Resonance production and $\pi \pi$ $S$-wave in ${\pi }^{-}+p\to {\pi }^{-}{\pi }^{-}{\pi }^{+}+p_{\mathbf{\text{recoil}}}$ at $190\mathrm{GeV}/c$

The COMPASS collaboration has collected the currently largest data set on diffractively produced ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ final states using a negative pion beam of $190\mathrm{GeV}/c$ momentum impinging on a stationary proton target. This data set allows for a systematic partial-wave analysis in 100 bins of three-pion mass, $0.5<m_{3\pi }<2.5\mathrm{GeV}/c^2$, and in 11 bins of the reduced four-momentum transfer squared, $0.1<t^{\prime }<1.0(\mathrm{GeV}/c{)}^2$. This two-dimensional analysis offers sensitivity to genuine one-step resonance production, i.e. the production of a state followed by its decay, as well as to more complex dynamical effects in nonresonant $3\pi$ production. In this paper, we present detailed studies on selected $3\pi$ partial waves with $J^{PC}=0^{-+}$, $1^{++}$, $2^{-+}$, $2^{++}$, and $4^{++}$. In these waves, we observe the well-known ground-state mesons as well as a new narrow axial-vector meson $a_1(1420)$ decaying into $f_0(980)\pi$. In addition, we present the results of a novel method to extract the amplitude of the ${\pi }^{-}{\pi }^{+}$ subsystem with $I^G{J}^{PC}=0^{+}{0}^{++}$ in various partial waves from the ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ data. Evidence is found for correlation of the $f_0(980)$ and $f_0(1500)$ appearing as intermediate ${\pi }^{-}{\pi }^{+}$ isobars in the decay of the known $\pi (1800)$ and ${\pi }_2(1880)$.

Figure 1

Diffractive dissociation of a beam pion on a target proton into the three-pion final state. The figure shows the excitation of an intermediate resonance $X^{-}$ via Pomeron exchange and its subsequent decay into $3\pi$.

Figure 2

Simplified scheme of the diffractive trigger. The main components are the beam trigger, which selects beam particles, and the RPD, which triggers on slow charged particles leaving the target. The veto system (red) rejects uninteresting events and consists of three parts: The veto hodoscopes, the sandwich, and the beam veto.

Figure 3

Panels (a) and (b) show the reconstructed beam energy $E_{\mathrm{beam}}$ after selection cuts (filled histograms). The open histogram in (a) represents the energy distribution without the RPD information. In the zoomed view (b), the vertical red lines indicate the accepted range.

Figure 4

Effect of the central-production (CP) veto. Panel (a): ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ invariant mass spectrum without the central-production veto (yellow histogram) together with the sample that is removed by the central-production veto (red histogram). The inset shows the same histogram with magnified ordinate scale. Note that the partial-wave analysis is performed only in the mass region of $0.5<m_{3\pi }<2.5\mathrm{GeV}/c^2$ indicated by the vertical red lines. Panel (b): ${\pi }^{-}{\pi }^{+}$ invariant mass distribution (two entries per event) of the sample that is cut away.

Figure 5

Final event sample after all selection cuts: (a) invariant mass spectrum of ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ in the range used in this analysis [see vertical lines in Fig. 4], (b) ${\pi }^{-}{\pi }^{+}$ mass distribution, (c) correlation of the two, (d) $t^{\prime }$ distribution with vertical lines indicating the range of $t^{\prime }$ values used in this analysis. The histograms in (b) and (c) have two entries per event. The labels indicate the position of major $3\pi$ and $2\pi$ resonances, the gray shaded areas in (a) the mass regions used to generate the Dalitz plots in Fig. 6.

Figure 6

(a) Dalitz plot in the mass regions of the $a_2(1320)$, which also includes the $a_1(1260)$, (b) around the ${\pi }_2(1670)$. The used $3\pi$ mass regions are indicated in Fig. 5. The dominant $\rho (770)$ $\pi$ decays of $a_1(1260)$ and $a_2(1320)$ are clearly visible. The ${\pi }_2(1670)$ region exhibits $\rho (770)$ $\pi$, $f_2(1270)$ $\pi$, and $f_0(980)$ $\pi$ decay modes.

Figure 7

The decay of $X^{-}$, as described in the isobar model, is assumed to proceed via an intermediate ${\pi }^{-}{\pi }^{+}$ state $\xi$, the so-called isobar.

Figure 8

Definition of the Gottfried-Jackson reference frame (GJ) in the $X$ rest system and of the helicity reference frame (HF) in the ${\xi }^0$ rest system as they are used to analyze the angular distributions of the decays $X^{-}\to {\xi }^0+{\pi }^{-}$ and ${\xi }^0\to {\pi }^{-}+{\pi }^{+}$, respectively. Unit vectors are indicated by a circumflex.

