J/ψ → φ(1020) π⁺ π⁻

Contents

J/ψ → φ(1020) π⁺ π⁻#

The decay $J / ψ \to ϕ (1020) π^{+} π^{-}$ has an initial state with spin and has one vector meson, $ϕ (1020)$ , in the final state. This is a follow-up to J/ψ → π⁰ π⁺ π⁻, where there is no spin in the final state.

Decay definition#

../_images/76181cd5fb09618fd2f36c3231d91ec42ffa9f5d33b19175afa7517f85f74be2.svg

index	name	LaTeX	$J^{P}$	mass (MeV)	width (MeV)
0	`J/psi(1S)`	$J / ψ (1 S)$	$1^{-}$	3,096	0
1	`phi(1020)`	$ϕ (1020)$	$1^{-}$	1,019	4
2	`pi-`	$π^{-}$	$0^{-}$	139	0
3	`pi+`	$π^{+}$	$0^{-}$	139	0

name	LaTeX	$J^{P}$	mass (MeV)	width (MeV)
`a(0)(1450)+`	$a_{0} (1450)^{+}$	$0^{+}$	1,439	258
`a(0)(1450)0`	$a_{0} (1450)^{0}$	$0^{+}$	1,439	258
`a(0)(1450)-`	$a_{0} (1450)^{-}$	$0^{+}$	1,439	258
`rho(1450)-`	$ρ (1450)^{-}$	$1^{-}$	1,465	400
`rho(1450)0`	$ρ (1450)^{0}$	$1^{-}$	1,465	400
`rho(1450)+`	$ρ (1450)^{+}$	$1^{-}$	1,465	400

\begin{array}{r} \begin{array}{c} J / ψ (1 S) [1^{-}] \to (a_{0} (1450)^{+} [0^{+}] \to ϕ (1020) [1^{-}] π^{+} [0^{-}]) π^{-} [0^{-}] \\ J / ψ (1 S) [1^{-}] \to (a_{0} (1450)^{-} [0^{+}] \to ϕ (1020) [1^{-}] π^{-} [0^{-}]) π^{+} [0^{-}] \\ J / ψ (1 S) [1^{-}] \to (a_{0} (1450)^{0} [0^{+}] \to π^{-} [0^{-}] π^{+} [0^{-}]) ϕ (1020) [1^{-}] \\ J / ψ (1 S) [1^{-}] \to (ρ (1450)^{+} [1^{-}] \to ϕ (1020) [1^{-}] π^{+} [0^{-}]) π^{-} [0^{-}] \\ J / ψ (1 S) [1^{-}] \to (ρ (1450)^{-} [1^{-}] \to ϕ (1020) [1^{-}] π^{-} [0^{-}]) π^{+} [0^{-}] \\ J / ψ (1 S) [1^{-}] \to (ρ (1450)^{0} [1^{-}] \to π^{-} [0^{-}] π^{+} [0^{-}]) ϕ (1020) [1^{-}] \end{array} \end{array}

Model formulation#

DPD model#

\sum_{λ_{0} = - 1}^{1} \sum_{λ_{1} = - 1}^{1} {| \sum_{λ_{0}^{'} = - 1}^{1} \sum_{λ_{1}^{'} = - 1}^{1} \sum_{λ_{2}^{'} = 0} \sum_{λ_{3}^{'} = 0} A_{λ_{0}^{'}, λ_{1}^{'}, λ_{2}^{'}, λ_{3}^{'}}^{1} d_{λ_{1}^{'}, λ_{1}}^{1} (ζ_{1 (1)}^{1}) d_{λ_{0}, λ_{0}^{'}}^{1} (ζ_{1 (1)}^{0}) + A_{λ_{0}^{'}, λ_{1}^{'}, λ_{2}^{'}, λ_{3}^{'}}^{2} d_{λ_{1}^{'}, λ_{1}}^{1} (ζ_{2 (1)}^{1}) d_{λ_{0}, λ_{0}^{'}}^{1} (ζ_{2 (1)}^{0}) + A_{λ_{0}^{'}, λ_{1}^{'}, λ_{2}^{'}, λ_{3}^{'}}^{3} d_{λ_{1}^{'}, λ_{1}}^{1} (ζ_{3 (1)}^{1}) d_{λ_{0}, λ_{0}^{'}}^{1} (ζ_{3 (1)}^{0}) |}^{2}

