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 0.62--1.05~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.
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 Sigma^- mean squared charge radius has been measured in the space-like Q^2 range 0.035-0.105 GeV^2/c^2 by elastic scattering of a Sigma^- beam off atomic electrons. The measurement was performed with the SELEX (E781) spectrometer using the Fermilab hyperon beam at a mean energy of 610 GeV/c. We obtain
Total systematic errors are given.
Exclusive differential cross sections of virtual bremsstrahlung in proton-proton scattering below the pion-production threshold have been measured for the first time. The total cross section integrated over photon invariant masses of 15 to 80MeV/c2 and integrated over the acceptance of the detector is 3.2±0.1±0.5pb. The data are compared to a low-energy calculation and a fully relativistic microscopic model, which predict a virtual bremsstrahlung cross section of 3.4 and 5.9 pb, respectively. Over the entire experimentally covered phase space, the low-energy calculation gives a better description of the data than the microscopic model.
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We have investigated the elastic scattering of high energy $\Sigma^-$ off electrons from carbon and copper targets using the CERN hyperon beam. Scattering events a
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Differential and total cross sections for the photoproduction of neutral pions from the proton have been measured for incident photon energies from 140–270 MeV, using the photon spectrometer TAPS at the tagged photon beam of the 855 MeV Mainz Microtron. The energy dependence of the s- and p-wave multipoles close to threshold was deduced from a multipole fit and a multipole analysis. The extracted s-wave amplitude E 0+ at threshold is found to be significantly smaller than the prediction of the classical low energy theorems, but is in reasonable agreement with the chiral perturbation theory.
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The 300 MeV electron linear accelerator of Mainz has been used to measure the angular dependence of the electron-proton elastic scattering cross sections at seven different energies for squared four-momentum transfers between 0.13 and 4.7 fm −2 . The proton form factors have been extracted from the cross sections by means of Rosenbluth plots and by fitting parametrized analytical functions directly to the cross sections. The best fit is compared to the data of other laboratories. The previously reported deviations from the dipole fit have been confirmed. From the form factors at q 2 <0.9 fm 2 the proton r.m.s. radius has been determined. A determination of the spectral function of the nucleon isovector form factor G E V in the time-like is obtained using a realistic ϱ resonance.
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Electron-proton elastic scattering cross sections were measured at low four-momentum transfers squared ( q 2 from 0.13 to 2.15 fm −2 ) at six different energies between 150 and 275 MeV. The electric ( G E ) and magnetic ( G M ) form factors of the proton have been determined by Rosenbluth plots and independently by using analytical functions for the form factors to fit the cross sections. The electric form factor is found to deviate significantly from the dipole fit. From the slope of the form factor functions at q 2 = 0 the rms radii of the charge and the magnetic moment distribution were determined. The charge rms radius is found to be more than 10% larger than the value given by the dipole fit.
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