"readJsonFile.py" of "The exotic meson $\pi_1(1600)$ with $J^{PC} = 1^{-+}$ and its decay into $\rho(770)\pi$"
Creators: name: COMPASS Collaboration
Publisher: name: HEPData
Year: 2021

description: Example Python 3 script that calculates the $\chi^2$ function in Eq. (B17) that needs to be minimized in order to resolve the zero mode in $\{\mathcal{T}_k^\text{fit}\}$ for a single $(m_{3\pi}, t^\prime)$ cell. The script also demonstrates the preparation of the covariance matrix of the $\{\mathcal{T}_k^\text{fit}\}$ according to Eq. (C4) and the zero-mode correction of the $\{\mathcal{T}_k^\text{fit}\}$.
description: The data provided here are the result of a freed-isobar partial-wave analysis (PWA) of the three-pion system produced via the diffractive reaction $\pi^- + p \to \pi^- \pi^- \pi^+ + p$ as presented in Section V. The freed-isobar PWA was performed independently in cells of the the three-pion mass $m_{3\pi}$ and the reduced four-momentum transfer squared $t^\prime$. To this end, the data were subdivided into 38 equidistant $m_{3\pi}$ bins, each $40\;\text{MeV}/c^2$ wide, in the range from $0.98$ to $2.50\;\text{GeV}/c^2$ and into 4 nonequidistant $t^\prime$ bins in the range from $0.1$ to $1.0\;(\text{GeV}/c)^2$ (see Table III). The PWA model included 12 partial waves with freed dynamic isobar amplitudes parametrized according to Eq. (7). These waves are listed in Table II.

Here, we provide data on the spin-exotic $1^{-+}1^+[\pi\pi]_{1^{-\,-}}\pi P$ wave, which is one of the 12 waves with a freed dynamic isobar amplitude. In this wave, the dynamic isobar amplitude of the $\pi^-\pi^+$ subsystem with $I^G\;J^{PC} = 1^-\;1^{--}$, i.e., $\rho$-like quantum numbers was determined from the data in bins of $m_{\pi^-\pi^+}$ indexed by $k$. The $m_{\pi^-\pi^+}$ bins are $40\;\text{MeV}/c^2$ wide, except in the range from $0.64$ to $0.92\;\text{GeV}/c^2$ around the $\rho(770)$, where we used a finer $20\;\text{MeV}/c^2$ binning. Due to phase space, the number of $m_{\pi^-\pi^+}$ bins depends on $m_{3\pi}$. The data are grouped according to the 4 $t^\prime$ bins:

 $t^\prime$ bin 0: $0.100 &lt; t^\prime &lt; 0.141\;(\text{GeV}/c)^2$
 $t^\prime$ bin 1: $0.141 &lt; t^\prime &lt; 0.194\;(\text{GeV}/c)^2$
 $t^\prime$ bin 2: $0.194 &lt; t^\prime &lt; 0.326\;(\text{GeV}/c)^2$
 $t^\prime$ bin 3: $0.326 &lt; t^\prime &lt; 1.000\;(\text{GeV}/c)^2$


For each $t^\prime$ bin, we provide tables with the intensity of the coherent sum of the transition amplitudes $\{\mathcal{T}_k^\text{fit}\}$ of the $1^{-+}1^+[\pi\pi]_{1^{-\,-}}\pi P$ wave [see Eqs. (9), (12), and (B15)] for each $m_{3\pi}$ bin. The coherent sums run over all $m_{\pi^-\pi^+}$ bins indexed by $k$. The values correspond to the orange points shown in Fig. 8 for the lowest and highest $t^\prime$ bins and in Fig. 15 in the supplemental material for the two intermediate $t^\prime$ bins. In the "Resources" section of each $t^\prime$ bin, we provide additional JSON files named <code>transition_amplitudes_tBin_&lt;#&gt;.json</code> for download, which contain for each $m_{3\pi}$ bin the values of the transition amplitudes $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, their covariances, and further information.

As explained in Appendices B and C, the transition amplitudes $\{\mathcal{T}_k^\text{fit}\}$ are affected by the zero-mode ambiguity. This ambiguity needs to be resolved at the amplitude level. We provide in the "Additional Resources" the example Python script <code>readJsonFile.py</code> that implements the zero-mode resolution described in Appendix B as well as the preparation of the covariance matrix of the $\{\mathcal{T}_k^\text{fit}\}$ discussed in Appendix C.