\begin{array}{r} \begin{array}{rcl} A_{- 1, - 1, 0, 0}^{1} & = & \sum_{λ_{R} = - 1}^{1} δ_{- 1, λ_{R} + 1} H_{ρ (1450)^{0}, 0, 0}^{decay} H_{ρ (1450)^{0}, λ_{R}, - 1}^{production} d_{λ_{R}, 0}^{1} (θ_{23}) + \sum_{λ_{R} = 0} δ_{- 1, λ_{R} + 1} H_{a_{0} (1450)^{0}, 0, 0}^{decay} H_{a_{0} (1450)^{0}, λ_{R}, - 1}^{production} d_{λ_{R}, 0}^{0} (θ_{23}) \\ A_{- 1, - 1, 0, 0}^{2} & = & \sum_{λ_{R} = - 1}^{1} δ_{- 1 λ_{R}} H_{ρ (1450)^{+}, 0, - 1}^{decay} H_{ρ (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{1} (θ_{31}) + \sum_{λ_{R} = 0} δ_{- 1 λ_{R}} H_{a_{0} (1450)^{+}, 0, - 1}^{decay} H_{a_{0} (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{0} (θ_{31}) \\ A_{- 1, - 1, 0, 0}^{3} & = & \sum_{λ_{R} = - 1}^{1} δ_{- 1 λ_{R}} H_{ρ (1450)^{-}, - 1, 0}^{decay} H_{ρ (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{1} (θ_{12}) + \sum_{λ_{R} = 0} δ_{- 1 λ_{R}} H_{a_{0} (1450)^{-}, - 1, 0}^{decay} H_{a_{0} (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{0} (θ_{12}) \\ A_{- 1, 0, 0, 0}^{1} & = & \sum_{λ_{R} = - 1}^{1} - δ_{- 1 λ_{R}} H_{ρ (1450)^{0}, 0, 0}^{decay} H_{ρ (1450)^{0}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{1} (θ_{23}) + \sum_{λ_{R} = 0} - δ_{- 1 λ_{R}} H_{a_{0} (1450)^{0}, 0, 0}^{decay} H_{a_{0} (1450)^{0}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{0} (θ_{23}) \\ A_{- 1, 0, 0, 0}^{2} & = & \sum_{λ_{R} = - 1}^{1} - δ_{- 1 λ_{R}} H_{ρ (1450)^{+}, 0, 0}^{decay} H_{ρ (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{1} (θ_{31}) + \sum_{λ_{R} = 0} - δ_{- 1 λ_{R}} H_{a_{0} (1450)^{+}, 0, 0}^{decay} H_{a_{0} (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{0} (θ_{31}) \\ A_{- 1, 0, 0, 0}^{3} & = & \sum_{λ_{R} = - 1}^{1} δ_{- 1 λ_{R}} H_{ρ (1450)^{-}, 0, 0}^{decay} H_{ρ (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{1} (θ_{12}) + \sum_{λ_{R} = 0} δ_{- 1 λ_{R}} H_{a_{0} (1450)^{-}, 0, 0}^{decay} H_{a_{0} (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{0} (θ_{12}) \\ A_{- 1, 1, 0, 0}^{1} & = & \sum_{λ_{R} = - 1}^{1} δ_{- 1, λ_{R} - 1} H_{ρ (1450)^{0}, 0, 0}^{decay} H_{ρ (1450)^{0}, λ_{R}, 1}^{production} d_{λ_{R}, 0}^{1} (θ_{23}) + \sum_{λ_{R} = 0} δ_{- 1, λ_{R} - 1} H_{a_{0} (1450)^{0}, 0, 0}^{decay} H_{a_{0} (1450)^{0}, λ_{R}, 1}^{production} d_{λ_{R}, 0}^{0} (θ_{23}) \\ A_{- 1, 1, 0, 0}^{2} & = & \sum_{λ_{R} = - 1}^{1} δ_{- 1 λ_{R}} H_{ρ (1450)^{+}, 0, 1}^{decay} H_{ρ (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{1} (θ_{31}) + \sum_{λ_{R} = 0} δ_{- 1 λ_{R}} H_{a_{0} (1450)^{+}, 0, 1}^{decay} H_{a_{0} (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{0} (θ_{31}) \\ A_{- 1, 1, 0, 0}^{3} & = & \sum_{λ_{R} = - 1}^{1} δ_{- 1 λ_{R}} H_{ρ (1450)^{-}, 1, 0}^{decay} H_{ρ (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{1} (θ_{12}) + \sum_{λ_{R} = 0} δ_{- 1 λ_{R}} H_{a_{0} (1450)^{-}, 1, 0}^{decay} H_{a_{0} (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{0} (θ_{12}) \\ A_{0, - 1, 0, 0}^{1} & = & \sum_{λ_{R} = - 1}^{1} δ_{0, λ_{R} + 1} H_{ρ (1450)^{0}, 0, 0}^{decay} H_{ρ (1450)^{0}, λ_{R}, - 1}^{production} d_{λ_{R}, 0}^{1} (θ_{23}) + \sum_{λ_{R} = 0} δ_{0, λ_{R} + 1} H_{a_{0} (1450)^{0}, 0, 0}^{decay} H_{a_{0} (1450)^{0}, λ_{R}, - 1}^{production} d_{λ_{R}, 0}^{0} (θ_{23}) \\ A_{0, - 1, 0, 0}^{2} & = & \sum_{λ_{R} = - 1}^{1} δ_{0 λ_{R}} H_{ρ (1450)^{+}, 0, - 1}^{decay} H_{ρ (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{1} (θ_{31}) + \sum_{λ_{R} = 0} δ_{0 λ_{R}} H_{a_{0} (1450)^{+}, 0, - 1}^{decay} H_{a_{0} (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{0} (θ_{31}) \\ A_{0, - 1, 0, 0}^{3} & = & \sum_{λ_{R} = - 1}^{1} δ_{0 λ_{R}} H_{ρ (1450)^{-}, - 1, 0}^{decay} H_{ρ (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{1} (θ_{12}) + \sum_{λ_{R} = 0} δ_{0 λ_{R}} H_{a_{0} (1450)^{-}, - 1, 0}^{decay} H_{a_{0} (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{0} (θ_{12}) \\ A_{0, 0, 0, 0}^{1} & = & \sum_{λ_{R} = - 1}^{1} - δ_{0 λ_{R}} H_{ρ (1450)^{0}, 0, 0}^{decay} H_{ρ (1450)^{0}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{1} (θ_{23}) + \sum_{λ_{R} = 0} - δ_{0 λ_{R}} H_{a_{0} (1450)^{0}, 0, 0}^{decay} H_{a_{0} (1450)^{0}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{0} (θ_{23}) \\ A_{0, 0, 0, 0}^{2} & = & \sum_{λ_{R} = - 1}^{1} - δ_{0 λ_{R}} H_{ρ (1450)^{+}, 0, 0}^{decay} H_{ρ (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{1} (θ_{31}) + \sum_{λ_{R} = 0} - δ_{0 λ_{R}} H_{a_{0} (1450)^{+}, 0, 0}^{decay} H_{a_{0} (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{0} (θ_{31}) \\ A_{0, 0, 0, 0}^{3} & = & \sum_{λ_{R} = - 1}^{1} δ_{0 λ_{R}} H_{ρ (1450)^{-}, 0, 0}^{decay} H_{ρ (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{1} (θ_{12}) + \sum_{λ_{R} = 0} δ_{0 λ_{R}} H_{a_{0} (1450)^{-}, 0, 0}^{decay} H_{a_{0} (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{0} (θ_{12}) \\ A_{0, 1, 0, 0}^{1} & = & \sum_{λ_{R} = - 1}^{1} δ_{0, λ_{R} - 1} H_{ρ (1450)^{0}, 0, 0}^{decay} H_{ρ (1450)^{0}, λ_{R}, 1}^{production} d_{λ_{R}, 0}^{1} (θ_{23}) + \sum_{λ_{R} = 0} δ_{0, λ_{R} - 1} H_{a_{0} (1450)^{0}, 0, 0}^{decay} H_{a_{0} (1450)^{0}, λ_{R}, 1}^{production} d_{λ_{R}, 0}^{0} (θ_{23}) \\ A_{0, 1, 0, 0}^{2} & = & \sum_{λ_{R} = - 1}^{1} δ_{0 λ_{R}} H_{ρ (1450)^{+}, 0, 1}^{decay} H_{ρ (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{1} (θ_{31}) + \sum_{λ_{R} = 0} δ_{0 λ_{R}} H_{a_{0} (1450)^{+}, 0, 1}^{decay} H_{a_{0} (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{0} (θ_{31}) \\ A_{0, 1, 0, 0}^{3} & = & \sum_{λ_{R} = - 1}^{1} δ_{0 λ_{R}} H_{ρ (1450)^{-}, 1, 0}^{decay} H_{ρ (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{1} (θ_{12}) + \sum_{λ_{R} = 0} δ_{0 λ_{R}} H_{a_{0} (1450)^{-}, 1, 0}^{decay} H_{a_{0} (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{0} (θ_{12}) \\ A_{1, - 1, 0, 0}^{1} & = & \sum_{λ_{R} = - 1}^{1} δ_{1, λ_{R} + 1} H_{ρ (1450)^{0}, 0, 0}^{decay} H_{ρ (1450)^{0}, λ_{R}, - 1}^{production} d_{λ_{R}, 0}^{1} (θ_{23}) + \sum_{λ_{R} = 0} δ_{1, λ_{R} + 1} H_{a_{0} (1450)^{0}, 0, 0}^{decay} H_{a_{0} (1450)^{0}, λ_{R}, - 1}^{production} d_{λ_{R}, 0}^{0} (θ_{23}) \\ A_{1, - 1, 0, 0}^{2} & = & \sum_{λ_{R} = - 1}^{1} δ_{1 λ_{R}} H_{ρ (1450)^{+}, 0, - 1}^{decay} H_{ρ (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{1} (θ_{31}) + \sum_{λ_{R} = 0} δ_{1 λ_{R}} H_{a_{0} (1450)^{+}, 0, - 1}^{decay} H_{a_{0} (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{0} (θ_{31}) \\ A_{1, - 1, 0, 0}^{3} & = & \sum_{λ_{R} = - 1}^{1} δ_{1 λ_{R}} H_{ρ (1450)^{-}, - 1, 0}^{decay} H_{ρ (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{1} (θ_{12}) + \sum_{λ_{R} = 0} δ_{1 λ_{R}} H_{a_{0} (1450)^{-}, - 1, 0}^{decay} H_{a_{0} (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{0} (θ_{12}) \\ A_{1, 0, 0, 0}^{1} & = & \sum_{λ_{R} = - 1}^{1} - δ_{1 λ_{R}} H_{ρ (1450)^{0}, 0, 0}^{decay} H_{ρ (1450)^{0}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{1} (θ_{23}) + \sum_{λ_{R} = 0} - δ_{1 λ_{R}} H_{a_{0} (1450)^{0}, 0, 0}^{decay} H_{a_{0} (1450)^{0}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{0} (θ_{23}) \\ A_{1, 0, 0, 0}^{2} & = & \sum_{λ_{R} = - 1}^{1} - δ_{1 λ_{R}} H_{ρ (1450)^{+}, 0, 0}^{decay} H_{ρ (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{1} (θ_{31}) + \sum_{λ_{R} = 0} - δ_{1 λ_{R}} H_{a_{0} (1450)^{+}, 0, 0}^{decay} H_{a_{0} (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{0} (θ_{31}) \\ A_{1, 0, 0, 0}^{3} & = & \sum_{λ_{R} = - 1}^{1} δ_{1 λ_{R}} H_{ρ (1450)^{-}, 0, 0}^{decay} H_{ρ (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{1} (θ_{12}) + \sum_{λ_{R} = 0} δ_{1 λ_{R}} H_{a_{0} (1450)^{-}, 0, 0}^{decay} H_{a_{0} (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 0}^{0} (θ_{12}) \\ A_{1, 1, 0, 0}^{1} & = & \sum_{λ_{R} = - 1}^{1} δ_{1, λ_{R} - 1} H_{ρ (1450)^{0}, 0, 0}^{decay} H_{ρ (1450)^{0}, λ_{R}, 1}^{production} d_{λ_{R}, 0}^{1} (θ_{23}) + \sum_{λ_{R} = 0} δ_{1, λ_{R} - 1} H_{a_{0} (1450)^{0}, 0, 0}^{decay} H_{a_{0} (1450)^{0}, λ_{R}, 1}^{production} d_{λ_{R}, 0}^{0} (θ_{23}) \\ A_{1, 1, 0, 0}^{2} & = & \sum_{λ_{R} = - 1}^{1} δ_{1 λ_{R}} H_{ρ (1450)^{+}, 0, 1}^{decay} H_{ρ (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{1} (θ_{31}) + \sum_{λ_{R} = 0} δ_{1 λ_{R}} H_{a_{0} (1450)^{+}, 0, 1}^{decay} H_{a_{0} (1450)^{+}, λ_{R}, 0}^{production} d_{λ_{R}, - 1}^{0} (θ_{31}) \\ A_{1, 1, 0, 0}^{3} & = & \sum_{λ_{R} = - 1}^{1} δ_{1 λ_{R}} H_{ρ (1450)^{-}, 1, 0}^{decay} H_{ρ (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{1} (θ_{12}) + \sum_{λ_{R} = 0} δ_{1 λ_{R}} H_{a_{0} (1450)^{-}, 1, 0}^{decay} H_{a_{0} (1450)^{-}, λ_{R}, 0}^{production} d_{λ_{R}, 1}^{0} (θ_{12}) \end{array} \end{array}

