J/ψ → φ(1020) π⁺ π⁻

J/ψ → φ(1020) π⁺ π⁻#

The decay \(J/\psi \to \phi(1020) \pi^+ \pi^-\) has an initial state with spin and has one vector meson, \(\phi(1020)\), 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)`	\(J/\psi(1S)\)	\(1^-\)	3,096	0
1	`phi(1020)`	\(\phi(1020)\)	\(1^-\)	1,019	4
2	`pi-`	\(\pi^{-}\)	\(0^-\)	139	0
3	`pi+`	\(\pi^{+}\)	\(0^-\)	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)+`	\(\rho(1450)^{+}\)	\(1^-\)	1,465	400
`rho(1450)-`	\(\rho(1450)^{-}\)	\(1^-\)	1,465	400
`rho(1450)0`	\(\rho(1450)^{0}\)	\(1^-\)	1,465	400

\[\begin{split}\begin{array}{c} J/\psi(1S) \to \left(a_{0}(1450)^{+} \to \phi(1020) \pi^{+}\right) \pi^{-} \\ J/\psi(1S) \to \left(a_{0}(1450)^{-} \to \phi(1020) \pi^{-}\right) \pi^{+} \\ J/\psi(1S) \to \left(a_{0}(1450)^{0} \to \pi^{-} \pi^{+}\right) \phi(1020) \\ J/\psi(1S) \to \left(\rho(1450)^{+} \to \phi(1020) \pi^{+}\right) \pi^{-} \\ J/\psi(1S) \to \left(\rho(1450)^{-} \to \phi(1020) \pi^{-}\right) \pi^{+} \\ J/\psi(1S) \to \left(\rho(1450)^{0} \to \pi^{-} \pi^{+}\right) \phi(1020) \\ \end{array}\end{split}\]

Model formulation#

DPD model#

\[\displaystyle \sum_{\lambda_{0}=-1}^{1} \sum_{\lambda_{1}=-1}^{1}{\left|{\sum_{\lambda_0^{\prime}=-1}^{1} \sum_{\lambda_1^{\prime}=-1}^{1} \sum_{\lambda_2^{\prime}=0} \sum_{\lambda_3^{\prime}=0}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{1}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{1(1)}\right) d^{1}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{1(1)}\right) + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{1}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{2(1)}\right) d^{1}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{2(1)}\right) + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{1}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{3(1)}\right) d^{1}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{3(1)}\right)}}\right|^{2}}\]

\[\begin{split}\begin{array}{rcl} A^{1}_{-1, -1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{-1, \lambda_{R} + 1} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{0}, \lambda_{R}, -1} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{\delta_{-1, \lambda_{R} + 1} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{0}, \lambda_{R}, -1} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{-1, -1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{+}, 0, -1} \mathcal{H}^\mathrm{production}_{\rho(1450)^{+}, \lambda_{R}, 0} d^{1}_{\lambda_{R},1}\left(\theta_{31}\right)} + \sum_{\lambda_{R}=0}{\delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{+}, 0, -1} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},1}\left(\theta_{31}\right)} \\ A^{3}_{-1, -1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{-}, -1, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{-}, \lambda_{R}, 0} d^{1}_{\lambda_{R},-1}\left(\theta_{12}\right)} + \sum_{\lambda_{R}=0}{\delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{-}, -1, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},-1}\left(\theta_{12}\right)} \\ A^{1}_{-1, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{- \delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{0}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{- \delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{0}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{-1, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{- \delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{+}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{+}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{31}\right)} + \sum_{\lambda_{R}=0}{- \delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{+}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{31}\right)} \\ A^{3}_{-1, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{-}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{-}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{12}\right)} + \sum_{\lambda_{R}=0}{\delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{-}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{12}\right)} \\ A^{1}_{-1, 1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{-1, \lambda_{R} - 1} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{0}, \lambda_{R}, 1} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{\delta_{-1, \lambda_{R} - 1} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{0}, \lambda_{R}, 1} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{-1, 1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{+}, 0, 1} \mathcal{H}^\mathrm{production}_{\rho(1450)^{+}, \lambda_{R}, 0} d^{1}_{\lambda_{R},-1}\left(\theta_{31}\right)} + \sum_{\lambda_{R}=0}{\delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{+}, 0, 1} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},-1}\left(\theta_{31}\right)} \\ A^{3}_{-1, 1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{-}, 1, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{-}, \lambda_{R}, 0} d^{1}_{\lambda_{R},1}\left(\theta_{12}\right)} + \sum_{\lambda_{R}=0}{\delta_{-1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{-}, 1, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},1}\left(\theta_{12}\right)} \\ A^{1}_{0, -1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{0, \lambda_{R} + 1} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{0}, \lambda_{R}, -1} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{\delta_{0, \lambda_{R} + 1} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{0}, \lambda_{R}, -1} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{0, -1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{+}, 0, -1} \mathcal{H}^\mathrm{production}_{\rho(1450)^{+}, \lambda_{R}, 0} d^{1}_{\lambda_{R},1}\left(\theta_{31}\right)} + \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{+}, 0, -1} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},1}\left(\theta_{31}\right)} \\ A^{3}_{0, -1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{-}, -1, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{-}, \lambda_{R}, 0} d^{1}_{\lambda_{R},-1}\left(\theta_{12}\right)} + \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{-}, -1, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},-1}\left(\theta_{12}\right)} \\ A^{1}_{0, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{- \delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{0}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{- \delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{0}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{0, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{- \delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{+}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{+}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{31}\right)} + \sum_{\lambda_{R}=0}{- \delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{+}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{31}\right)} \\ A^{3}_{0, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{-}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{-}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{12}\right)} + \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{-}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{12}\right)} \\ A^{1}_{0, 1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{0, \lambda_{R} - 1} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{0}, \lambda_{R}, 1} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{\delta_{0, \lambda_{R} - 1} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{0}, \lambda_{R}, 1} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{0, 1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{+}, 0, 1} \mathcal{H}^\mathrm{production}_{\rho(1450)^{+}, \lambda_{R}, 0} d^{1}_{\lambda_{R},-1}\left(\theta_{31}\right)} + \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{+}, 0, 1} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},-1}\left(\theta_{31}\right)} \\ A^{3}_{0, 1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{-}, 1, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{-}, \lambda_{R}, 0} d^{1}_{\lambda_{R},1}\left(\theta_{12}\right)} + \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{-}, 1, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},1}\left(\theta_{12}\right)} \\ A^{1}_{1, -1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{1, \lambda_{R} + 1} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{0}, \lambda_{R}, -1} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{\delta_{1, \lambda_{R} + 1} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{0}, \lambda_{R}, -1} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{1, -1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{+}, 0, -1} \mathcal{H}^\mathrm{production}_{\rho(1450)^{+}, \lambda_{R}, 0} d^{1}_{\lambda_{R},1}\left(\theta_{31}\right)} + \sum_{\lambda_{R}=0}{\delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{+}, 0, -1} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},1}\left(\theta_{31}\right)} \\ A^{3}_{1, -1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{-}, -1, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{-}, \lambda_{R}, 0} d^{1}_{\lambda_{R},-1}\left(\theta_{12}\right)} + \sum_{\lambda_{R}=0}{\delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{-}, -1, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},-1}\left(\theta_{12}\right)} \\ A^{1}_{1, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{- \delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{0}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{- \delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{0}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{1, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{- \delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{+}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{+}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{31}\right)} + \sum_{\lambda_{R}=0}{- \delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{+}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{31}\right)} \\ A^{3}_{1, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{-}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{-}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{12}\right)} + \sum_{\lambda_{R}=0}{\delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{-}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{12}\right)} \\ A^{1}_{1, 1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{1, \lambda_{R} - 1} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{0}, \lambda_{R}, 1} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{\delta_{1, \lambda_{R} - 1} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{0}, \lambda_{R}, 1} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{1, 1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{+}, 0, 1} \mathcal{H}^\mathrm{production}_{\rho(1450)^{+}, \lambda_{R}, 0} d^{1}_{\lambda_{R},-1}\left(\theta_{31}\right)} + \sum_{\lambda_{R}=0}{\delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{+}, 0, 1} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},-1}\left(\theta_{31}\right)} \\ A^{3}_{1, 1, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\rho(1450)^{-}, 1, 0} \mathcal{H}^\mathrm{production}_{\rho(1450)^{-}, \lambda_{R}, 0} d^{1}_{\lambda_{R},1}\left(\theta_{12}\right)} + \sum_{\lambda_{R}=0}{\delta_{1 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(1450)^{-}, 1, 0} \mathcal{H}^\mathrm{production}_{a_{0}(1450)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},1}\left(\theta_{12}\right)} \\ \end{array}\end{split}\]

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

\[\begin{split}\begin{array}{rcl} \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)} \\ \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)} \\ \zeta^0_{1(1)} &=& 0 \\ \zeta^1_{1(1)} &=& 0 \\ \zeta^0_{2(1)} &=& - \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)} \\ \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^0_{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)} \\ \zeta^1_{3(1)} &=& - \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)} \\ \end{array}\end{split}\]

