Triangle diagram in heavy-baryon chiral perturbation theory

TL;DR

This paper examines the pion-baryon triangle diagram near a specific kinematic region where the momentum transfer is close to four times the pion mass squared, revealing that standard approximations break down and resummation is needed.

Contribution

It identifies a special kinematic window where the static-limit approximation fails and demonstrates the necessity of resumming baryon recoil effects in heavy-baryon chiral perturbation theory.

Findings

Static-limit approximation is invalid near the specified kinematic region.

Resummation of baryon recoil effects is necessary in this window.

No overall enhancement of power counting occurs despite resummation.

Abstract

The pion-baryon triangle diagram is inspected for a special kinematic region where the squared momentum transfer $t$ is close to $4m_\pi^2$: $|t - 4m_\pi^2| \lesssim m_\pi^4/m_N^2$. Instead of arguing on the ground of anomalous threshold, we investigate possible impacts on power counting. The pion can have very small energies, as opposed to $\sim m_\pi$ in the physical region, which allows all three propagators to be extremely near their mass shell and contributes significantly to the loop integral. We find that in this narrow kinematic window the static-limit approximation for the baryon propagator is invalid and the resummation of the kinetic energy is necessary. Interestingly, in contrast to low-energy two-baryon processes, this resummation of baryon recoils does not lead to overall enhancement of power counting of the diagram.