There is an isobar Wigner- $d$ function, which takes the following helicity angles as argument:

\begin{array}{r} \begin{array}{rcl} θ_{23} & = & acos (\frac{2 σ_{1} (- m_{1}^{2} - m_{2}^{2} + σ_{3}) - (m_{0}^{2} - m_{1}^{2} - σ_{1}) (m_{2}^{2} - m_{3}^{2} + σ_{1})}{\sqrt{λ (m_{0}^{2}, m_{1}^{2}, σ_{1})} \sqrt{λ (σ_{1}, m_{2}^{2}, m_{3}^{2})}}) \\ θ_{31} & = & acos (\frac{2 σ_{2} (- m_{2}^{2} - m_{3}^{2} + σ_{1}) - (m_{0}^{2} - m_{2}^{2} - σ_{2}) (- m_{1}^{2} + m_{3}^{2} + σ_{2})}{\sqrt{λ (m_{0}^{2}, m_{2}^{2}, σ_{2})} \sqrt{λ (σ_{2}, m_{3}^{2}, m_{1}^{2})}}) \\ θ_{12} & = & acos (\frac{2 σ_{3} (- m_{1}^{2} - m_{3}^{2} + σ_{2}) - (m_{0}^{2} - m_{3}^{2} - σ_{3}) (m_{1}^{2} - m_{2}^{2} + σ_{3})}{\sqrt{λ (m_{0}^{2}, m_{3}^{2}, σ_{3})} \sqrt{λ (σ_{3}, m_{1}^{2}, m_{2}^{2})}}) \\ ζ_{1 (1)}^{0} & = & 0 \\ ζ_{1 (1)}^{1} & = & 0 \\ ζ_{2 (1)}^{0} & = & - acos (\frac{- 2 m_{0}^{2} (- m_{1}^{2} - m_{2}^{2} + σ_{3}) + (m_{0}^{2} + m_{1}^{2} - σ_{1}) (m_{0}^{2} + m_{2}^{2} - σ_{2})}{\sqrt{λ (m_{0}^{2}, m_{2}^{2}, σ_{2})} \sqrt{λ (m_{0}^{2}, σ_{1}, m_{1}^{2})}}) \\ ζ_{2 (1)}^{1} & = & acos (\frac{2 m_{1}^{2} (- m_{0}^{2} - m_{3}^{2} + σ_{3}) + (m_{0}^{2} + m_{1}^{2} - σ_{1}) (- m_{1}^{2} - m_{3}^{2} + σ_{2})}{\sqrt{λ (m_{0}^{2}, m_{1}^{2}, σ_{1})} \sqrt{λ (σ_{2}, m_{1}^{2}, m_{3}^{2})}}) \\ ζ_{3 (1)}^{0} & = & acos (\frac{- 2 m_{0}^{2} (- m_{1}^{2} - m_{3}^{2} + σ_{2}) + (m_{0}^{2} + m_{1}^{2} - σ_{1}) (m_{0}^{2} + m_{3}^{2} - σ_{3})}{\sqrt{λ (m_{0}^{2}, m_{1}^{2}, σ_{1})} \sqrt{λ (m_{0}^{2}, σ_{3}, m_{3}^{2})}}) \\ ζ_{3 (1)}^{1} & = & - acos (\frac{2 m_{1}^{2} (- m_{0}^{2} - m_{2}^{2} + σ_{2}) + (m_{0}^{2} + m_{1}^{2} - σ_{1}) (- m_{1}^{2} - m_{2}^{2} + σ_{3})}{\sqrt{λ (m_{0}^{2}, m_{1}^{2}, σ_{1})} \sqrt{λ (σ_{3}, m_{1}^{2}, m_{2}^{2})}}) \end{array} \end{array}

