Minimally nonlocal nucleon-nucleon potentials with chiral two-pion exchange including<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi></mml:math>resonances

Abstract

We construct a coordinate-space chiral potential, including $\ensuremath{\Delta}$-isobar intermediate states in its two-pion-exchange component up to order ${Q}^{3}$ ($Q$ denotes generically the low momentum scale). The contact interactions entering at next-to-leading and next-to-next-to-next-to-leading orders (${Q}^{2}$ and ${Q}^{4}$, respectively) are rearranged by Fierz transformations to yield terms at most quadratic in the relative momentum operator of the two nucleons. The low-energy constants multiplying these contact interactions are fitted to the 2013 Granada database, consisting of $2309 pp$ and $2982 np$ data (including, respectively, 148 and 218 normalizations) in the laboratory-energy range 0--300 MeV. For the total $5291 pp$ and $np$ data in this range, we obtain a ${\ensuremath{\chi}}^{2}/\mathrm{datum}$ of roughly 1.3 for a set of three models characterized by long- and short-range cutoffs, ${R}_{\mathrm{L}}$ and ${R}_{\mathrm{S}}$, respectively, ranging from $({R}_{\mathrm{L}},{R}_{\mathrm{S}})=(1.2,0.8)$ fm down to $(0.8,0.6)$ fm. The long-range (short-range) cutoff regularizes the one- and two-pion exchange (contact) part of the potential.