Short-range three-nucleon interaction from $A=3$ data and its hierarchical structure

We construct accurate models of the three-nucleon ($3N$) interaction by fitting, in a hybrid phenomenological approach, the low-energy constants parametrizing the subleading $3N$ contact operators to the triton binding energy, $n\text{−}d$ scattering lengths, cross section, and polarization observables of $p\text{−}d$ scattering at 2 MeV center-of-mass energy. These models lead to a satisfactory description of polarized $p\text{−}d$ scattering data in the whole energy range below the deuteron breakup threshold. In particular, the longstanding $A_y$ puzzle seems to be solved thanks to the new terms considered in the $3N$ force. Two types of hierarchies among the subleading contact operators are also derived, based on the large-$N_c$ counting and on a recently proposed relativistic counting. We test these hierarchies against the same experimental data and show that they are respected at a reasonable level.

Figure 1

Fitted curves, including only the tensor and spin-orbit subleading contact operator on the top of the AV18+UIX interaction, to a set of cross-section and polarization observables in $p\text{−}d$ scattering at 3 MeV proton energy [27], for $\mathrm{\Lambda }=200-500$ MeV (red bands) as compared to the purely two-body AV18 interaction (dashed black lines) and to the AV18+UIX two- and three-nucleon interaction (dashed-dotted, blue lines).

Figure 2

Same as Fig. 1 but including all the $T=1/2$ subleading contact operators.

Figure 3

Predictions of the models determined at $\mathrm{\Lambda }$ between 200 and 500 MeV (red bands) for a set of cross section and polarization $p\text{−}d$ observables at $E_p=1$ MeV energy, as compared to the purely two-body AV18 interaction (dashed, black lines) and to the AV18+UIX two- and three-nucleon interaction (dashed-dotted, blue lines), in comparison to experimental data [44].

Figure 4

Predictions of the model determined at $\mathrm{\Lambda }=300$ MeV for a set of cross section and polarization $p\text{−}d$ observables at $E_p=2.5$ MeV proton energy (solid, red lines) as compared to the purely two-body AV18 interaction (dashed, black lines) and to the AV18+UIX two- and three-nucleon interaction (dashed-dotted, blue lines), in comparison to experimental data [27].

Figure 5

Same as Fig. 4 but for $E_p=2$ MeV proton energy. Data are from Ref. [44].

Figure 6

Same as Fig. 4 but for proton energy $E_p=0.647$ MeV. Data are from Ref. [45].

Figure 7

Prediction of the model determined at $\mathrm{\Lambda }=300$ MeV (red stars connected by solid lines) for the center-of-mass energy dependence of the vector polarization observable $i{T}_{11}$ at $\theta ={88}^{\circ }$ as compared to available experimental data of Ref. [45]. Also shown is the prediction of the simplified model corresponding to the second row of Table 1 (red stars, connected by dotted lines), of the purely two-body AV18 interaction (black stars connected by dashed lines) and to the AV18+UIX nuclear interaction (blues stars connected by dashed-dotted lines).

Figure 8

Same as Fig. 1 but for the models that include all the $T=1/2$ subleading contact operators, without the UIX TNI.

Figure 9

Fit results in the leading order of the large-$N_c$ limit to a set of cross section and polarization $p\text{−}d$ observables at 3 MeV proton energy, for $\mathrm{\Lambda }=200\text{–}500$ MeV (red bands) as compared to the purely two-body AV18 interaction (dashed, black lines) and to the AV18+UIX two- and three-nucleon interaction (dashed-dotted, blue lines).

Figure 10

Fit results in the leading order of the relativistic counting to a set of cross section and polarization $p\text{−}d$ observables at 3 MeV proton energy, for $\mathrm{\Lambda }=200\text{–}500$ MeV (red bands) as compared to the purely two-body AV18 interaction (dashed, black lines) and to the AV18+UIX two- and three-nucleon interaction (dashed-dotted, blue lines).