Minimally nonlocal nucleon-nucleon potentials with chiral two-pion exchange including $\Delta$ resonances

We construct a coordinate-space chiral potential, including $\Delta$-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 ${\chi }^2/\mathrm{datum}$ 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.

Figure 1

OPE and TPE contributions at LO (a), NLO (b)–(g), and N2LO (h)–(p). Nucleons, $\Delta$ isobars, and pions are denoted, respectively, by the solid, thick solid, and dashed lines; both direct and crossed box contributions are retained in diagrams (d), (f)–(g), (k), and (n)–(p). The open circles denote $\pi N$ and $\pi N\Delta$ couplings from the subleading chiral Lagrangians ${\mathcal{L}}_{\pi N}^{\left(2\right)}$ [46] and ${\mathcal{L}}_{\pi N\Delta }^{\left(2\right)}$ [43]. Note that relativistic $1/M_N$ corrections ($M_N$ is the nucleon mass) included in ${\mathcal{L}}_{\pi N}^{\left(2\right)}$ Lagrangian are not considered here. In particular the contributions of diagrams (i), (k), and (n) are neglected.

Figure 2

S-wave, P-wave, and D-wave phase shifts in the $npT=0$ channel, obtained in the Nijmegen [3, 4], Gross and Stadler [9], and Navarro Pérez et al. [10] partial-wave analyses, are compared to those of models a, b, and c, indicated by the band. For the mixing angle ${\epsilon }_1$ (phase shift ${}^3{\mathrm{D}}_3)$ the lower limit of the band corresponds to model a (model b) and the upper limit to model c (model c).

Figure 3

Same as in Fig. 2, but for the S-wave, P-wave, and D-wave phase shifts in the $npT=1$ channel. For the mixing angle ${\epsilon }_2$ the lower limit of the band corresponds to model c and the upper limit to model b.

Figure 4

S-wave, P-wave, and D-wave phase shifts in the $ppT=1$ channel, obtained in the Nijmegen and Navarro Pérez et al. partial-wave analyses, are compared to those of models a, b, and c, indicated by the band.

Figure 5

The $pp,np$, and $nn{}^1{\mathrm{S}}_0$ and the $pp$ and $np{}^3{\mathrm{P}}_0,{}^3{\mathrm{P}}_1$, and ${}^3{\mathrm{P}}_2$ phase shifts obtained with potential model b, including the full electromagnetic component.

Figure 6

The $pp,np$, and $nn{}^1{\mathrm{S}}_0$ up to laboratory energy of 50 MeV including (panel left) and ignoring (panel right) the full electromagnetic component of potential model b.

Figure 7

The $S$-wave and $D$-wave components of the deuteron wave function corresponding to models a (dashed lines), b (dotted-dashed lines), and c (dotted-dashed-dotted lines) are compared with those corresponding to the AV18 (solid lines).

Figure 8

The effective range function of Eq. (D10) for the potential model b with $(R_{\mathrm{L}},R_{\mathrm{S}})=(1.0,0.7)$ fm. The dashed line is a straight line fit.

Figure 9

Central components of the long-range potential $v_{12}^{\mathrm{L}}$ in pair spin-isospin channels $ST=00$ and 11.

Figure 10

Same as in Fig. 9, but for the short-range charge-independent potential $v_{12}^{\mathrm{S},\mathrm{CI}}$.

Figure 11

Same as in Fig. 9 but in pair spin-isospin channels $ST=01$ and 10.

Figure 12

Same as in Fig. 10 but in pair spin-isospin channels $ST=01$ and 10.

Figure 13

Tensor components of the long-range potential $v_{12}^{\mathrm{L}}$ in pair isospin channels $T=0$ and 1.

Figure 14

Same as in Fig. 13 but for the short-range charge-independent potential $v_{12}^{\mathrm{S},\mathrm{CI}}$.

Figure 15

Spin-orbit components of the short-range charge-independent potential $v_{12}^{\mathrm{S},\mathrm{CI}}$ in pair isospin channels $T=0$ and 1.

Figure 16

Spin and isospin independent quadratic spin-orbit components of the short-range charge-independent potential $v_{12}^{\mathrm{S},\mathrm{CI}}$.

Figure 17

Quadratic orbital angular momentum components of the short-range charge-independent potential $v_{12}^{\mathrm{S},\mathrm{CI}}$ in pair spin channels $S=0$ and 1.

Figure 18

Quadratic relative momentum components of the short-range charge-independent potential $v_{12}^{\mathrm{S},\mathrm{CI}}$ in pair spin channels $S=0$ and 1.

Figure 19

Quadratic-relative-momentum-tensor components of the short-range charge-independent potential $v_{12}^{\mathrm{S},\mathrm{CI}}$ in pair isospin channels $T=0$ and 1.