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 $\sqrt{{\mathrm{Mm}}_{\pi }}$ characteristic for pion production in nucleon-nucleon collisions we calculate all diagrams for the reaction $N\overrightarrow{N}\mathrm{NN}\pi$ 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 $p\overrightarrow{p}\mathrm{pp}{\pi }^0$ 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 $p\overrightarrow{p}\mathrm{pp}{\pi }^0,pn{\pi }^{+}.$