np.linspace(M1+M2, 2, 500)
>>> s_plot = E_plot**2
>>> plt.plot(E_plot, momentum_cms(s_plot, M1, M2))
.. plot::
import numpy as np
import matplotlib.pyplot as plt
from dispersionrelations.kinematics import momentum_cms
M1 = 0.3
M2 = 0.5
E_plot = np.linspace(M1+M2, 2, 500)
s_plot = E_plot**2
plt.plot(E_plot, momentum_cms(s_plot, M1, M2))
See Also
--------
dispersionrelations.kinematics.Kallen
"""
return np.sqrt(Kallen(s, M1**2, M2**2) / s + 0j) / 2
[docs]
def phase_space_twobody(s, M1, M2):
r"""
Phase space function for a two-particle system.
Parameters
----------
s : array_like
Four-momentum squared of the two-body system.
M1 : float, complex
Mass of the first particle.
M2 : float, complex
Mass of the second particle.
Returns
-------
ρ : array_like
The same shape as input `s`.
Notes
-----
The phase space function is defined as
.. math:: \rho(s, M_1, M_2) = \frac{1}{16\pi}\frac{2q}{\sqrt{s}} = \frac{1}{16\pi}\frac{\sqrt{\lambda(s, M_1, M_2)}}{s},
where :math:`q` is the CMS momentum of the particles.
Examples
-------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from dispersionrelations.kinematics import phase_space_twobody
>>> M1 = 0.3
>>> M2 = 0.5
>>> E_plot = np.linspace(M1+M2, 2, 500)
>>> s_plot = E_plot**2
>>> plt.plot(E_plot, phase_space_twobody(s_plot, M1, M2))
.. plot::
import numpy as np
import matplotlib.pyplot as plt
from dispersionrelations.kinematics import phase_space_twobody
M1 = 0.3
M2 = 0.5
E_plot = np.linspace(M1+M2, 2, 500)
s_plot = E_plot**2
plt.plot(E_plot, phase_space_twobody(s_plot, M1, M2))
See Also
--------
dispersionrelations.kinematics.momentum_cms, dispersionrelations.kinematics.Kallen
"""
return 1 / (16 * np.pi) * np.sqrt(Kallen(s, M1**2, M2**2) / s**2 + 0j)
