DOI: 10.17182/hepdata.82958.v1/t1
title: "Transition Amplitudes" of "Light isovector resonances in $\pi^- p \to \pi^-\pi^-\pi^+ p$ at 190 GeV/${\it c}$"

description: Real and imaginary parts of the normalized transition amplitudes $\mathcal{T}_a$ of the 14 selected partial waves in the 1100 $(m_{3\pi}, t')$ cells (see Eq. (12) in the paper). The wave index $a$ represents the quantum numbers that uniquely define the partial wave. The quantum numbers are given by the shorthand notation $J^{PC} M^\varepsilon [$isobar$] \pi L$. We use this notation to label the transition amplitudes in the column headers. The $m_{3\pi}$ values that are given in the first column correspond to the bin centers. Each of the 100 $m_{3\pi}$ bins is 20 MeV/$c^2$ wide. Since the 11 $t'$ bins are non-equidistant, the lower and upper bounds of each $t'$ bin are given in the column headers.

The transition amplitudes define the spin-density matrix elements $\varrho_{ab}$ for waves $a$ and $b$ according to Eq. (18). The spin-density matrix enters the resonance-model fit via Eqs. (33) and (34). The transition amplitudes are normalized via Eqs. (9), (16), and (17) such that the partial-wave intensities $\varrho_{aa} = |\mathcal{T}_a|^2$ are given in units of acceptance-corrected number of events. The relative phase $\Delta\phi_{ab}$ between two waves $a$ and $b$ is given by $\arg(\varrho_{ab}) = \arg(\mathcal{T}_a) - \arg(\mathcal{T}_b)$. Note that only relative phases are well-defined. The phase of the $1^{++}0^+ \rho(770) \pi S$ wave was set to $0^\circ$ so that the corresponding transition amplitudes are real-valued. In the PWA model, some waves are excluded in the region of low $m_{3\pi}$ (see paper and [Phys. Rev. D 95, 032004 (2017)] for a detailed description of the PWA model). For these waves, the transition amplitudes are set to zero.

The tables with the covariance matrices of the transition amplitudes for all 1100 $(m_{3\pi}, t')$ cells can be downloaded via the 'Additional Resources' for this table.
description: Selected partial-wave amplitudes from a partial-wave analysis (PWA) of exclusive events from the diffractive-dissociation reaction $\pi^- p \to \pi^- \pi^- \pi^+ p$ with a 191 GeV/$c$ pion beam and a stationary proton target. The PWA was performed in 100 bins of the three-pion mass $m_{3\pi}$ in the range between 0.5 and 2.5 GeV/$c^2$, and simultaneously in 11 non-equidistant bins of the reduced four-momentum transfer squared $t'$ (see Eq. (2) in the paper) in the range between 0.1 and 1.0 (GeV/$c$)$^2$. The PWA model consists of 88 waves (see paper and [Phys. Rev. D 95, 032004 (2017)] for a detailed description of the PWA model). The waves are uniquely defined by the quantum numbers of the 3-pion system (spin $J$, parity $P$, $C$-parity, spin projection $M \geq 0$, and reflectivity $\varepsilon = \pm 1$) and the decay chain ($\pi^-\pi^+$ isobar and orbital angular momentum $L$ between the isobar and the bachelor pion). To identify waves, we use the shorthand notation $J^{PC} M^\varepsilon [$isobar$] \pi L$.

Out of the 88 waves, the following subset of 14 waves was selected for a resonance-model fit:
 $0^{-+}0^+ f_0(980) \pi S$
 $1^{++}0^+ \rho(770) \pi S$
 $1^{++}0^+ f_0(980) \pi P$
 $1^{++}0^+ f_2(1270) \pi P$
 $1^{-+}1^+ \rho(770) \pi P$
 $2^{++}1^+ \rho(770) \pi D$
 $2^{++}2^+ \rho(770) \pi D$
 $2^{++}1^+ f_2(1270) \pi P$
 $2^{-+}0^+ \rho(770) \pi F$
 $2^{-+}0^+ f_2(1270) \pi S$
 $2^{-+}1^+ f_2(1270) \pi S$
 $2^{-+}0^+ f_2(1270) \pi D$
 $4^{++}1^+ \rho(770) \pi G$
 $4^{++}1^+ f_2(1270) \pi F$
The resonance-model fit uses Breit-Wigner amplitudes to describe the $m_{3\pi}$ dependence of the amplitudes of the 14 waves simultaneously in the 11 $t'$ bins using 11 iso-vector light-meson states with $J^{PC} = 0^{-+}$, $1^{++}$, $2^{++}$, $2^{-+}$, $4^{++}$, and spin-exotic $1^{-+}$ quantum numbers (see paper for details).

The provided data tables contain all information that is necessary to repeat the resonance-model fit. The data set consists of three kinds of data tables:

(i) a table with the real and imaginary parts of the normalized transition amplitudes $\mathcal{T}_a$ of the 14 selected partial waves in the 1100 $(m_{3\pi}, t')$ cells (see Eq. (12) in the paper). Together with the reflectivity quantum number $\varepsilon$, which is $+1$ for all 14 waves, the wave index $a = J^{PC} M [$isobar$] \pi L$ uniquely identifies a partial wave. The transition amplitudes define the spin-density matrix elements $\varrho_{ab}$ for waves $a$ and $b$ according to Eq. (18). The transition amplitudes are normalized via Eqs. (9), (16), and (17) such that the partial-wave intensities $\varrho_{aa} = |\mathcal{T}_a|^2$ are given in units of acceptance-corrected number of events. In the paper, partial-wave intensities and relative phases are shown as a function of $m_{3\pi}$ to discuss resonance signals and to illustrate the agreement of the model with the data (see e.g. Fig. 10ff. and the supplemental material of the paper).

(ii) tables with the covariance matrices of the real and imaginary parts of the transition amplitudes $\mathcal{T}_a$; one matrix for each of the 1100 $(m_{3\pi}, t')$ cells. For each covariance matrix, the center of the $m_{3\pi}$ bin and the lower and upper bounds of the $t'$ bin are given in the table header. The partial-wave labels used for the covariance matrix elements are identical to the labels in the column headers of the table of the transition amplitudes in (i). The tables are stored as yaml files in the HEPData format in the 'covariance_matrices.tar.gz' file, which is provided as an additional resource for the table of the transition amplitudes in (i). In addition to the 1100 files for the covariance-matrix tables, the 'covariance_matrices.tar.gz' file contains the file 'submission.yaml', which lists the meta-information for the covariance-matrix tables.

(iii) a table with the decay phase-space volume $I_{aa}$ (see Eq. (6)) for the 14 selected partial waves in the $m_{3\pi}$ range from 0.5 to 2.5 GeV/$c^2$ in steps of 10 MeV/$c^2$. The values are given in arbitrary units, normalized such that $I_{aa}(m_{3\pi} = 2.5~\text{GeV}/c^2) = 1$. $I_{aa}$ appears in the resonance model in Eqs. (19) and (20). The partial-wave labels used for the decay phase-space volume are identical to the labels in the column headers of the table of the transition amplitudes in (i).