DOI: 10.17182/hepdata.114098.v1/t4
title: "$t^\prime$ Bin 3" of "The exotic meson $\pi_1(1600)$ with $J^{PC} = 1^{-+}$ and its decay into $\rho(770)\pi$"

description: Results for the spin-exotic $1^{-+}1^+[\pi\pi]_{1^{-\,-}}\pi P$ wave from the free-isobar partial-wave analysis performed in the fourth $t^\prime$ bin from $0.326$ to $1.000\;(\text{GeV}/c)^2$. The plotted values represent the intensity of the coherent sum of the dynamic isobar amplitudes $\{\mathcal{T}_k^\text{fit}\}$ as a function of $m_{3\pi}$, where the coherent sums run over all $m_{\pi^-\pi^+}$ bins indexed by $k$. These intensity values are given in number of events per $40\;\text{MeV}/c^2$ $m_{3\pi}$ interval and correspond to the orange points in Fig. 8(b).

In the "Resources" section of this $t^\prime$ bin, we provide the JSON file named <code>transition_amplitudes_tBin_3.json</code> for download, which contains 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. The data in this JSON file are organized in independent bins of $m_{3\pi}$. The information in these bins can be accessed via the key <code>m3pi_bin_&lt;#&gt;_t_prime_bin_3</code>. Each independent $m_{3\pi}$ bin contains the kinematic ranges of the $(m_{3\pi}, t^\prime)$ cell, which are accessible via the keys <code>m3pi_lower_limit</code>, <code>m3pi_upper_limit</code>, <code>t_prime_lower_limit</code>, and <code>t_prime_upper_limit</code>. the $m_{\pi^-\pi^+}$ bin borders, which are accessible via the keys <code>m2pi_lower_limits</code> and <code>m2pi_upper_limits</code>. the real and imaginary parts of the transition amplitudes $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, which are accessible via the keys <code>transition_amplitudes_real_part</code> and <code>transition_amplitudes_imag_part</code>, respectively. the covariance matrix of the real and imaginary parts of the $\{\mathcal{T}_k^\text{fit}\}$ for all $m_{\pi^-\pi^+}$ bins, which is accessible via the key <code>covariance_matrix</code>. Note that this matrix is real-valued and that its rows and columns are indexed such that $(\Re,\Im)$ pairs of the transition amplitudes are arranged with increasing $k$. the normalization factors $\mathcal{N}_a$ in Eq. (13) for all $m_{\pi^-\pi^+}$ bins, which are accessible via the key <code>normalization_factors</code>. the shape of the zero mode, i.e., the values of $\tilde\Delta_k$ for all $m_{\pi^-\pi^+}$ bins, which is accessible via the key <code>zero_mode_shape</code>. the reference wave, which is accessible via the key <code>reference_wave</code>. Note that this is always the $4^{++}1^+\rho(770)\pi G$ wave.
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.