Pions in Nuclear Effective Field Theory: How They Behave Differently at Different Scales and How They Decouple at Very Low Energies

We explain how the Wilsonian renormalization group (RG) can determine the power counting of the nuclear effective field theory (NEFT) including pions. We emphasize that the separation of pion exchange into the short-distance part and the long-distance part is essential since they behave differently in the RG analysis; we found that the latter is perturbative whereas the a part of the former is nonperturbative. As for the contact interactions power counting turns out to be the same as that for pionless NEFT: pion exchange does not affect the scaling property of contact operators. Our RG equations for NEFT including pions connect smoothly with those for the pionless NEFT: pions decouple at very low energies as we expect.