AmpForm model#

\[\displaystyle \sum_{m_{A}=-1}^{1} \sum_{m_{0}=-1}^{1} \sum_{m_{1}=0} \sum_{m_{2}=0}{\left|{A^{01}_{m_{A}, m_{0}, m_{1}, m_{2}} + A^{02}_{m_{A}, m_{0}, m_{1}, m_{2}} + A^{12}_{m_{A}, m_{0}, m_{1}, m_{2}}}\right|^{2}}\]

\[\begin{split}\begin{array}{rcl} A^{01}_{0, 0, 0, 0} &=& - C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{0,-1}\left(- \phi_{01},\theta_{01},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{1}_{0,1}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{1,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{0}; \rho(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{1}_{0,0}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{0,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{-}}_{0} \pi^{+}_{0}; a_{0}(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{0,0}\left(- \phi_{01},\theta_{01},0\right) \\ A^{01}_{0, -1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{-1,-1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{0,-1}\left(- \phi_{01},\theta_{01},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{0,1}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{1,-1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) - C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{0}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{0,-1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{0,0}\left(- \phi_{01},\theta_{01},0\right) \\ A^{01}_{0, 1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{-1,1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{0,-1}\left(- \phi_{01},\theta_{01},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{0,1}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{1,1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{0}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{0,0}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{0,1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) \\ A^{01}_{-1, 0, 0, 0} &=& - C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{1}_{-1,-1}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{-1,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{1}_{-1,1}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{1,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{0}; \rho(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{0,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{-}}_{0} \pi^{+}_{0}; a_{0}(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{-1,0}\left(- \phi_{01},\theta_{01},0\right) \\ A^{01}_{1, 0, 0, 0} &=& - C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{1,-1}\left(- \phi_{01},\theta_{01},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{1}_{1,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{1,1}\left(- \phi_{01},\theta_{01},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{0}; \rho(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{1}_{0,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{1,0}\left(- \phi_{01},\theta_{01},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{-}}_{0} \pi^{+}_{0}; a_{0}(1450)^{-} \to \phi(1020)_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{1,0}\left(- \phi_{01},\theta_{01},0\right) \\ A^{01}_{-1, -1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{-1,-1}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{-1,-1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{-1,1}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{1,-1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) - C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{0}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{0,-1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) \\ A^{01}_{-1, 1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{-1,-1}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{-1,1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{-1,1}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{1,1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{0}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{0,1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) \\ A^{01}_{1, -1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{-1,-1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{1,-1}\left(- \phi_{01},\theta_{01},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{1,-1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{1,1}\left(- \phi_{01},\theta_{01},0\right) - C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{0}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{0,-1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{1,0}\left(- \phi_{01},\theta_{01},0\right) \\ A^{01}_{1, 1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{-1,1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{1,-1}\left(- \phi_{01},\theta_{01},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{+1}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{1,1}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{1,1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{J/\psi(1S) \to \pi^{+}_{0} \rho(1450)^{-}_{0}; \rho(1450)^{-} \to \phi(1020)_{+1} \pi^{-}_{0}} D^{1}_{0,1}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) D^{1}_{1,0}\left(- \phi_{01},\theta_{01},0\right) \\ A^{02}_{0, 0, 0, 0} &=& - C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{1}_{-1,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{0,-1}\left(- \phi_{02},\theta_{02},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{1}_{0,1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{1,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{0}; \rho(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{1}_{0,0}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{0,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{+}}_{0} \pi^{-}_{0}; a_{0}(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{0}_{0,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{0,0}\left(- \phi_{02},\theta_{02},0\right) \\ A^{02}_{0, -1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{-1,-1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{0,-1}\left(- \phi_{02},\theta_{02},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{0,1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{1,-1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) - C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{0}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{0,-1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{0,0}\left(- \phi_{02},\theta_{02},0\right) \\ A^{02}_{0, 1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{-1,1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{0,-1}\left(- \phi_{02},\theta_{02},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{0,1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{1,1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{0}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{0,0}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{0,1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) \\ A^{02}_{-1, 0, 0, 0} &=& - C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{1}_{-1,-1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{-1,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{1}_{-1,1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{1,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{0}; \rho(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{1}_{-1,0}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{0,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{+}}_{0} \pi^{-}_{0}; a_{0}(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{0}_{0,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{-1,0}\left(- \phi_{02},\theta_{02},0\right) \\ A^{02}_{1, 0, 0, 