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A complete calculation of the pion-nucleon loops that contribute to thetransition operator for $NN\to NN\pi$ up-to-and-includingnext-to-next-to-leading order (N$^2$LO) in chiral effective field theory nearthreshold is presented. The evaluation is based on the so-called momentumcounting scheme, which takes into account the relatively large momentum of theinitial nucleons inherent in pion-production reactions. We show that thesignificant cancellations between the loops found at next-to-leading order(NLO) in the earlier studies are also operative at N$^2$LO. In particular, the$1/m_N$ corrections (with $m_N$ being the nucleon mass) to loop diagrams cancelat N$^2$LO, as do the contributions of the pion loops involving the low-energyconstants $c_i$, i=1...4. In contrast to the NLO calculation however, thecancellation of loops at N$^2$LO is incomplete, yielding a non-vanishingcontribution to the transition amplitude. Together with the one-pion exchangetree-level operators, the loop contributions provide the long-range part of theproduction operator. Finally, we discuss the phenomenological implications ofthese findings. In particular, we find that the amplitudes generated by theN$^2$LO pion loops yield contributions comparable in size with the mostimportant phenomenological heavy-meson exchange amplitudes.

Pion production in nucleon-nucleon collisions in chiral effective field theory: next-to-next-to-leading order contributions