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

TL;DR

This paper presents a detailed next-to-next-to-leading order calculation of pion production in nucleon-nucleon collisions using chiral effective field theory, revealing significant cancellations and non-vanishing loop contributions with important phenomenological implications.

Contribution

It provides the first complete N$^2$LO calculation of $NN\to NN\pi$ transition operators, highlighting cancellations and the significance of loop contributions at this order.

Findings

Loop cancellations at NLO also occur at N$^2$LO

Non-zero loop contributions at N$^2$LO influence the transition amplitude

N$^2$LO pion loops have comparable effects to heavy-meson exchanges

Abstract

A complete calculation of the pion-nucleon loops that contribute to the transition operator for $NN\to NN\pi$ up-to-and-including next-to-next-to-leading order (N$^2$LO) 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 N$^2$LO. In particular, the $1/m_N$ corrections (with $m_N$ being the nucleon mass) to loop diagrams cancel at N$^2$LO, 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 N$^2$LO 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 N$^2$LO pion loops yield contributions comparable in size with the most important phenomenological heavy-meson exchange amplitudes.