0} &=& - C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{1}_{-1,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{1,-1}\left(- \phi_{02},\theta_{02},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{1}_{1,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{1,1}\left(- \phi_{02},\theta_{02},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{0}; \rho(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{1}_{0,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{1,0}\left(- \phi_{02},\theta_{02},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{+}}_{0} \pi^{-}_{0}; a_{0}(1450)^{+} \to \phi(1020)_{0} \pi^{+}_{0}} D^{0}_{0,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{1,0}\left(- \phi_{02},\theta_{02},0\right) \\ A^{02}_{-1, -1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{-1,-1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{-1,-1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{-1,1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{1,-1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) - C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{0}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{-1,0}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{0,-1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) \\ A^{02}_{-1, 1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{-1,-1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{-1,1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{-1,1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{1,1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{0}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{-1,0}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{0,1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) \\ A^{02}_{1, -1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{-1,-1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{1,-1}\left(- \phi_{02},\theta_{02},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{1,-1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{1,1}\left(- \phi_{02},\theta_{02},0\right) - C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{0}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{0,-1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{1,0}\left(- \phi_{02},\theta_{02},0\right) \\ A^{02}_{1, 1, 0, 0} &=& C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{-1,1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{1,-1}\left(- \phi_{02},\theta_{02},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{+1}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{1,1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{1,1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{J/\psi(1S) \to \pi^{-}_{0} \rho(1450)^{+}_{0}; \rho(1450)^{+} \to \phi(1020)_{+1} \pi^{+}_{0}} D^{1}_{0,1}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) D^{1}_{1,0}\left(- \phi_{02},\theta_{02},0\right) \\ A^{12}_{0, 0, 0, 0} &=& C_{J/\psi(1S) \to \phi(1020)_{0} \rho(1450)^{0}_{+1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{0,-1}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to \phi(1020)_{0} \rho(1450)^{0}_{-1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{0,1}\left(- \phi_{0},\theta_{0},0\right) + C_{J/\psi(1S) \to \phi(1020)_{0} \rho(1450)^{0}_{0}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{0,0}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{0}}_{0} \phi(1020)_{0}; a_{0}(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{0,0}\left(- \phi_{0},\theta_{0},0\right) \\ A^{12}_{0, -1, 0, 0} &=& C_{J/\psi(1S) \to \phi(1020)_{-1} \rho(1450)^{0}_{-1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{0,0}\left(- \phi_{0},\theta_{0},0\right) + C_{J/\psi(1S) \to \phi(1020)_{-1} \rho(1450)^{0}_{0}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{0,-1}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{0}}_{0} \phi(1020)_{+1}; a_{0}(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{0,-1}\left(- \phi_{0},\theta_{0},0\right) \\ A^{12}_{0, 1, 0, 0} &=& C_{J/\psi(1S) \to \phi(1020)_{+1} \rho(1450)^{0}_{+1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{0,0}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to \phi(1020)_{+1} \rho(1450)^{0}_{0}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{0,1}\left(- \phi_{0},\theta_{0},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{0}}_{0} \phi(1020)_{+1}; a_{0}(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{0,1}\left(- \phi_{0},\theta_{0},0\right) \\ A^{12}_{-1, 0, 0, 0} &=& C_{J/\psi(1S) \to \phi(1020)_{0} \rho(1450)^{0}_{+1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,-1}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to \phi(1020)_{0} \rho(1450)^{0}_{-1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{-1,1}\left(- \phi_{0},\theta_{0},0\right) + C_{J/\psi(1S) \to \phi(1020)_{0} \rho(1450)^{0}_{0}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{0}}_{0} \phi(1020)_{0}; a_{0}(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{-1,0}\left(- \phi_{0},\theta_{0},0\right) \\ A^{12}_{1, 0, 0, 0} &=& C_{J/\psi(1S) \to \phi(1020)_{0} \rho(1450)^{0}_{+1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{1,-1}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to \phi(1020)_{0} \rho(1450)^{0}_{-1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{1,1}\left(- \phi_{0},\theta_{0},0\right) + C_{J/\psi(1S) \to \phi(1020)_{0} \rho(1450)^{0}_{0}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{1,0}\left(- \phi_{0},\theta_{0},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{0}}_{0} \phi(1020)_{0}; a_{0}(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{1,0}\left(- \phi_{0},\theta_{0},0\right) \\ A^{12}_{-1, -1, 0, 0} &=& C_{J/\psi(1S) \to \phi(1020)_{-1} \rho(1450)^{0}_{-1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{-1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to \phi(1020)_{-1} \rho(1450)^{0}_{0}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,-1}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{0}}_{0} \phi(1020)_{+1}; a_{0}(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{-1,-1}\left(- \phi_{0},\theta_{0},0\right) \\ A^{12}_{-1, 1, 0, 0} &=& C_{J/\psi(1S) \to \phi(1020)_{+1} \rho(1450)^{0}_{+1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to \phi(1020)_{+1} \rho(1450)^{0}_{0}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,1}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{0}}_{0} \phi(1020)_{+1}; a_{0}(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{-1,1}\left(- \phi_{0},\theta_{0},0\right) \\ A^{12}_{1, -1, 0, 0} &=& C_{J/\psi(1S) \to \phi(1020)_{-1} \rho(1450)^{0}_{-1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{-1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{1,0}\left(- \phi_{0},\theta_{0},0\right) + C_{J/\psi(1S) \to \phi(1020)_{-1} \rho(1450)^{0}_{0}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{1,-1}\left(- \phi_{0},\theta_{0},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{0}}_{0} \phi(1020)_{+1}; a_{0}(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{1,-1}\left(- \phi_{0},\theta_{0},0\right) \\ A^{12}_{1, 1, 0, 0} &=& C_{J/\psi(1S) \to \phi(1020)_{+1} \rho(1450)^{0}_{+1}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{1,0}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{1,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{J/\psi(1S) \to \phi(1020)_{+1} \rho(1450)^{0}_{0}; \rho(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{1}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{1,1}\left(- \phi_{0},\theta_{0},0\right) + C_{J/\psi(1S) \to {a_{0}(1450)^{0}}_{0} \phi(1020)_{+1}; a_{0}(1450)^{0} \to \pi^{+}_{0} \pi^{-}_{0}} D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) D^{1}_{1,1}\left(- \phi_{0},\theta_{0},0\right) \\ \end{array}\end{split}\]

