We report a measurement of the $e^+e^- \to \pi^+\pi^-\pi^0$ cross section in the energy range from 0.62 to 3.50 GeV using an initial-state radiation technique. We use an $e^+e^-$ data sample corresponding to 191 $\text{fb}^{-1}$ of integrated luminosity, collected at a center-of-mass energy at or near the $\Upsilon{(4S)}$ resonance with the Belle II detector at the SuperKEKB collider. Signal yields are extracted by fitting the two-photon mass distribution in $e^+e^- \to \pi^+\pi^-\pi^0\gamma$ events, which involve a $\pi^0 \to \gamma\gamma$ decay and an energetic photon radiated from the initial state. Signal efficiency corrections with an accuracy of 1.6% are obtained from several control data samples. The uncertainty on the cross section at the $\omega$ and $\phi$ resonances is dominated by the systematic uncertainty of 2.2%. The resulting cross sections in the 0.62-1.80 GeV energy range yield $ a_\mu^{3\pi} = [48.91 \pm 0.23~(\mathrm{stat}) \pm 1.07~(\mathrm{syst})] \times 10^{-10} $ for the leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment. This result differs by $2.5$ standard deviations from the most precise current determination.
Energy bin range ($\sqrt{s'}$), number of events after unfolding ($N_{\mathrm{unf}}$), corrected efficiency ($\varepsilon$), and cross section ($\sigma_{3\pi}$) for $e^{+}e^{-} \to \pi^{+} \pi^{-} \pi^{0}$ in energy range 1.05--3.50~GeV. The two uncertainties in the cross section are the statistical and systematic contributions. The statistical uncertainties for the unfolding and cross section are square roots of the diagonal components of the unfolding covariance matrix. The image shows Figure 23 in the PRD paper, and the points with error bars indicate the cross section in the table.
The statistic covariance matrix for the $e^+e^- \to \pi^+ \pi^- \pi^0$ cross section measurement at the Belle II. The 212 x 212 matrix of the energy ranges from 0.62 to 3.50 GeV. This covariance matrix, obtained by propagating the covariance matrix in the unfolding procedure, shows the total statistical uncertainties for the cross section results.
The systematic covariance matrix for the $e^+e^- \to \pi^+ \pi^- \pi^0$ cross section measurement at the Belle II. The 212 x 212 matrix of the energy ranges from 0.62 to 3.50 GeV. This covariance matrix includes all systematic uncertainty given in Table I. The total covariance matrix for the measured cross section can be obtained by adding statistical and systematic covariance matrices.
The dynamics of the process $ e^+e^- \to \pi^+\pi^-\pi^0 $ is studied in the energy region from 1.15 to 2.00 GeV using data accumulated with the SND detector at the VEPP-2000 $e^+e^-$ collider. The Dalitz plot distribution and $\pi^+\pi^-$ mass spectrum are analyzed in a model including the intermediate states $\rho(770)\pi$, $\rho(1450)\pi$, and $\omega\pi^0$. As a result, the energy dependences of the $\rho(770)\pi$ and $\rho(1450)\pi$ cross sections and the relative phases between the $\rho(770)\pi$ amplitude and the $\rho(1450)\pi $ and $\omega\pi^0$ amplitudes are obtained. The $\rho(1450)\pi$ cross section has a peak in the energy region of the $\omega(1650)$ resonance (1.55-1.75 GeV). In this energy range the contributions of the $\rho(770)\pi$ and $\rho(1450)\pi$ states are of the same order of magnitude. No resonance structure near 1.65 GeV is observed in the $\rho(770)\pi$ cross section. We conclude that the intermediate state $\rho(1450)\pi$ gives a significant contribution to the decay of $\omega (1650)\to\pi^+\pi^-\pi^0$, whereas the $\rho(770)\pi$ mechanism dominates in the decay $\omega(1420)\to\pi^+\pi^-\pi^0$.
The Born cross section of the process e+e- -> pi+pi-pi0, scan 2012.
The Born cross section of the process e+e- -> pi+pi-pi0, scan 2011.
The cross section of intermediate states rho pi0, rho' pi0, omega pi0 in the process e+e- -> pi+pi-pi0 extracted by the Dalitz plot analysis.