AmpForm model#

\sum_{m_{A} = - 1}^{1} \sum_{m_{0} = - 1}^{1} \sum_{m_{1} = 0} \sum_{m_{2} = 0} {| A_{m_{A}, m_{0}, m_{1}, m_{2}}^{01} + A_{m_{A}, m_{0}, m_{1}, m_{2}}^{02} + A_{m_{A}, m_{0}, m_{1}, m_{2}}^{12} |}^{2}

\begin{array}{r} \begin{array}{rcl} A_{0, 0, 0, 0}^{01} & = & - C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{0, - 1}^{1} (- ϕ_{01}, θ_{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{0, 1}^{1} (- ϕ_{01}, θ_{01}, 0) D_{1, 0}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{0}^{-}; ρ (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{0, 0}^{1} (- ϕ_{01}, θ_{01}, 0) D_{0, 0}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{-}}_{0} π_{0}^{+}; a_{0} (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{0, 0}^{1} (- ϕ_{01}, θ_{01}, 0) \\ A_{0, - 1, 0, 0}^{01} & = & C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{- 1, - 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{0, - 1}^{1} (- ϕ_{01}, θ_{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{0, 1}^{1} (- ϕ_{01}, θ_{01}, 0) D_{1, - 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) - C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{0}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{0, - 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{0, 0}^{1} (- ϕ_{01}, θ_{01}, 0) \\ A_{0, 1, 0, 0}^{01} & = & C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{- 1, 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{0, - 1}^{1} (- ϕ_{01}, θ_{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{0, 1}^{1} (- ϕ_{01}, θ_{01}, 0) D_{1, 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{0}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{0, 0}^{1} (- ϕ_{01}, θ_{01}, 0) D_{0, 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) \\ A_{- 1, 0, 0, 0}^{01} & = & - C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{- 1, - 1}^{1} (- ϕ_{01}, θ_{01}, 0) D_{- 1, 0}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{- 1, 1}^{1} (- ϕ_{01}, θ_{01}, 0) D_{1, 0}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{0}^{-}; ρ (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{01}, θ_{01}, 0) D_{0, 0}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{-}}_{0} π_{0}^{+}; a_{0} (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{- 1, 0}^{1} (- ϕ_{01}, θ_{01}, 0) \\ A_{1, 0, 0, 0}^{01} & = & - C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{1, - 1}^{1} (- ϕ_{01}, θ_{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{1, 0}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{1, 1}^{1} (- ϕ_{01}, θ_{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{0}^{-}; ρ (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{0, 0}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{1, 0}^{1} (- ϕ_{01}, θ_{01}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{-}}_{0} π_{0}^{+}; a_{0} (1450)^{-} \to ϕ (1020)_{0} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{1, 0}^{1} (- ϕ_{01}, θ_{01}, 0) \\ A_{- 1, - 1, 0, 0}^{01} & = & C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{- 1, - 1}^{1} (- ϕ_{01}, θ_{01}, 0) D_{- 1, - 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{- 1, 1}^{1} (- ϕ_{01}, θ_{01}, 0) D_{1, - 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) - C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{0}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{01}, θ_{01}, 0) D_{0, - 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) \\ A_{- 1, 1, 0, 0}^{01} & = & C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{- 1, - 1}^{1} (- ϕ_{01}, θ_{01}, 0) D_{- 1, 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{- 1, 1}^{1} (- ϕ_{01}, θ_{01}, 0) D_{1, 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{0}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{01}, θ_{01}, 0) D_{0, 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) \\ A_{1, - 1, 0, 0}^{01} & = & C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{- 1, - 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{1, - 1}^{1} (- ϕ_{01}, θ_{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{1, - 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{1, 1}^{1} (- ϕ_{01}, θ_{01}, 0) - C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{0}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{0, - 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{1, 0}^{1} (- ϕ_{01}, θ_{01}, 0) \\ A_{1, 1, 0, 0}^{01} & = & C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{- 1, 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{1, - 1}^{1} (- ϕ_{01}, θ_{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{+ 1}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{1, 1}^{1} (- ϕ_{01}, θ_{01}, 0) D_{1, 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) + C_{J / ψ (1 S) \to π_{0}^{+} ρ (1450)_{0}^{-}; ρ (1450)^{-} \to ϕ (1020)_{+ 1} π_{0}^{-}} D_{0, 1}^{1} (- ϕ_{0}^{01}, θ_{0}^{01}, 0) D_{1, 0}^{1} (- ϕ_{01}, θ_{01}, 0) \\ A_{0, 0, 0, 0}^{02} & = & - C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{- 1, 0}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{0, - 1}^{1} (- ϕ_{02}, θ_{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{0, 1}^{1} (- ϕ_{02}, θ_{02}, 0) D_{1, 0}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{0}^{+}; ρ (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{0, 0}^{1} (- ϕ_{02}, θ_{02}, 0) D_{0, 0}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{+}}_{0} π_{0}^{-}; a_{0} (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{0, 0}^{0} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{0, 0}^{1} (- ϕ_{02}, θ_{02}, 0) \\ A_{0, - 1, 0, 0}^{02} & = & C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{- 1, - 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{0, - 1}^{1} (- ϕ_{02}, θ_{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{0, 1}^{1} (- ϕ_{02}, θ_{02}, 0) D_{1, - 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) - C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{0}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{0, - 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{0, 0}^{1} (- ϕ_{02}, θ_{02}, 0) \\ A_{0, 1, 0, 0}^{02} & = & C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{- 1, 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{0, - 1}^{1} (- ϕ_{02}, θ_{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{0, 1}^{1} (- ϕ_{02}, θ_{02}, 0) D_{1, 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{0}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{0, 0}^{1} (- ϕ_{02}, θ_{02}, 0) D_{0, 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) \\ A_{- 1, 0, 0, 0}^{02} & = & - C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{- 1, - 1}^{1} (- ϕ_{02}, θ_{02}, 0) D_{- 1, 0}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{- 1, 1}^{1} (- ϕ_{02}, θ_{02}, 0) D_{1, 0}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{0}^{+}; ρ (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{- 1, 0}^{1} (- ϕ_{02}, θ_{02}, 0) D_{0, 0}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{+}}_{0} π_{0}^{-}; a_{0} (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{0, 0}^{0} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{- 1, 0}^{1} (- ϕ_{02}, θ_{02}, 0) \\ A_{1, 0, 0, 0}^{02} & = & - C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{- 1, 0}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{1, - 1}^{1} (- ϕ_{02}, θ_{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{1, 