Data accumulated recently for the exclusive measurement of the $pp\to pp\pi^+\pi^-$ reaction at a beam energy of 0.793 GeV using the COSY-TOF spectrometer have been analyzed with respect to possible events from the $pp \to nn\pi^+\pi^+$ reaction channel. The latter is expected to be the only $\pi\pi$ production channel, which contains no major contributions from resonance excitation close to threshold and hence should be a good testing ground for chiral dynamics in the $\pi\pi$ production process. No single event has been found, which meets all conditions for being a candidate for the $pp \to nn \pi^+\pi^+$ reaction. This gives an upper limit for the cross section of 0.16 $\mu$b (90% C.L.), which is more than an order of magnitude smaller than the cross sections of the other two-pion production channels at the same incident energy.
Upper limit to the P P --> N N PI+ PI+ cross section.
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