7.2. DPD angles#

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from polarimetry.amplitude.angles import (
    formulate_scattering_angle,
    formulate_theta_hat_angle,
    formulate_zeta_angle,
)
from polarimetry.io import display_latex

Equation (A1) from [2]:

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angles = [
    formulate_scattering_angle(1, 2),
    formulate_scattering_angle(2, 3),
    formulate_scattering_angle(3, 1),
]
display_latex(dict(angles))
\[\begin{split}\displaystyle \begin{array}{rcl} \theta_{12} &=& \operatorname{acos}{\left(\frac{2 \sigma_{3} \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right) - \left(m_{0}^{2} - m_{3}^{2} - \sigma_{3}\right) \left(m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{3}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{1}^{2}, m_{2}^{2}\right)}} \right)} \\ \theta_{23} &=& \operatorname{acos}{\left(\frac{2 \sigma_{1} \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) - \left(m_{0}^{2} - m_{1}^{2} - \sigma_{1}\right) \left(m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{1}, m_{2}^{2}, m_{3}^{2}\right)}} \right)} \\ \theta_{31} &=& \operatorname{acos}{\left(\frac{2 \sigma_{2} \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right) - \left(m_{0}^{2} - m_{2}^{2} - \sigma_{2}\right) \left(- m_{1}^{2} + m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\ \end{array}\end{split}\]

Equation (A2):

for i in [1, 2, 3]:
    _, θii = formulate_theta_hat_angle(i, i)
    assert θii == 0

Equation (A3):

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angles = [
    formulate_theta_hat_angle(3, 1),
    formulate_theta_hat_angle(1, 2),
    formulate_theta_hat_angle(2, 3),
]
display_latex(dict(angles))
\[\begin{split}\displaystyle \begin{array}{rcl} \hat\theta_{3(1)} &=& \operatorname{acos}{\left(\frac{- 2 m_{0}^{2} \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(m_{0}^{2} + m_{3}^{2} - \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(m_{0}^{2}, \sigma_{3}, m_{3}^{2}\right)}} \right)} \\ \hat\theta_{1(2)} &=& \operatorname{acos}{\left(\frac{- 2 m_{0}^{2} \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(m_{0}^{2} + m_{2}^{2} - \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(m_{0}^{2}, \sigma_{1}, m_{1}^{2}\right)}} \right)} \\ \hat\theta_{2(3)} &=& \operatorname{acos}{\left(\frac{- 2 m_{0}^{2} \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right) + \left(m_{0}^{2} + m_{2}^{2} - \sigma_{2}\right) \left(m_{0}^{2} + m_{3}^{2} - \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{3}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(m_{0}^{2}, \sigma_{2}, m_{2}^{2}\right)}} \right)} \\ \end{array}\end{split}\]

Equations (A4-5):

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θ31hat = formulate_theta_hat_angle(3, 1)[1]
θ13hat = formulate_theta_hat_angle(1, 3)[1]
θ12hat = formulate_theta_hat_angle(1, 2)[1]
θ21hat = formulate_theta_hat_angle(2, 1)[1]
θ23hat = formulate_theta_hat_angle(2, 3)[1]
θ32hat = formulate_theta_hat_angle(3, 2)[1]

assert θ31hat == -θ13hat
assert θ12hat == -θ21hat
assert θ23hat == -θ32hat

Equations (A6):

for i in [1, 2, 3]:
    for k in [1, 2, 3]:
        _, ζi_k0 = formulate_zeta_angle(i, k, 0)
        _, ζi_ki = formulate_zeta_angle(i, k, i)
        _, ζi_kk = formulate_zeta_angle(i, k, k)
        assert ζi_ki == ζi_k0
        assert ζi_kk == 0

Equations (A7):

