Partial wave analysis of the ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ and ${\pi }^{-}{\pi }^0{\pi }^0$ systems and the search for a $J^{PC}=1^{-+}$ meson

A partial wave analysis (PWA) of the ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ and ${\pi }^{-}{\pi }^0{\pi }^0$ systems produced in the reaction ${\pi }^{-}p\to (3\pi {)}^{-}p$ at $18\mathrm{GeV}/c$ was carried out using an isobar model assumption. This analysis is based on 3.0 M ${\pi }^{-}{\pi }^0{\pi }^0$ events and 2.6 M ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ events and shows production of the $a_1(1260)$, $a_2(1320)$, ${\pi }_2(1670)$, and $a_4(2040)$ resonances. Results of detailed studies of the stability of partial wave fits are presented. An earlier analysis of 250 K ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ events from the same experiment showed possible evidence for a $J^{PC}=1^{-+}$ exotic meson with a mass of $\sim 1.6\mathrm{GeV}/c^2$ decaying into $\rho \pi$. In this analysis of a higher statistics sample of the $(3\pi {)}^{-}$ system in two charged modes we find no evidence of an exotic meson.

Figure 1

E852 apparatus and the MPS at BNL.

Figure 2

Scatterplot of one di-photon mass against the other di-photon mass (three such pairings per event) for events satisfying the ${\pi }^{-}{\pi }^0{\pi }^{0}p$ final state trigger and with four reconstructed photons in the LGD. The inset shows the di-photon mass spectrum with a fit to a Gaussian plus a polynomial. The ${\pi }^0$ mass resolution is $10.4\mathrm{MeV}/c^2$.

Figure 3

Distribution in missing mass squared recoiling against the $3\pi$ system for the (left) ${\pi }^{-}{\pi }^0{\pi }^0$ mode and the (right) ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ mode. The unshaded distribution is before any cuts. The shaded distribution is for events that survived the confidence level cuts and all the cuts listed in Tables . The missing mass is calculated before the kinematic fitting. Results of fits of the shaded distributions to a Gaussian are shown.

Figure 4

Distribution in confidence level after kinematic fitting for the (left) ${\pi }^{-}{\pi }^0{\pi }^0$ mode and the (right) ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ mode. Events were required to have a confidence level greater than 0.20.

Figure 5

Distribution in the ${\pi }^{+}p$ effective mass for events in the charged mode satisfying the initial cuts. The shaded distribution shows the events accepted by the $m_{{\pi }^{+}p}>1.5\mathrm{GeV}/c^2$ requirement.

Figure 6

(a) Uncorrected distribution in the square of the four-momentum transfer $t$, from the incoming ${\pi }^{-}$ to the outgoing $(3\pi {)}^{-}$; (b) $(3\pi {)}^{-}$ effective mass distribution; (c) ${\pi }^{-}{\pi }^0$ effective mass distribution; and (d) ${\pi }^0{\pi }^0$ effective mass distribution for the neutral mode.

Figure 7

(a) Uncorrected distribution in the square of the four-momentum transfer $t$, from the incoming ${\pi }^{-}$ to the outgoing $(3\pi {)}^{-}$; (b) $(3\pi {)}^{-}$ effective mass distribution; (c) ${\pi }^{-}{\pi }^{-}$ effective mass distribution; and (d) ${\pi }^{-}{\pi }^{+}$ effective mass distribution for the charged mode.

Figure 8

Uncorrected $(3\pi {)}^{-}$ mass distribution for the (a) neutral mode and (b) charged mode for three $t$ regions: $t_1$, $t_6$, and $t_8$ as defined in Table .

Figure 9

Uncorrected $2\pi$ mass distributions for selected regions of the $3\pi$ effective mass. (a) ${\pi }^{-}{\pi }^0$ mass in the $a_2$ region; (b) ${\pi }^0{\pi }^0$ mass in the ${\pi }_2$ region; (c) ${\pi }^{-}{\pi }^{+}$ mass in the $a_2$ region; and (d) ${\pi }^{-}{\pi }^{+}$ mass in the ${\pi }_2$ region.

