Complete next-to-leading-order calculation for pion production in nucleon-nucleon collisions at threshold

Based on a counting scheme that explicitly takes into account the large momentum $\sqrt{{\mathrm{Mm}}_{\ensuremath{\pi}}}$ characteristic for pion production in nucleon-nucleon collisions we calculate all diagrams for the reaction $N\stackrel{\ensuremath{\rightarrow}}{N}\mathrm{NN}\ensuremath{\pi}$ at threshold up to next-to-leading-order. At this order there are no free parameters and the size of the next-to-leading- order contributions is in line with the expectation from power counting. The sum of loop corrections at that order vanishes for the process $p\stackrel{\ensuremath{\rightarrow}}{p}\mathrm{pp}{\ensuremath{\pi}}^{0}$ at threshold. The total contribution at next-to-leading-order from loop diagrams that include the delta degree of freedom vanishes at threshold in both reaction channels $p\stackrel{\ensuremath{\rightarrow}}{p}\mathrm{pp}{\ensuremath{\pi}}^{0},pn{\ensuremath{\pi}}^{+}.$

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