Study of the process $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0$ using initial state radiation with BABAR

The process $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0\gamma$ is studied at a center-of-mass energy near the $\mathrm{\Upsilon }(4S)$ resonance using a data sample of $469{\mathrm{fb}}^{-1}$ collected with the BABAR detector at the PEP-II collider. We have performed a precise measurement of the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0$ cross section in the center-of-mass energy range from 0.62 to 3.5 GeV. In the energy regions of the $\omega$ and $\varphi$ resonances, the cross section is measured with a systematic uncertainty of 1.3%. The leading-order hadronic contribution to the muon magnetic anomaly calculated using the measured $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0$ cross section from threshold to 2.0 GeV is $(45.86\pm 0.14\pm 0.58)\times {10}^{-10}$. From the fit to the measured $3\pi$ mass spectrum we have determined the resonance parameters $\mathrm{\Gamma }(\omega \to e^{+}{e}^{-})\mathcal{B}(\omega \to {\pi }^{+}{\pi }^{-}{\pi }^0)=(0.5698\pm 0.0031\pm 0.0082)\mathrm{keV}$, $\mathrm{\Gamma }(\varphi \to e^{+}{e}^{-})\mathcal{B}(\varphi \to {\pi }^{+}{\pi }^{-}{\pi }^0)=(0.1841\pm 0.0021\pm 0.0080)\mathrm{keV}$, and $\mathcal{B}(\rho \to 3\pi )=(0.88\pm 0.23\pm 0.30)\times {10}^{-4}$. The significance of the $\rho \to 3\pi$ signal is greater than $6\sigma$. For the $J/\psi$ resonance we have measured the product $\mathrm{\Gamma }(J/\psi \to e^{+}{e}^{-})\mathcal{B}(J/\psi \to 3\pi )=(0.1248\pm 0.0019\pm 0.0026)\mathrm{keV}$.

Figure 1

The ${\chi }_{3\pi \gamma }^2$ distributions for data (points with error bars) and simulated (histogram) signal plus background events from the $\omega$ mass region. The shaded (green) histogram shows the distribution for simulated background events. The dashed histogram is the background distribution multiplied by a factor of 25.

Figure 2

The ${\chi }_{3\pi \gamma }^2$ distributions for data from the mass range $1.05<M_{3\pi }<3.00\mathrm{GeV}/c^2$. The shaded (green) histogram shows events rejected by background suppression requirements, as explained in the text.

Figure 3

The $3\pi$ invariant mass spectrum for data events with $50<{\chi }_{3\pi \gamma }^2<500$ and $M_{\mathrm{rec}}^2>10{\mathrm{GeV}}^2/c^4$. The solid histogram represents the result of the fit with a sum of signal and background distributions. The light-shaded (green) area represents the fitted $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0\gamma$ contribution, while the dark-shaded (blue) histograms are the spectrum for all other background processes.

Figure 4

The $M_{\gamma \gamma }^{*}$ distribution for data events from the region $0.6<M_{3\pi }<3.5\mathrm{GeV}/c^2$ selected with our standard selection criteria except for the condition that $M_{\gamma \gamma }^{*}$ be outside the window $0.10–0.17\mathrm{GeV}/c^2$. The curve is the result of the fit by the sum of a Gaussian and a linear function. The linear function is shown separately by the dashed line.

Figure 5

The $M_{3\pi }$ spectra for data (points with error bars) and simulated (histogram) $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ events obtained from the fits to the $M_{\gamma \gamma }^{*}$ distributions as described in the text.

Figure 6

The $M_{3\pi }$ spectrum for data (points with error bars) and simulated (histogram) $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ events with the finer binning.

Figure 7

The $M_{3\pi }$ spectrum for the $4\pi$ (open circles) and non-$4\pi q\overline{q}$ (filled circles) background events selected with the standard criteria. The spectra are obtained using reweighted $q\overline{q}$ simulation.

Figure 8

The $M_{3\pi }$ spectrum for selected data events in the range from 0.6 to $1.1\mathrm{GeV}/c^2$ (histogram). The points with error bars represent the estimated background contribution.

Figure 9

Left panel: The $M_{3\pi }$ spectrum for background events (points with error bars). The filled (blue) histogram represents the contribution of all background sources except $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0\gamma$ and $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}\gamma$. The open histogram is a sum of the filled histogram and the spectrum for $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}\gamma$ events. Right panel: The ratio of the background spectrum to the spectrum for selected data events.

Figure 10

Left panel: The $M_{3\pi }$ spectrum for selected data events with $1.1<M_{3\pi }<3.5\mathrm{GeV}/c^2$ (open histogram). The dashed histogram at $3.0<M_{3\pi }<3.2\mathrm{GeV}/c^2$ represents the spectrum after subtraction of the $J/\psi$ resonance contribution (see Sec. 11). The points with error bars show the calculated spectrum for background events. The filled histogram represents the background spectrum with the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ contribution subtracted. Right panel: The $M_{3\pi }$ spectrum for background events from all sources except $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ and $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^0\gamma$ (points with error bars). The histogram represents the same spectrum with the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0\gamma$ contribution subtracted.

Figure 11

The $M_{3\pi }$ dependence of the ${\alpha }_{\mathrm{sig}}$ coefficient. The dependence is fitted by the function $y=p_1[1-p_2\exp (-p_3{M}_{3\pi })]$, in which two parameters $p_i$ are determined from the relations $y(m_{\varphi })={\alpha }_{\mathrm{sig}}(m_{\varphi })$ and $y(m_{J/\psi })={\alpha }_{\mathrm{sig}}(m_{J/\psi })$.

