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

A complete calculation of the pion-nucleon loops that contribute to the transition operator for NN -> NN pi up-to-and-including next-to-next-to-leading order ((NLO)-L-2) in chiral effective field theory near threshold is presented. The evaluation is based on the so-called momentum counting scheme, which takes into account the relatively large momentum of the initial nucleons inherent in pion-production reactions. We show that the significant cancellations between the loops found at next-to-leading order (NLO) in the earlier studies are also operative at (NLO)-L-2. In particular, the 1/m(N) corrections (with m(N) being the nucleon mass) to loop diagrams cancel at (NLO)-L-2, as do the contributions of the pion loops involving the low-energy constants c(i), i = 1, ..., 4. In contrast to the NLO calculation however, the cancellation of loops at (NLO)-L-2 is incomplete, yielding a nonvanishing contribution to the transition amplitude. Together with the one-pion exchange tree-level operators, the loop contributions provide the long-range part of the production operator. Finally, we discuss the phenomenological implications of these findings. In particular, we find that the amplitudes generated by the (NLO)-L-2 pion loops yield contributions comparable in size with the most important phenomenological heavy-meson exchange amplitudes.