Figure 10

(a) $3\pi$ mass acceptance as a function of $3\pi$ mass; (b) $t$ acceptance as a function of $t$; (c) $\delta m$—mass resolution as a function of $3\pi$ mass; and (d) $\delta t$—$t$ resolution as a function of $t$ mass—all for both neutral and charged modes.

Figure 11

Acceptance-corrected ${\pi }^{-}{\pi }^0{\pi }^0$ and ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ effective mass distributions for two regions in $t$: (a) $t1$ and (b) $t8$. These $t$ regions are defined in Table .

Figure 12

Partial wave analysis of the $3\pi$ system in the isobar model. The state $X$ is characterized by spin ($J$), parity ($P$), and charge conjugation ($C$). It decays at point 1 into a di-pion resonance $R_{\pi \pi }$ (the isobar) and a bachelor $\pi$. The di-pion has spin $S$. The angular momentum between the isobar and the bachelor $\pi$ is $L$. At point 2 the di-pion resonance decays into ${\pi }_a$ and ${\pi }_b$. More details are given in the text.

Figure 13

The decay angular structure for a particular partial wave where the resonance ($X$) has $J^{PC}=2^{-+}$ and decays into $f_2(1275)\pi$, where the isobar is the $f_2(1275)$ and the angular momentum between the isobar and bachelor $\pi$ is $L=2$. (a) The correlation between the decay angles (${\theta }_{GJ}$, ${\varphi }_{GJ}$) of the resonance in the Gottfried-Jackson frame and (b) The correlation between the decay angles (${\theta }_H$, ${\varphi }_H$) of the isobar in the helicity frame.

Figure 14

The acceptance in decay angles for the $3\pi$ effective mass in the ${\pi }_2(1670)$ mass region. The angles $\theta$ and $\varphi$ are measured in the Gottfried-Jackson frame for the resonance decay and in the helicity frame for the isobar decay. Acceptances are shown for (a) $\cos \theta$ and (b) $\varphi$ for both the charged and neutral modes of the $3\pi$ system.

Figure 15

Partial wave intensity (filled circles) as a function of $3\pi$ mass (lower plot) shown along with the change in $\ln \mathcal{L}$ (shown above the intensity plot—read the right axis scale) that results when this partial wave is removed from the parent set of partial waves. (a) $2^{-+}{1}^{+}{f}_2\pi D$ wave—a significant wave and (b) $2^{-+}{1}^{-}{f}_2\pi D$ wave—an insignificant wave.

Figure 16

Comparison of the (a) $H(201)$ and (b) $H(420)$ moments as computed directly from data and from PWA fits for the low and high-wave sets for the ${\pi }^{-}{\pi }^0{\pi }^0$ channel. In (c) the ${\Delta }^2$ (differences squared divided by errors squared summed over all mass bins and divided by the number of mass bins) is shown for various moments.

Figure 17

Comparison of the (a) $H(201)$ and (b) $H(420)$ moments as computed directly from data and from PWA fits for the low and high-wave sets for the ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ channel. In (c) the ${\Delta }^2$ (differences squared divided by errors squared summed over all mass bins and divided by the number of mass bins) is shown for various moments.

Figure 18

The sum of partial waves and the background wave for both the charged and neutral modes.

Figure 19

Coherent sum of the $1^{++}$ $\rho \pi$-waves for the (upper) ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ and (lower) ${\pi }^{-}{\pi }^0{\pi }^0$ modes. For each the results from the high-wave set and low-wave set are shown.

Figure 20

$2^{++}{1}^{+}$ $D$-wave $\rho \pi$ partial wave intensity for the (upper) ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ and (lower) ${\pi }^{-}{\pi }^0{\pi }^0$ modes. For each the results from the high-wave set and low-wave set are shown.

Figure 21

$2^{-+}{0}^{+}$ $S$-wave $f_2\pi$ partial wave intensity for the (upper) ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ and (lower) ${\pi }^{-}{\pi }^0{\pi }^0$ modes. For each the results from the high-wave set and low-wave set are shown.

Figure 22

$2^{-+}{0}^{+}$ $D$-wave $f_2\pi$ partial wave intensity for the (upper) ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ and (lower) ${\pi }^{-}{\pi }^0{\pi }^0$ modes. For each the results from the high-wave set and low-wave set are shown.