0}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{1, 1}^{1} (- ϕ_{02}, θ_{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{0}^{+}; ρ (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{0, 0}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{1, 0}^{1} (- ϕ_{02}, θ_{02}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{+}}_{0} π_{0}^{-}; a_{0} (1450)^{+} \to ϕ (1020)_{0} π_{0}^{+}} D_{0, 0}^{0} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{1, 0}^{1} (- ϕ_{02}, θ_{02}, 0) \\ A_{- 1, - 1, 0, 0}^{02} & = & C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{- 1, - 1}^{1} (- ϕ_{02}, θ_{02}, 0) D_{- 1, - 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{- 1, 1}^{1} (- ϕ_{02}, θ_{02}, 0) D_{1, - 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) - C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{0}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{- 1, 0}^{1} (- ϕ_{02}, θ_{02}, 0) D_{0, - 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) \\ A_{- 1, 1, 0, 0}^{02} & = & C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{- 1, - 1}^{1} (- ϕ_{02}, θ_{02}, 0) D_{- 1, 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{- 1, 1}^{1} (- ϕ_{02}, θ_{02}, 0) D_{1, 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{0}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{- 1, 0}^{1} (- ϕ_{02}, θ_{02}, 0) D_{0, 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) \\ A_{1, - 1, 0, 0}^{02} & = & C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{- 1, - 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{1, - 1}^{1} (- ϕ_{02}, θ_{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{1, - 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{1, 1}^{1} (- ϕ_{02}, θ_{02}, 0) - C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{0}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{0, - 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{1, 0}^{1} (- ϕ_{02}, θ_{02}, 0) \\ A_{1, 1, 0, 0}^{02} & = & C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{- 1, 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{1, - 1}^{1} (- ϕ_{02}, θ_{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{+ 1}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{1, 1}^{1} (- ϕ_{02}, θ_{02}, 0) D_{1, 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) + C_{J / ψ (1 S) \to π_{0}^{-} ρ (1450)_{0}^{+}; ρ (1450)^{+} \to ϕ (1020)_{+ 1} π_{0}^{+}} D_{0, 1}^{1} (- ϕ_{0}^{02}, θ_{0}^{02}, 0) D_{1, 0}^{1} (- ϕ_{02}, θ_{02}, 0) \\ A_{0, 0, 0, 0}^{12} & = & C_{J / ψ (1 S) \to ϕ (1020)_{0} ρ (1450)_{+ 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, - 1}^{1} (- ϕ_{0}, θ_{0}, 0) D_{1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{0} ρ (1450)_{- 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{0, 1}^{1} (- ϕ_{0}, θ_{0}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{0} ρ (1450)_{0}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{1} (- ϕ_{0}, θ_{0}, 0) D_{0, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{0}}_{0} ϕ (1020)_{0}; a_{0} (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{0, 0}^{1} (- ϕ_{0}, θ_{0}, 0) \\ A_{0, - 1, 0, 0}^{12} & = & C_{J / ψ (1 S) \to ϕ (1020)_{- 1} ρ (1450)_{- 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{0, 0}^{1} (- ϕ_{0}, θ_{0}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{- 1} ρ (1450)_{0}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, - 1}^{1} (- ϕ_{0}, θ_{0}, 0) D_{0, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{0}}_{0} ϕ (1020)_{+ 1}; a_{0} (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{0, - 1}^{1} (- ϕ_{0}, θ_{0}, 0) \\ A_{0, 1, 0, 0}^{12} & = & C_{J / ψ (1 S) \to ϕ (1020)_{+ 1} ρ (1450)_{+ 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{1} (- ϕ_{0}, θ_{0}, 0) D_{1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{+ 1} ρ (1450)_{0}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{0, 1}^{1} (- ϕ_{0}, θ_{0}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{0}}_{0} ϕ (1020)_{+ 1}; a_{0} (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{0, 1}^{1} (- ϕ_{0}, θ_{0}, 0) \\ A_{- 1, 0, 0, 0}^{12} & = & C_{J / ψ (1 S) \to ϕ (1020)_{0} ρ (1450)_{+ 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, - 1}^{1} (- ϕ_{0}, θ_{0}, 0) D_{1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{0} ρ (1450)_{- 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{- 1, 1}^{1} (- ϕ_{0}, θ_{0}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{0} ρ (1450)_{0}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{0}, θ_{0}, 0) D_{0, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{0}}_{0} ϕ (1020)_{0}; a_{0} (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{- 1, 0}^{1} (- ϕ_{0}, θ_{0}, 0) \\ A_{1, 0, 0, 0}^{12} & = & C_{J / ψ (1 S) \to ϕ (1020)_{0} ρ (1450)_{+ 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{1, - 1}^{1} (- ϕ_{0}, θ_{0}, 0) D_{1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{0} ρ (1450)_{- 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{1, 1}^{1} (- ϕ_{0}, θ_{0}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{0} ρ (1450)_{0}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{1, 0}^{1} (- ϕ_{0}, θ_{0}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{0}}_{0} ϕ (1020)_{0}; a_{0} (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{1, 0}^{1} (- ϕ_{0}, θ_{0}, 0) \\ A_{- 1, - 1, 0, 0}^{12} & = & C_{J / ψ (1 S) \to ϕ (1020)_{- 1} ρ (1450)_{- 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{0}, θ_{0}, 0) D_{- 1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{- 1} ρ (1450)_{0}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, - 1}^{1} (- ϕ_{0}, θ_{0}, 0) D_{0, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{0}}_{0} ϕ (1020)_{+ 1}; a_{0} (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{- 1, - 1}^{1} (- ϕ_{0}, θ_{0}, 0) \\ A_{- 1, 1, 0, 0}^{12} & = & C_{J / ψ (1 S) \to ϕ (1020)_{+ 1} ρ (1450)_{+ 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{0}, θ_{0}, 0) D_{1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{+ 1} ρ (1450)_{0}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, 1}^{1} (- ϕ_{0}, θ_{0}, 0) D_{0, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{0}}_{0} ϕ (1020)_{+ 1}; a_{0} (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{- 1, 1}^{1} (- ϕ_{0}, θ_{0}, 0) \\ A_{1, - 1, 0, 0}^{12} & = & C_{J / ψ (1 S) \to ϕ (1020)_{- 1} ρ (1450)_{- 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{- 1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{1, 0}^{1} (- ϕ_{0}, θ_{0}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{- 1} ρ (1450)_{0}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{1, - 1}^{1} (- ϕ_{0}, θ_{0}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{0}}_{0} ϕ (1020)_{+ 1}; a_{0} (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{1, - 1}^{1} (- ϕ_{0}, θ_{0}, 0) \\ A_{1, 1, 0, 0}^{12} & = & C_{J / ψ (1 S) \to ϕ (1020)_{+ 1} ρ (1450)_{+ 1}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{1, 0}^{1} (- ϕ_{0}, θ_{0}, 0) D_{1, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) + C_{J / ψ (1 S) \to ϕ (1020)_{+ 1} ρ (1450)_{0}^{0}; ρ (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{1} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{1, 1}^{1} (- ϕ_{0}, θ_{0}, 0) + C_{J / ψ (1 S) \to {a_{0} (1450)^{0}}_{0} ϕ (1020)_{+ 1}; a_{0} (1450)^{0} \to π_{0}^{+} π_{0}^{-}} D_{0, 0}^{0} (- ϕ_{1}^{12}, θ_{1}^{12}, 0) D_{1, 1}^{1} (- ϕ_{0}, θ_{0}, 0) \end{array} \end{array}