Figure 9

Example of a $3\pi$ angular distribution observed in the mass region $1.6<m_{3\pi }<1.7\mathrm{GeV}/c^2$ around the ${\pi }_2(1670)$ indicated by vertical red lines in the upper left panel. The main decay of this resonance is through the $f_2(1270)$ isobar, which is a $J^{PC}=2^{++}$ state, decaying into ${\pi }^{-}{\pi }^{+}$ in a $D$-wave. The $f_2(1270)$ and the bachelor pion are in a relative $S$ or $D$-wave.

Figure 10

Parametrization of the $[\pi \pi {]}_S$ isobar amplitude based on the $M$ solution described in Ref. [41]. Panel (a) shows the intensity, (b) the phase, and (c) the corresponding Argand diagram. The open circles in the latter are evenly spaced in $m_{{\pi }^{-}{\pi }^{+}}$ in $20\mathrm{MeV}/c^2$ intervals. Arbitrary units are denoted by “a. u.”.

Figure 11

Correlation of the reduced four-momentum transfer squared $t^{\prime }$ and the invariant mass $m_{3\pi }$ of the $3\pi$ system in the analyzed kinematic region. The partial-wave analysis is performed independently in 100 equidistant $m_{3\pi }$ bins with a width of $20\mathrm{MeV}/c^2$, each of which is subdivided into eleven nonequidistant $t^{\prime }$ bins. The latter are indicated by the dashed horizontal lines. The numerical values for the $t^{\prime }$ bins are listed in Table 4.

Figure 12

The $t^{\prime }$-summed intensity of the coherent sum of all negative-reflectivity waves (a) and, for comparison, together with the total intensity of all partial waves (b).

Figure 13

The $t^{\prime }$-summed intensity of the flat wave (a) and, for comparison, together with the total intensity of all partial waves (b).

Figure 14

The $t^{\prime }$-summed intensities of major waves with spin projection $M=0$ showing in (a) the $a_1(1260)$ and in (b) the ${\pi }_2(1670)$. The shaded regions indicate the mass intervals that are integrated over to generate the $t^{\prime }$ spectra [see Fig. 32].

Figure 15

The $t^{\prime }$-summed intensity of waves with spin projection $M=1$ showing in (a) the $a_1(1260)$, in (b) and (c) the $a_2(1320)$, in (d) the ${\pi }_2(1670)$, and in (e) and (f) the $a_4(2040)$. The shaded regions indicate the mass intervals that are integrated over to generate the $t^{\prime }$ spectra [see Figs. 32, 33, 33, and 34].

Figure 16

Panel (a): $t^{\prime }$-summed intensity of $2^{++}{2}^{+}\rho (770)$ $\pi$ $D$ wave with spin projection $M=2$ and the $a_2(1320)$ peak. Panel (b): $t^{\prime }$-summed intensity of $1^{++}{0}^{+}{f}_2(1270)$ $\pi$ $P$ wave with spin projection $M=0$. In this wave, the mass regions below $1.5\mathrm{GeV}/c^2$ and above $2.0\mathrm{GeV}/c^2$ (shown by gray points) are sensitive to the truncation of the partial-wave expansion series (see Sec. 4f). In both waves, the shaded regions indicate the mass intervals that are integrated over to generate the $t^{\prime }$ spectra [see Fig. 33].

Figure 17

The intensities of the $1^{++}{0}^{+}\rho (770)$ $\pi$ $S$ and $2^{++}{1}^{+}\rho (770)$ $\pi$ $D$ waves in two different $t^{\prime }$ regions. Upper row: low $t^{\prime }$; Lower row: high $t^{\prime }$. The shaded regions indicate the mass intervals that are integrated over to generate the $t^{\prime }$ spectra.

Figure 18

Same as Fig. 17, but for the $2^{-+}{0}^{+}{f}_2(1270)$ $\pi$ $S$ and $4^{++}{1}^{+}\rho (770)$ $\pi$ $G$ waves.

Figure 19

Example for a nonresonant production process for the $3\pi$ final state as proposed by Deck [25].

Figure 20

Same as Fig. 17, but for the $1^{++}{0}^{+}{f}_2(1270)$ $\pi$ $P$ and $2^{++}{1}^{+}{f}_2(1270)$ $\pi$ $P$ waves. In the former wave, the mass regions below $1.5\mathrm{GeV}/c^2$ and above $2.0\mathrm{GeV}/c^2$ (shown by gray points) are sensitive to the truncation of the partial-wave expansion series (see Sec. 4f).