\[\begin{split}\begin{array}{rcl} m_{01} &=& m_{{p}_{01}} \\ m_{02} &=& m_{{p}_{02}} \\ m_{12} &=& m_{{p}_{12}} \\ \phi_{0} &=& \phi\left({p}_{12}\right) \\ \phi^{01}_{0} &=& \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{01}}\right|}{E\left({p}_{01}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{01}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{01}\right)\right) p_{0}\right) \\ \phi^{02}_{0} &=& \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{02}}\right|}{E\left({p}_{02}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{02}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{02}\right)\right) p_{0}\right) \\ \phi_{01} &=& \phi\left({p}_{01}\right) \\ \phi^{12}_{1} &=& \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{12}}\right|}{E\left({p}_{12}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{12}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{12}\right)\right) p_{1}\right) \\ \phi_{02} &=& \phi\left({p}_{02}\right) \\ \theta_{0} &=& \theta\left({p}_{12}\right) \\ \theta^{01}_{0} &=& \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{01}}\right|}{E\left({p}_{01}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{01}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{01}\right)\right) p_{0}\right) \\ \theta^{02}_{0} &=& \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{02}}\right|}{E\left({p}_{02}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{02}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{02}\right)\right) p_{0}\right) \\ \theta_{01} &=& \theta\left({p}_{01}\right) \\ \theta^{12}_{1} &=& \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{12}}\right|}{E\left({p}_{12}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{12}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{12}\right)\right) p_{1}\right) \\ \theta_{02} &=& \theta\left({p}_{02}\right) \\ \end{array}\end{split}\]

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
I0000 00:00:1714472709.594407    3648 service.cc:145] XLA service 0x7fd62c18cab0 initialized for platform Host (this does not guarantee that XLA will be used). Devices:
I0000 00:00:1714472709.594458    3648 service.cc:153]   StreamExecutor device (0): Host, Default Version
I0000 00:00:1714472709.615310    3648 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#

def unfold_intensity(model: HelicityModel) -> sp.Expr:
    unfolded_intensity = perform_cached_doit(model.intensity)
    unfolded_amplitudes = {
        symbol: perform_cached_doit(expr) for symbol, expr in model.amplitudes.items()
    }
    return unfolded_intensity.xreplace(unfolded_amplitudes)


ampform_intensity_expr = unfold_intensity(AMPFORM_MODEL)
dpd_intensity_expr = unfold_intensity(DPD_MODEL)

ampform_func = perform_cached_lambdify(
    ampform_intensity_expr,
    parameters=AMPFORM_MODEL.parameter_defaults,
)
dpd_func = perform_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],
    ),
])