Nucleon-nucleon potentials from $\mathrm{\Delta }$-full chiral effective-field-theory and implications

We closely investigate $NN$ potentials based upon the $\mathrm{\Delta }$-full version of chiral effective-field-theory. We find that recently constructed $NN$ potentials of this kind, which (when applied together with three-nucleon forces) were presented as predicting accurate binding energies and radii for a range of nuclei from $A=16$ to $A=132$ and providing accurate equations of state for nuclear matter, yield a ${\chi }^2/\mathrm{datum}$ of 60 for the reproduction of the $pp$ data below 100 MeV laboratory energy. This ${\chi }^2$ is more than three times what the Hamada-Johnston potential of the year of 1962 achieved already some 60 years ago. We perceive this historical fact as concerning in view of the current emphasis on precision. We are able to trace the very large ${\chi }^2$ as well as the apparent success of the potentials in nuclear structure to unrealistic predictions for $P$-wave states, in which the $\mathrm{\Delta }$-full next-to-next-to-leading order (NNLO) potentials are off by up to 40 times the NNLO truncation errors. In fact, we show that the worse the description of the $P$-wave states, the better the predictions in nuclear structure. Thus, these potentials cannot be seen as the solution to the outstanding problems in current microscopic nuclear structure physics.

Figure 1

Chiral 2NF without and with $\mathrm{\Delta }$-isobar degrees of freedom. Arrows indicate the shift of strength when explicit $\mathrm{\Delta }$'s are added to the theory. Note that the $\mathrm{\Delta }$-full theory consists of the diagrams involving $\mathrm{\Delta }$'s plus the $\mathrm{\Delta }$-less ones. Solid lines represent nucleons, double lines $\mathrm{\Delta }$ isobars, and dashed lines pions. Small dots, large solid dots, solid squares, and diamonds denote vertices of index ${\delta }_i=$ 0, 1, 2, and 4, respectively. ${\mathrm{\Lambda }}_b$ denotes the breakdown scale. Further explanations are given in the text.

Figure 2

Neutron-proton phase parameters as predicted by the Gőteborg-Oak Ridge (GO) potentials [29] [solid red line $\mathrm{\Delta }\mathrm{NNLO}{\left(450\right)}_{\mathbf{GO}}$, dashed red $\mathrm{\Delta }\mathrm{NNLO}{\left(394\right)}_{\mathbf{GO}}$] and by our refit (Rf) potentials [solid blue line $\mathrm{\Delta }\mathrm{NNLO}{\left(450\right)}_{\mathbf{Rf}}$, dashed blue $\mathrm{\Delta }\mathrm{NNLO}{\left(394\right)}_{\mathbf{Rf}}$]. Partial waves and mixing parameters with total angular momentum $J\le 2$ are displayed for laboratory energies up to 200 MeV. The filled and open circles represent the results from the Nijmegen [35] and the Granada [36] $np$ phase-shift analyses, respectively.

Figure 3

Energy per nucleon in symmetric nuclear matter, $E$, as a function of density, $\rho$, as generated by some two-body forces. Notation for the $\mathrm{\Delta }\mathrm{NNLO}$ potentials as in Fig. 2. Magic (solid green line) refers to the 1.8/2.0 (EM) potential of Ref. [16]. The shaded band includes the theoretical uncertainties associated with the predictions by the Rf potentials (blue lines) [57]. Note that this shaded band also covers the predictions by the $\mathrm{\Delta }$-less NNLO and ${\mathrm{N}}^3\mathrm{LO}$ potentials of Ref. [23] applied in Ref. [25] to intermediate-mass nuclei. The gray box outlines the area where nuclear saturation is expected to occur.

Figure 4

Neutron-proton phase shifts below 100 MeV for three critical $P$ waves. Notation as in Fig. 2.