\begin{array}{r} \begin{array}{rcl} m_{01} & = & m_{p_{01}} \\ m_{02} & = & m_{p_{02}} \\ m_{12} & = & m_{p_{12}} \\ ϕ_{0} & = & ϕ (p_{12}) \\ ϕ_{0}^{01} & = & ϕ (B_{z} (\frac{| \vec{p_{01}} |}{E (p_{01})}) R_{y} (- θ (p_{01})) R_{z} (- ϕ (p_{01})) p_{0}) \\ ϕ_{0}^{02} & = & ϕ (B_{z} (\frac{| \vec{p_{02}} |}{E (p_{02})}) R_{y} (- θ (p_{02})) R_{z} (- ϕ (p_{02})) p_{0}) \\ ϕ_{01} & = & ϕ (p_{01}) \\ ϕ_{1}^{12} & = & ϕ (B_{z} (\frac{| \vec{p_{12}} |}{E (p_{12})}) R_{y} (- θ (p_{12})) R_{z} (- ϕ (p_{12})) p_{1}) \\ ϕ_{02} & = & ϕ (p_{02}) \\ θ_{0} & = & θ (p_{12}) \\ θ_{0}^{01} & = & θ (B_{z} (\frac{| \vec{p_{01}} |}{E (p_{01})}) R_{y} (- θ (p_{01})) R_{z} (- ϕ (p_{01})) p_{0}) \\ θ_{0}^{02} & = & θ (B_{z} (\frac{| \vec{p_{02}} |}{E (p_{02})}) R_{y} (- θ (p_{02})) R_{z} (- ϕ (p_{02})) p_{0}) \\ θ_{01} & = & θ (p_{01}) \\ θ_{1}^{12} & = & θ (B_{z} (\frac{| \vec{p_{12}} |}{E (p_{12})}) R_{y} (- θ (p_{12})) R_{z} (- ϕ (p_{12})) p_{1}) \\ θ_{02} & = & θ (p_{02}) \end{array} \end{array}

