Two-pion exchange nucleon-nucleon potentials with Roper resonance excitation

Motivated by the recent paper arXiv:2602.11815 [nucl-th], we calculate in these notes the spectral functions (i.e. imaginary parts) of the NN-potentials as they arise from $2\pi$-exchange with single and double Roper resonance excitation. In contrast to the full one-loop calculation, the spectral functions of the isoscalar and isovector central and tensor potentials can be given in simple analytical form. The pertinent momentum-space potentials $V_C(q), W_C(q), V_T(q), W_T(q)$ are obtained via subtracted dispersion relations. This representation allows also to include a regulator function, that tames high-momentum components of the chiral $2\pi$-exchange. The calculation is extended to $2\pi$-exchange with combined $\Delta(1232)$-isobar and Roper resonance excitation.