Figure 12

The $M_{3\pi }$ spectrum for background events extracted from data in each mass bin by solving the system of equations (3) (point with error bars) compared with the spectrum obtained using simulation (filled rectangles).

Figure 13

The $M_{3\pi }$ spectrum for signal events extracted from data in each mass bin by solving the system of equations (3).

Figure 14

The expected $M_{3\pi }$ spectrum from the FSR processes $e^{+}{e}^{-}\to M\gamma \to {\pi }^{+}{\pi }^{-}{\pi }^0\gamma$, where $M=a_1(1260)\gamma$, $a_2(1320)\gamma$, $a_1(1640)$, and $a_2(1700)$. The solid histogram represents the incoherent sum of the processes. The dotted (dashed) histogram demonstrates the effect of constructive (destructive) interference between the $a_1(1260)$ and $a_1(1640)$ amplitudes, and the $a_2(1320)$ and $a_2(1700)$ amplitudes.

Figure 15

The $3\pi$ mass dependence of the detection efficiency obtained using MC simulation. The curve is the result of the fit described in the text.

Figure 16

The $3\pi$ mass distributions for events selected without using the photons from the ${\pi }^0$ decay. The left (right) plot is for events with ${\chi }_{3\pi \gamma }^2<40$ (${\chi }_{3\pi \gamma }^2>40$). The points with error bars show the data distribution. The solid histogram is the result of the fit described in the text. The light shaded (green) region represents the simulated $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0\gamma$ contribution. The dark shaded (blue) histogram is the fitted background contribution from other sources.

Figure 17

The distributions of the ${\chi }^2$ of the kinematic fit for selected data (histogram) and simulated (points with error bars) $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}\gamma$ events with the ${\mu }^{+}{\mu }^{-}$ invariant mass in the range $0.2–1.1\mathrm{GeV}/c^2$.

Figure 18

The $\mathrm{\Delta }{\phi }_{\pm }$ distribution for simulated signal events from the $\omega$ mass region.

Figure 19

Left panel: The distributions of the true (solid histogram) and reconstructed (dashed histogram) $3\pi$ mass for simulated signal events. Right panel: The distribution of the difference between the reconstructed and true mass for simulated signal events with $20<{\chi }_{3\pi \gamma }^2<40$ from the $\omega$ peak. The curve is the result of the fit described in the text. The dashed histogram represents the fitted Lorentzian contribution.

Figure 20

The $3\pi$ mass spectrum in six regions: (a) $0.62–0.70\mathrm{GeV}/c^2$, (b) $0.70–0.74\mathrm{GeV}/c^2$, (c) $0.74–0.82\mathrm{GeV}/c^2$, (d) $0.82–0.98\mathrm{GeV}/c^2$, (e) $0.98–1.06\mathrm{GeV}/c^2$, and (f) $1.06–1.80\mathrm{GeV}/c^2$. The solid, dashed, and dotted histograms represent the results of the fit in Models 1, 2, and 3 listed in Table 2, respectively.

Figure 21

The $3\pi$ mass spectrum in the regions (a) $0.62–0.72\mathrm{GeV}/c^2$ and (b) $0.82–0.98\mathrm{GeV}/c^2$. The solid and dashed histograms represent the results of the fit in Models 1 and 4 listed in Table 2, respectively.

Figure 22

The relative difference between SND [2, 3] and CMD-2 [4, 5] data on the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0$ cross section and the cross section calculated using Eq. (11) with parameters obtained from the fit to the $3\pi$ mass spectrum. The uncertainties shown for the SND and CMD2-2 data are statistical. The systematic uncertainty is 3.4% for the SND data at the $\omega$ [3], 1.3% for the CMD-2 data at the $\omega$ [4], 5% for the SND data at the $\varphi$ [3], and 2.5% for the CMD-2 data at the $\varphi$ [5]. The systematic uncertainty in the BABAR cross section is about 1.5%.

Figure 23

The mass-transfer matrix giving the number of events generated with a (true) mass $M_{3\pi }^{\mathrm{true}}$ in a bin $j$ and reconstructed with a (measured) mass $M_{3\pi }$ in a bin $i$.

Figure 24

The relative difference between the two model spectra described in the text (curve) and the corresponding unfolded spectra (points with error bars).

Figure 25

The comparison of the unfolded data spectrum with the result of the fit to the $3\pi$ mass spectrum discussed in Sec. 7. The error bars correspond to the diagonal elements of the covariance matrix for the unfolded spectrum.

Figure 26

The relative systematic uncertainty in the cross section as a function of the $3\pi$ mass. The dashed, dash-dotted, and dotted histograms represent the contributions to the uncertainty due to background subtraction, the unfolding procedure, and Lorentzian plus Gaussian smearing.

Figure 27

The $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^0$ cross section measured in this work (circles) in comparison with the SND result [7] (triangles). In the $M_{3\pi }$ interval $3.0–3.2\mathrm{GeV}/c^2$, the nonresonant cross section obtained after subtraction of the $J/\psi$ contribution (see Sec. 11) is shown. For the BABAR data, the error bar represents the statistical uncertainty, while the shaded box shows the systematic uncertainty. For the SND data, only the statistical uncertainty is shown; the systematic uncertainty is 4.4%.

Figure 28

The $3\pi$ mass spectrum for selected events in the $J/\psi$ mass region. The curve is the result of the fit described in the text.