Phase space sample#

def generate_phase_space(reaction: ReactionInfo, size: int) -> dict[str, jnp.ndarray]:
    rng = TFUniformRealNumberGenerator(seed=0)
    phsp_generator = TFPhaseSpaceGenerator(
        initial_state_mass=reaction.initial_state[-1].mass,
        final_state_masses={i: p.mass for i, p in reaction.final_state.items()},
    )
    return phsp_generator.generate(size, rng)


phsp = generate_phase_space(AMPFORM_MODEL.reaction_info, size=100_000)
ampform_phsp = ampform_transformer(phsp)
dpd_phsp = dpd_transformer(phsp)

WARNING: All log messages before absl::InitializeLog() is called are written to STDERR
E0000 00:00:1743423244.808296    3511 cuda_dnn.cc:8310] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered
E0000 00:00:1743423244.812497    3511 cuda_blas.cc:1418] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered

WARNING: All log messages before absl::InitializeLog() is called are written to STDERR
I0000 00:00:1743423247.173969    3590 service.cc:148] XLA service 0x7fae4418c5e0 initialized for platform Host (this does not guarantee that XLA will be used). Devices:
I0000 00:00:1743423247.174010    3590 service.cc:156]   StreamExecutor device (0): Host, Default Version
I0000 00:00:1743423247.195877    3591 device_compiler.h:188] Compiled cluster using XLA!  This line is logged at most once for the lifetime of the process.

Convert to numerical functions#

ampform_intensity_expr = cached.unfold(AMPFORM_MODEL)
dpd_intensity_expr = cached.unfold(DPD_MODEL)

ampform_func = cached.lambdify(
    ampform_intensity_expr,
    parameters=AMPFORM_MODEL.parameter_defaults,
)
dpd_func = cached.lambdify(
    dpd_intensity_expr,
    parameters=DPD_MODEL.parameter_defaults,
)

Visualization#

Define widget UI Hide code cell source

def create_sliders() -> dict[str, ToggleButtons]:
    all_parameters = dict(AMPFORM_MODEL.parameter_defaults.items())
    all_parameters.update(DPD_MODEL.parameter_defaults)
    sliders = {}
    for symbol, value in all_parameters.items():
        value = "+1"
        if (
            symbol.name.startswith(R"\mathcal{H}^\mathrm{decay}") and "+" in symbol.name
        ) and any(s in symbol.name for s in ["{1}", "*", "rho"]):
            value = "-1"
        sliders[symbol.name] = ToggleButtons(
            description=Rf"\({sp.latex(symbol)}\)",
            options=["-1", "0", "+1"],
            value=value,
            continuous_update=False,
        )
    return sliders


def to_unicode(particle: Particle) -> str:
    unicode = particle.name
    unicode = unicode.replace("pi", "π")
    unicode = unicode.replace("rho", "p")
    unicode = unicode.replace("Sigma", "Σ")
    unicode = unicode.replace("~", "")
    unicode = unicode.replace("Σ", "~Σ")
    unicode = unicode.replace("+", "⁺")
    unicode = unicode.replace("-", "⁻")
    unicode = unicode.replace("(0)", "₀")
    unicode = unicode.replace("(1)", "₁")
    return unicode.replace(")0", ")⁰")


sliders = create_sliders()
resonance_selector = SelectMultiple(
    description="Resonance",
    options={to_unicode(p): p.latex for p in resonances},
    value=[resonances[0].latex, resonances[1].latex],
    layout=Layout(
        height=f"{14 * (len(resonances) + 1)}pt",
        width="auto",
    ),
)
hide_expressions = Checkbox(description="Hide expressions", value=True)
simplify_expressions = Checkbox(description="Simplify", value=True)
ipywidgets.link((hide_expressions, "value"), (simplify_expressions, "disabled"))

package_names = ("AmpForm", "AmpForm-DPD")
ui = HBox([
    VBox([resonance_selector, hide_expressions, simplify_expressions]),
    Tab(
        children=[
            Accordion(
                children=[
                    GridBox([
                        sliders[key]
                        for key in sorted(sliders)
                        if p.latex in key
                        if (
                            key[0] in {"C", "H"}
                            if package == "AmpForm"
                            else key.startswith(R"\mathcal{H}")
                        )
                    ])
                    for package in package_names
                ],
                selected_index=1,
                titles=package_names,
            )
            for p in resonances
        ],
        titles=[to_unicode(p) for p in resonances],
    ),
])