Figure 23

The $4^{++}{1}^{+}$ $G$-wave $\rho \pi$ partial wave intensity for the ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ and ${\pi }^{-}{\pi }^0{\pi }^0$ modes.

Figure 24

(a) The $1^{-+}{1}^{+}$ $P$-wave $\rho \pi$ partial wave in the neutral mode (${\pi }^{-}{\pi }^0{\pi }^0$) for the high-wave set PWA and the low-wave set PWA and (b) the phase difference $\Delta \Phi$ between the $2^{++}$ and $1^{-+}$ for the two wave sets.

Figure 25

(a) The $1^{-+}{1}^{+}$ $P$-wave $\rho \pi$ partial wave in the charged mode (${\pi }^{-}{\pi }^{-}{\pi }^{+}$) for the high-wave set PWA and the low-wave set PWA and (b) the phase difference $\Delta \Phi$ between the $2^{++}$ and $1^{-+}$ for the two wave sets.

Figure 26

The intensities for the $2^{-+}{0}^{+}$ $F$-wave $\rho \pi$ (open circles), $2^{-+}{1}^{+}$ $F$-wave $\rho \pi$ (filled black circles), and $2^{-+}{1}^{+}$ $P$-wave $\rho \pi$ (filled gray circles) waves for the neutral and charged modes. These waves were included in the high-wave set but not in the low-wave set.

Figure 27

The positive reflectivity $1^{-+}$ $P$-wave $\rho \pi$ intensity for the neutral mode for the high-wave set (filled circles), the modified high-wave set (filled squares), and the low-wave set (open circles). In the modified high-wave set the two $2^{-+}$ $F$-wave $\rho \pi$ waves and the $2^{-+}{1}^{+}$ $P$-wave $\rho \pi$ wave were removed.

Figure 28

The positive reflectivity $1^{-+}$ $P$-wave $\rho \pi$ intensity for the charged mode for the high-wave set (filled circles), the modified high-wave set (filled squares), and the low-wave set (open circles). In the modified high-wave set the two $2^{-+}$ $F$-wave $\rho \pi$ waves and the $2^{-+}{1}^{+}$ $P$-wave $\rho \pi$ wave were removed.

Figure 29

The $2^{++}$ intensity for charged $3\pi$ mode for the high-wave set (filled circles) and low-wave set (open circles). The $4^{++}$ intensity (filled triangles) for the high-wave set is also shown.

Figure 30

The change in $\ln \mathcal{L}$ resulting from the removal of the $1^{-+}{1}^{+}(\rho \pi )$ $P$-wave partial wave (solid line) and other $2^{-+}$ waves (dashed line) for the neutral mode (top) and charged mode (bottom). See the text for the sign convention and discussion.

Figure 31

Variation of the $1^{-+}{1}^{+}$ $P$-wave $\rho \pi$ intensity for the charged mode for the $12t$-bins of Table  increasing in $|t|$ from left to right and top to bottom.

Figure 32

Variation of the $2^{-+}{0}^{+}$ $S$-wave $f_2\pi$ intensity for the charged mode for the $12t$-bins of Table , increasing in $|t|$ from left to right and top to bottom.

Figure 33

Variation of the phase difference between the $1^{-+}$ and $2^{-+}$ amplitudes, whose intensities are shown in Figs. 31, 32, respectively, for the charged mode for the $12t$-bins of Table , increasing in $|t|$ from left to right and top to bottom.

Figure 34

Variation of the $2^{++}{1}^{+}$ $D$-wave $\rho \pi$ intensity for the charged mode for the $12t$-bins of Table , increasing in $|t|$ from left to right and top to bottom.

Figure 35

Variation of the $4^{++}{1}^{+}$ $G$-wave $\rho \pi$ intensity for the charged mode for the $12t$-bins of Table , increasing in $|t|$ from left to right and top to bottom.

Figure 36

$t$ dependence of the $a_2(1320)$ as measured in this experiment for the charged mode (filled circles) and neutral mode (open circles) along with measurements of V. Chabaud et al. [25] of ${\pi }^{-}p\to a_2^{-}(1320)p$ at $18.8\mathrm{GeV}/c$.