Anomalous mapping between pionfull and pionless EFT's

The pion contributions to the coupling $C_0$ of pionless EFT are studied via both non-relativistic and relativistic forms of chiral effective field theory for nuclear forces. A definite item in the $2N$-reducible component of the box diagram is shown to be dominant over the $2N$-irreducible (potential) ones due to the pinching of low-lying nucleon poles, and this anomalous mapping between pionless and pionfull EFT's occurs right within the non-relativistic regime. A natural strategy for renormalization of the pionfull theory emerges as a byproduct through the interactive analysis of the box diagram. Such mapping perspective may shed some light on the efficient organization of the pionfull effective field theory for nuclear forces.