Two-pion exchange nucleon-nucleon potential:O(q⁴) relativistic chiral expansion

We present a relativistic procedure for the chiral expansion of the two-pion exchange component of the $NN$ potential, which emphasizes the role of intermediate $\pi N$ subamplitudes. The relationship between power counting in $\pi N$ and $NN$ processes is discussed and results are expressed directly in terms of observable subthreshold coefficients. Interactions are determined by one- and two-loop diagrams, involving pions, nucleons, and other degrees of freedom, frozen into empirical subthreshold coefficients. The full evaluation of these diagrams produces amplitudes containing many different loop integrals. Their simplification by means of relations among these integrals leads to a set of intermediate results. Subsequent truncation to $O(q^4)$ yields the relativistic potential, which depends on six loop integrals, representing bubble, triangle, crossed box, and box diagrams. The bubble and triangle integrals are the same as in $\pi N$ scattering and we have shown that they also determine the chiral structures of box and crossed box integrals. Relativistic threshold effects make our results to be not equivalent with those of the heavy baryon approach. Performing a formal expansion of our results in inverse powers of the nucleon mass, even in regions where this expansion is not valid, we recover most of the standard heavy baryon results. The main differences are due to the Goldberger-Treiman discrepancy and terms of $O(q^3)$, possibly associated with the iteration of the one-pion exchange potential.