Figure 21

Same as Fig. 17, but for the $2^{-+}{0}^{+}\rho (770)$ $\pi$ $F$ and $4^{++}{1}^{+}{f}_2(1270)$ $\pi$ $F$ waves.

Figure 22

The $t^{\prime }$-summed intensities of $2^{-+}{M}^{+}{f}_2(1270)$ $\pi$ $L$ waves in panels (a), (b), and (c) and of $2^{-+}{0}^{+}\rho (770)$ $\pi$ $F$ wave in panel (d). The ${\pi }_2(1670)$ dominates the waves shown in (a) and (c), the ${\pi }_2(1880)$ the one shown in (b). Both resonances appear in the wave shown in (d). Upper row: comparison of same decay mode but different orbital angular momenta $L=0$ and 2 in the decay; Left column: comparison of same decay mode but different spin projections $M=0$ and 1. The shaded regions indicate the mass intervals that are integrated over to generate the $t^{\prime }$ spectra discussed in Sec. 5b.

Figure 23

Intensity of the coherent sum of all $f_0(980)$ $\pi$ waves with positive reflectivity, summed over all $t^{\prime }$ bins.

Figure 24

Intensity of the $0^{-+}{0}^{+}[\pi \pi {]}_S\pi S$ wave in two different $t^{\prime }$ regions. (a) low $t^{\prime }$; (b) high $t^{\prime }$. The shaded regions indicate the mass intervals that are integrated over to generate the $t^{\prime }$ spectra.

Figure 25

The $t^{\prime }$-summed intensity for selected waves with $\pi \pi S$-wave isobars. Left column: waves with the narrow $f_0(980)$ isobar; Right column: waves with the broad $[\pi \pi {]}_S$ isobar. The $J^{PC}=0^{-+}$ waves in the top row show the $\pi (1800)$. The structure at $1.2\mathrm{GeV}/c^2$ is probably mainly of nonresonant origin. The $2^{-+}$ intensities in the center row exhibit a complicated destructive interference pattern around $1.8\mathrm{GeV}/c^2$. The bottom row shows an enhancement in the region of the $a_1(1260)$ in the $1^{++}{0}^{+}[\pi \pi {]}_S\pi P$ wave and a new state, the $a_1(1420)$, in $1^{++}{0}^{+}{f}_0(980)$ $\pi$ $P$. The shaded regions indicate the mass intervals that are integrated over to generate the $t^{\prime }$ spectra (see Figs. 35 and 36).

Figure 26

Distributions of the five phase-space variables used to calculate the decay amplitudes shown for different $3\pi$ mass bins in the region $0.127<t^{\prime }<0.144(\mathrm{GeV}/c{)}^2$. Each distribution is shown for real data (blue points) and for weighted Monte Carlo events (red bands), which are generated according to the fit result. Each distribution is normalized to its maximum deviation from its average $y$ value. Along the ordinate, the average $y$ values for the distributions (indicated by gray lines) are shifted equidistantly with respect to one another. The $3\pi$ mass ranges from 1.0 to $2.4\mathrm{GeV}/c^2$ and is subdivided into $40\mathrm{MeV}/c^2$ wide bins. The central values of selected $3\pi$ mass bins are given as labels in the $\cos {\vartheta }_{\mathrm{GJ}}$ distribution.

Figure 27

Comparison of kinematic distributions of weighted Monte Carlo events, generated according to the fit result, with the corresponding real-data distributions in the low-$t^{\prime }$ region $0.127<t^{\prime }<0.144(\mathrm{GeV}/c{)}^2$. Panel (a) shows the acceptance-corrected $3\pi$ invariant mass distribution. The other panels show kinematic distributions in the mass interval $1.6<m_{3\pi }<1.8\mathrm{GeV}/c^2$ around the ${\pi }_2(1670)$, which is indicated by vertical red lines in (a). (b) invariant mass spectrum of the ${\pi }^{-}{\pi }^{+}$ subsystem; (c) distribution of the Gottfried-Jackson angles for real data; (e) ratio of the real-data distribution in (c) and that of the weighted Monte Carlo. Panels (d) and (f) show the respective distributions for the helicity angles. Note that (b), (c), and (d) have two entries per event.

Figure 28

Same as Fig. 26, but for the high-$t^{\prime }$ region $0.449<t^{\prime }<0.724(\mathrm{GeV}/c{)}^2$.

Figure 29

Same as Fig. 27, but for the high-$t^{\prime }$ region $0.449<t^{\prime }<0.724(\mathrm{GeV}/c{)}^2$ and the mass slice $1.2<m_{3\pi }<1.4\mathrm{GeV}/c^2$ around the $a_2(1320)$.

