Measurement of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mi>p</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mi>n</mml:mi></mml:mrow></mml:math>from threshold to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>750</mml:mn><mml:mspace width="0.3em"/><mml:mtext>MeV</mml:mtext><mml:mo>∕</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:math>

Reaction ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}n$ has been measured with high statistics in the beam momentum range $270--750\phantom{\rule{0.3em}{0ex}}\text{MeV}∕c$. The data were obtained using the Crystal Ball multiphoton spectrometer, which has $93%$ of $4\ensuremath{\pi}$ solid angle coverage. The dynamics of the ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}n$ reaction and the dependence on the beam energy are displayed in total cross sections, Dalitz plots, invariant-mass spectra, and production angular distributions. Special attention is paid to the evaluation of the acceptance that is needed for the precision determination of the total cross section ${\ensuremath{\sigma}}_{t}({\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}n)$. The energy dependence of ${\ensuremath{\sigma}}_{t}({\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}n)$ shows a shoulder at the Roper resonance [i.e., the $N(1440){\frac{1}{2}}^{+}$], and there is also a maximum near the $N(1520){\frac{3}{2}}^{\ensuremath{-}}$. It illustrates the importance of these two resonances to the ${\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ production process. The Dalitz plots are highly nonuniform; they indicate that the ${\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}n$ final state is dominantly produced via the ${\ensuremath{\pi}}^{0}{\ensuremath{\Delta}}^{0}(1232)$ intermediate state. The invariant-mass spectra differ much from the phase-space distributions. The production angular distributions are also different from the isotropic distribution, and their structure depends on the beam energy. For beam momenta above $550\phantom{\rule{0.3em}{0ex}}\text{MeV}∕c$, the density distribution in the Dalitz plots strongly depends on the angle of the outgoing dipion system (or equivalently on the neutron angle). The role of the ${f}_{0}(600)$ meson (also known as the $\ensuremath{\sigma}$) in ${\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}n$ production remains controversial.

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