Kinematics

import sympy as sp
from IPython.display import Math

from ampform.io import aslatex

This notebook originates from Phase space for a three-body decay.

Kinematics for a three-body decay \(0 \to 123\) can be fully described by two Mandelstam variables \(\sigma_1, \sigma_2\), because the third variable \(\sigma_3\) can be expressed in terms \(\sigma_1, \sigma_2\), the mass \(m_0\) of the initial state, and the masses \(m_1, m_2, m_3\) of the final state. As can be seen, the roles of \(\sigma_1, \sigma_2, \sigma_3\) are interchangeable.

See Eq. (1.2) in [10]

from ampform.kinematics.phasespace import compute_third_mandelstam

m0, m1, m2, m3 = sp.symbols("m:4")
s1, s2, s3 = sp.symbols("sigma1:4")
s3_expr = compute_third_mandelstam(s1, s2, m0, m1, m2, m3)

latex = aslatex({s3: s3_expr})
Math(latex)

\[\begin{split}\displaystyle \begin{array}{rcl} \sigma_{3} &=& m_{0}^{2} + m_{1}^{2} + m_{2}^{2} + m_{3}^{2} - \sigma_{1} - \sigma_{2} \\ \end{array}\end{split}\]

The phase space is defined by the closed area that satisfies the condition \(\phi(\sigma_1,\sigma_2) \leq 0\), where \(\phi\) is a Kibble function:

See §V.2 in [10]

from ampform.kinematics.phasespace import Kibble

kibble = Kibble(s1, s2, s3, m0, m1, m2, m3)

latex = aslatex({kibble: kibble.evaluate()})
Math(latex)

\[\begin{split}\displaystyle \begin{array}{rcl} \phi\left(\sigma_{1}, \sigma_{2}\right) &=& \lambda\left(\lambda\left(\sigma_{2}, m_{2}^{2}, m_{0}^{2}\right), \lambda\left(\sigma_{3}, m_{3}^{2}, m_{0}^{2}\right), \lambda\left(\sigma_{1}, m_{1}^{2}, m_{0}^{2}\right)\right) \\ \end{array}\end{split}\]

and \(\lambda\) is the Källén function:

from ampform.kinematics.phasespace import Kallen

x, y, z = sp.symbols("x:z")
kallen = Kallen(x, y, z)

latex = aslatex({kallen: kallen.evaluate()})
Math(latex)

\[\begin{split}\displaystyle \begin{array}{rcl} \lambda\left(x, y, z\right) &=& x^{2} - 2 x y - 2 x z + y^{2} - 2 y z + z^{2} \\ \end{array}\end{split}\]

Any distribution over the phase space can now be defined using a two-dimensional grid over a Mandelstam pair \(\sigma_1,\sigma_2\) of choice, with the condition \(\phi(\sigma_1,\sigma_2)<0\) selecting the values that are physically allowed.

from ampform.kinematics.phasespace import is_within_phasespace

is_within_phasespace(s1, s2, m0, m1, m2, m3)

\[\begin{split}\displaystyle \begin{cases} 1 & \text{for}\: \phi\left(\sigma_{1}, \sigma_{2}\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\end{split}\]

See Phase space for a three-body decay for an interactive visualization of the phase space region and an analytic expression for the phase space boundary.