Figure 30

The $t^{\prime }$ dependence of the measured $3\pi$ invariant mass spectrum and vice versa.

Figure 31

Result of a fit to the $t^{\prime }$ dependence of events from the diffractive-dissociation reaction ${\pi }^{-}+p\to {\pi }^{-}{\pi }^{-}{\pi }^{+}+p$, as measured by COMPASS (filled circles) and by the ACCMOR collaboration [16] (open circles). For each $m_{3\pi }$ bin, the $t^{\prime }$ spectrum was fit using a double-exponential model [see Eq. (44)]. Panel (a) shows the mass dependence of the two slope parameters $b_1$ (upper points) and $b_2$ (lower points). Note the extended mass range of the present measurement as compared to the ACCMOR data. Panel (b) shows the ratio $I_1/I_2$ of the integrated exponential contributions, see Eq. (45).

Figure 32

The $t^{\prime }$ dependence of the intensity of the $1^{++}{M}^{+}\rho (770)$ $\pi$ $S$ waves with spin projections $M=0$ (a) and $M=1$ (b), integrated over the mass region around the $a_1(1260)$ as indicated by the shaded regions in Figs. 14 and 15. The solid red curve in (a) represents a double-exponential fit using Eq. (46), the one in (b) a single-exponential fit with parameter $A_2=0$. In (a), the two exponential components are shown by the dotted lines. In (b), the fitted $t^{\prime }$ range is indicated by the solid curve; the extrapolation is shown as dashed curve. See text for details.

Figure 33

The $t^{\prime }$ dependence of the intensity of three $2^{++}$ waves with different isobars and different spin-projections $M$, integrated over the mass region around the $a_2(1320)$ as indicated by the shaded regions in Figs. 15, 15, and 16. The solid red curves represent double-exponential fits using Eq. (46); the fit components are shown as dotted curves. See text for details.

Figure 34

The $t^{\prime }$ dependence of the intensity of the $4^{++}{1}^{+}\rho (770)$ $\pi$ $G$ and $f_2(1270)\pi F$ waves, integrated over the mass interval around the $a_4(2040)$ as indicated by the shaded regions in Figs. 15 and 15. The solid red curves represent double-exponential fits using Eq. (46); the fit components are shown as dotted curves. See text for details.

Figure 35

The $t^{\prime }$ dependence of the intensity of the $0^{-+}{0}^{+}[\pi \pi {]}_S\pi S$ wave in two different mass regions, which correspond to the two peaks that are indicated by the shaded bands in Fig. 25. The red curves represent single-exponential fits using Eq. (46) with $A_2=0$. The fitted $t^{\prime }$ ranges are indicated by the solid curves; the extrapolations are shown as dashed curves. See text for details.

Figure 36

The $t^{\prime }$ dependences of the intensities of the $1^{++}{0}^{+}$ waves with $[\pi \pi {]}_S$ (a) and $f_0(980)$ (b) isobars taken around the peak region in the respective wave as indicated by the shaded bands in Figs. 25 and 25. The red curves represent single-exponential fits using Eq. (46) with $A_2=0$. The fitted $t^{\prime }$ ranges are indicated by the solid curves; the extrapolations are shown as dashed curves. See text for details.

Figure 37

Intensities of the coherent sum of partial waves with the same quantum numbers but different $0^{++}$ isobars, as obtained in the conventional analysis scheme with fixed isobar amplitudes (blue), and the corresponding intensities obtained from the freed-isobar fit (red). Left column: low $t^{\prime }$; Right column: high $t^{\prime }$.

Figure 38

Intensity of the $[\pi \pi {]}_{0^{++}}$ $\pi$ $S$-wave component of the $J^{PC}{M}^{ϵ}=0^{-+}{0}^{+}$ amplitude resulting from the freed-isobar fits in four $t^{\prime }$ bins. Left column: two-dimensional representation of the intensity of the $0^{-+}{0}^{+}[\pi \pi {]}_{0^{++}}$ $\pi$ $S$ wave as a function of $m_{{\pi }^{-}{\pi }^{+}}$ and $m_{3\pi }$. Central and right columns: intensity as a function of $m_{3\pi }$ summed over selected $m_{{\pi }^{-}{\pi }^{+}}$ intervals around the $f_0(980)$ (center) and around the $f_0(1500)$ (right) as indicated by pairs of horizontal dashed lines in the left column. The vertical dashed lines indicate the centers of the $m_{3\pi }$ bins discussed in Sec. 6d.