The total 1r- -p interaction cross sections (of) were measured with an accuracy of 1.5-2% for about 50 pion energies between 140 and 360 Mev. The pion energy was known to within ± 1%. No anomalies in the energy dependence of Of were found which could indicate the existence of a p0meson with a mass in the range of 270 to 410 Mev/c2• The data are inconsistent with the energy value E2 = 650 Mev for the second maximum of Of found by Frisch et al. 7 but agree with the conclusion drawn by Brisson et al. 8 that it should be located at a lower energy ( E2 :::::: 610 Mev). The data are in agreement with the dispersion relations for 1r- -p scattering. It is thus demonstrated that the PuppiStanghellini problem as such no longer exists and that it arose only as a result of an inaccurate knowledge of the total 1r--p interaction cross section.
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CROSS SECTION ON NEUTRON CALCULATED FROM DEUTERIUM MEASUREMENTS USING THE NUCLEON SPECTATOR MODEL.
Data are presented for the reaction ep → ep π 0 at a nominal four-momentum transfer squared of 0.5 (GeV/ c ) 2 . The data were obtained using an extracted electron beam from NINA and two magnetic spectrometers for coincidence detection of the electron and proton. Details are given of the experimental method and the results are given for isobar masses in the range 1.19 – 1.73 GeV/ c 2 .
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Backward cross sections.
In the study of the reaction $e^+e^-\to K_{S}K_{L}$ at the VEPP-2M $e^+e^-$ collider with the SND detector the nuclear interaction length of $K_{L}$ meson in NaI(Tl) has been measured. Its value is found to be 30--50 cm in the $K_{L}$ momentum range 0.11--0.48 GeV/$c$. The results are compared with the values used in the simulation programs GEANT4 and UNIMOD.
The energy interval ($\sqrt{s}$), integrated luminosity ($IL$), number of selected events ($N$), number of background events ($N_{\rm bkg}$), number of events with five or more photons ($N_{5\gamma}$, $N_{5\gamma,{\rm bkg}}$), and the measured $K_L$ nuclear interaction length in NaI(Tl) ($\lambda_{\rm int}$).
The cross section for the e$^{+}$ e$^{–}$ → π$^{+}$π$^{–}$π$^{0}$ process in the energy range 1.05–2.00 GeV has been measured using the data collected in the experiment with the Spherical Neutral Detector (SND) at the VEPP-2000 e$^{+}$ e$^{–}$ collider. The obtained results on the cross section are in good agreement with previous measurements by the SND at the VEPP-2M collider and BABAR, but have a better accuracy.
The Born cross section of the process e+e- -> pi+pi-pi0.
Some cross-sections for the photo-production of ~z~ from hydrogen for pion c.m. angles in the range 60~ ~ are presented. The data have been obtained by measuring proton yields from a hydrogen target, thus permitting separation of single-pion production from the strong background caused by double-pion production. The values, which extend from 360 to 938 MeV, show reasonable agreement with the results of a recent phase-shift analysis
No description provided.
In the experiment with the SND detector at the VEPP-2000 $e^+e^-$ collider the cross section for the process $e^+e^-\to\eta\pi^+\pi^-$ has been measured in the center-of-mass energy range from 1.22 to 2.00 GeV. Obtained results are in agreement with previous measurements and have better accuracy. The energy dependence of the $e^+e^-\to\eta\pi^+\pi^-$ cross section has been fitted with the vector-meson dominance model. From this fit the product of the branching fractions $B(\rho(1450)\to\eta\pi^+\pi^-)B(\rho(1450)\to e^+e^-)$ has been extracted and compared with the same products for $\rho(1450)\to\omega\pi^0$ and $\rho(1450)\to\pi^+\pi^-$ decays. The obtained cross section data have been also used to test the conservation of vector current hypothesis.
The c.m. energy ($\sqrt{s}$), integrated luminosity ($L$), detection efficiency ($\varepsilon$), number of selected signal events ($N$), radiative-correction factor ($1 + \delta$), measured $e^+e^- \to \eta \pi^+\pi^-$ Born cross section ($\sigma_B$). For the number of events and cross section the statistical error is quoted. The systematic uncertainty on the cross section is 8.3% at $\sqrt{s}<1.45$ GeV, 5.0% at $1.45<\sqrt{s}<1.60$ GeV, and 7.8% at $\sqrt{s}>1.60$ GeV.
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