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angles = [
    formulate_zeta_angle(1, 1, 3),
    formulate_zeta_angle(1, 2, 1),
    formulate_zeta_angle(2, 2, 1),
    formulate_zeta_angle(2, 3, 2),
    formulate_zeta_angle(3, 3, 2),
    formulate_zeta_angle(3, 1, 3),
]
display_latex(dict(angles))
\[\begin{split}\displaystyle \begin{array}{rcl} \zeta^1_{1(3)} &=& \operatorname{acos}{\left(\frac{2 m_{1}^{2} \left(- m_{0}^{2} - m_{2}^{2} + \sigma_{2}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{1}^{2}, m_{2}^{2}\right)}} \right)} \\ \zeta^1_{2(1)} &=& \operatorname{acos}{\left(\frac{2 m_{1}^{2} \left(- m_{0}^{2} - m_{3}^{2} + \sigma_{3}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{1}^{2}, m_{3}^{2}\right)}} \right)} \\ \zeta^2_{2(1)} &=& \operatorname{acos}{\left(\frac{2 m_{2}^{2} \left(- m_{0}^{2} - m_{3}^{2} + \sigma_{3}\right) + \left(m_{0}^{2} + m_{2}^{2} - \sigma_{2}\right) \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{1}, m_{2}^{2}, m_{3}^{2}\right)}} \right)} \\ \zeta^2_{3(2)} &=& \operatorname{acos}{\left(\frac{2 m_{2}^{2} \left(- m_{0}^{2} - m_{1}^{2} + \sigma_{1}\right) + \left(m_{0}^{2} + m_{2}^{2} - \sigma_{2}\right) \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{2}^{2}, m_{1}^{2}\right)}} \right)} \\ \zeta^3_{3(2)} &=& \operatorname{acos}{\left(\frac{2 m_{3}^{2} \left(- m_{0}^{2} - m_{1}^{2} + \sigma_{1}\right) + \left(m_{0}^{2} + m_{3}^{2} - \sigma_{3}\right) \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{3}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\ \zeta^3_{1(3)} &=& \operatorname{acos}{\left(\frac{2 m_{3}^{2} \left(- m_{0}^{2} - m_{2}^{2} + \sigma_{2}\right) + \left(m_{0}^{2} + m_{3}^{2} - \sigma_{3}\right) \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{3}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(\sigma_{1}, m_{3}^{2}, m_{2}^{2}\right)}} \right)} \\ \end{array}\end{split}\]

Equations (A8):

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ζ1_12 = formulate_zeta_angle(1, 1, 2)[1]
ζ1_21 = formulate_zeta_angle(1, 2, 1)[1]
ζ2_23 = formulate_zeta_angle(2, 2, 3)[1]
ζ2_32 = formulate_zeta_angle(2, 3, 2)[1]
ζ3_31 = formulate_zeta_angle(3, 3, 1)[1]
ζ3_13 = formulate_zeta_angle(3, 1, 3)[1]

assert ζ1_12 == -ζ1_21
assert ζ2_23 == -ζ2_32
assert ζ3_31 == -ζ3_13

Equations (A10):

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angles = [
    formulate_zeta_angle(1, 2, 3),
    formulate_zeta_angle(2, 3, 1),
    formulate_zeta_angle(3, 1, 2),
]
display_latex(dict(angles))
\[\begin{split}\displaystyle \begin{array}{rcl} \zeta^1_{2(3)} &=& \operatorname{acos}{\left(\frac{2 m_{1}^{2} \left(m_{2}^{2} + m_{3}^{2} - \sigma_{1}\right) + \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{1}^{2}, m_{2}^{2}\right)}} \right)} \\ \zeta^2_{3(1)} &=& \operatorname{acos}{\left(\frac{2 m_{2}^{2} \left(m_{1}^{2} + m_{3}^{2} - \sigma_{2}\right) + \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right)}{\sqrt{\lambda\left(\sigma_{1}, m_{2}^{2}, m_{3}^{2}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{1}^{2}, m_{2}^{2}\right)}} \right)} \\ \zeta^3_{1(2)} &=& \operatorname{acos}{\left(\frac{2 m_{3}^{2} \left(m_{1}^{2} + m_{2}^{2} - \sigma_{3}\right) + \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right) \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right)}{\sqrt{\lambda\left(\sigma_{1}, m_{2}^{2}, m_{3}^{2}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\ \end{array}\end{split}\]

Equations (A11):

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ζ1_32 = formulate_zeta_angle(1, 3, 2)[1]
ζ1_23 = formulate_zeta_angle(1, 2, 3)[1]
ζ2_13 = formulate_zeta_angle(2, 1, 3)[1]
ζ2_31 = formulate_zeta_angle(2, 3, 1)[1]
ζ3_21 = formulate_zeta_angle(3, 2, 1)[1]
ζ3_12 = formulate_zeta_angle(3, 1, 2)[1]

assert ζ1_32 == -ζ1_23
assert ζ2_13 == -ζ2_31
assert ζ3_21 == -ζ3_12