Figure 39

Same as Fig. 38, but for the $1^{++}{0}^{+}[\pi \pi {]}_{0^{++}}$ $\pi$ $P$ wave, except that the right column shows the $m_{{\pi }^{-}{\pi }^{+}}$ interval around the $f_0(980)$.

Figure 40

Same as Fig. 38, but for the $2^{-+}{0}^{+}[\pi \pi {]}_{0^{++}}$ $\pi$ $D$ wave.

Figure 41

The freed $[\pi \pi {]}_{0^{++}}$ amplitude in the $0^{-+}{0}^{+}[\pi \pi {]}_{0^{++}}$ $\pi$ $S$ wave in the range $0.194<t^{\prime }<0.326(\mathrm{GeV}/c{)}^2$ for three intervals in $m_{3\pi }$; below (top row), at (center row), and above the $\pi (1800)$ (bottom row). Left column: intensities as a function of $m_{{\pi }^{-}{\pi }^{+}}$; Right column: Argand diagrams. The crosses with error bars are the result of the mass-independent fit. The numbers in the Argand diagrams show the corresponding mass value of the ${\pi }^{-}{\pi }^{+}$ system. The data points are connected by lines in order to indicate the order. The line segments highlighted in blue correspond to the $m_{{\pi }^{-}{\pi }^{+}}$ ranges around the $f_0(980)$ from 960 to $1000\mathrm{MeV}/c^2$ and, if phase space permits, around the $f_0(1500)$ from 1400 to $1560\mathrm{MeV}/c^2$. The $2\pi$ mass is binned in $10\mathrm{MeV}/c^2$ wide intervals around the $f_0(980)$ and in $40\mathrm{MeV}/c^2$ wide slices elsewhere. The phase of the Argand diagrams is fixed by the $1^{++}{0}^{+}\rho (770)$ $\pi$ $S$ wave.

Figure 42

Same as Fig. 41, but for the $1^{++}{0}^{+}[\pi \pi {]}_{0^{++}}$ $\pi$ $P$ wave in three $m_{3\pi }$ bins around the $a_1(1420)$.

Figure 43

Same as Fig. 41, but for the $2^{-+}{0}^{+}[\pi \pi {]}_{0^{++}}$ $\pi$ $D$ wave in three $m_{3\pi }$ bins around the ${\pi }_2(1880)$.

Figure 44

Comparison of $t^{\prime }$-summed partial-wave intensities obtained from the standard mass-independent fit with rank $N_r=1$ of the spin-density matrix for waves with $ϵ=+1$ and rank 2 for the ones with $ϵ=-1$ (blue/black) with the intensities from a fit with rank 2 for both sectors (red/gray).

Figure 45

Comparison of $t^{\prime }$-summed partial-wave intensities obtained from the mass-independent fits with (blue/black) and without (red/gray) $ϵ=-1$ waves.

Figure 46

Comparison of $t^{\prime }$-summed partial-wave intensities obtained from mass-independent fits using two different parametrizations for the $\rho (770)$ isobar amplitude: a Breit-Wigner with Eq. (40) (blue/black) and one with Eq. (32) (red/gray).

Figure 47

Comparison of $t^{\prime }$-summed partial-wave intensities obtained from mass-independent fits using two different parametrizations for the $f_0(980)$ isobar amplitude: Flatté parametrization [Eq. (43), blue/black] and modified $S$-wave Breit-Wigner [Eq. (b1) with Eq. (b2), red/gray].

Figure 48

Comparison of $t^{\prime }$-summed partial-wave intensities obtained from mass-independent fits for standard event selection (blue/black) and a looser event selection (red/gray), where particle identification and central-production rejection have not been applied.

Figure 49

Comparison of $t^{\prime }$-summed partial-wave intensities obtained from mass-independent fits using two different $t^{\prime }$ binnings: 11 $t^{\prime }$ bins (blue/black) and 22 $t^{\prime }$ bins (red/gray).

Figure 50

Detection efficiency for ${\pi }^{-}{\pi }^{-}{\pi }^{+}$ phase-space events in different regions of $m_{3\pi }$ and $t^{\prime }$. Each graph shows the detection efficiency as a function of the angles, $\cos {\vartheta }_{\mathrm{GJ}}$ (abscissa, from $-1$ to $+1$) and ${\varphi }_{\mathrm{TY}}$ (ordinate, from $-180°$ to $+180°$), of the isobar in the Gottfried-Jackson frame.

Figure 51

Same as Fig. 50, but for the detection efficiency as a function of the angles, $\cos {\vartheta }_{\mathrm{HF}}$ and ${\varphi }_{\mathrm{HF}}$, of the ${\pi }^{-}$ in the helicity frame.