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

A. A. Filin, 2 V. Baru, 2 E. Epelbaum, C. Hanhart, H. Krebs, A. E. Kudryavtsev, and F. Myhrer Institut für Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum, Germany Institute for Theoretical and Experimental Physics, 117218, B. Cheremushkinskaya 25, Moscow, Russia Institut für Kernphysik, (Theorie) and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA Abstract A complete calculation of the pion-nucleon loops that contribute to the transition operator for NN → NNπ up-to-and-including next-to-next-to-leading order (N2LO) 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 N2LO. In particular, the 1/mN corrections (with mN being the nucleon mass) to loop diagrams cancel at N 2LO, as do the contributions of the pion loops involving the low-energy constants ci, i = 1 . . . 4. In contrast to the NLO calculation however, the cancellation of loops at N2LO is incomplete, yielding a non-vanishing 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 N2LO pion loops yield contributions comparable in size with the most important phenomenological heavy-meson exchange amplitudes.