Three-nucleon system: Irreducible and reducible contributions of the three-nucleon force

The three-nucleon bound and scattering equations are solved in momentum space for a coupled-channels Hamiltonian. The Hamiltonian couples the purely nucleonic sector of Hilbert space with a sector in which one nucleon is excited to a $\Delta$ isobar. The interaction consists of irreducible two-baryon and irreducible three-baryon potentials. The calculation keeps only the purely nucleonic one among the irreducible three-baryon potentials. The coupled-channels two-baryon potential yields additional reducible contributions to the three-nucleon force. The Coulomb interaction between the two protons is included using the method of screening and renormalization. Three-nucleon force effects on the bound-state energies and on observables of elastic nucleon-deuteron scattering and breakup are studied.

Figure 1

Hilbert space for the description of nuclear phenomena at energies below the $\pi$-production threshold. The shown example is for baryon number 3. Compared with a purely nucleonic Hilbert space, it is extended by a sector, in which one nucleon (thin line) is turned into a $\Delta$ isobar (thick line). The deuteron and two-nucleon reactions are described in the corresponding Hilbert space of baryon number 2.

Figure 2

Interactions describing the nuclear dynamics in the Hilbert space of Fig. 1. They are irreducible coupled-channels potentials of two-baryon (upper row) and three-baryon (lower row) nature. Besides their purely nucleonic parts, they couple the purely nucleonic sector with the one containing a $\Delta$ isobar, and they act directly in the sector containing a $\Delta$ isobar. Only selected examples of the potentials are shown; the Hermitian-conjugate pieces have to be added as well as those obtained by permuting the $\Delta$ isobar in the initial and final states to other positions in the diagrams.

Figure 3

Irreducible contributions and selected examples of reducible contributions to the $2N$ (upper row) and to the $3N$ (lower row) interactions. The two reducible $3N$ processes shown are, in sequence, the Fujita-Miyazawa and the $\Delta$-ring processes.

Figure 4

Neutron analyzing power $A_y$ for neutron-deuteron elastic scattering at 10-MeV neutron energy as function of the center-of-mass scattering angle ${\Theta }_{\mathrm{c}}.\mathrm{m}.$. Results of CD Bonn (dotted curve), CD $\mathrm{Bonn}+\Delta$ (dashed curve), CD $\mathrm{Bonn}+\Delta +\text{U1}$ (solid curve), CD $\mathrm{Bonn}+\Delta +\text{U2}$ (double-dotted-dashed curve), and CD $\mathrm{Bonn}+\Delta +\text{U3}$ (dotted-dashed curve) are compared with the experimental data from Ref. [29].

Figure 5

Differential cross section and deuteron vector analyzing power for proton-deuteron elastic scattering at 135-MeV proton energy as function of the center-of-mass scattering angle ${\Theta }_{\mathrm{c}}.\mathrm{m}.$. Curves as in Fig. 4. The theoretical predictions of this figure are obtained omitting the Coulomb repulsion between the protons for reasons of calculational simplicity; at this energy Coulomb does not yield any noticeable effect at angles ${\Theta }_{\mathrm{c}}.\mathrm{m}.>{20}^{\circ }$ [20]; its inclusion is therefore immaterial for the objectives of this subsection. The experimental data are from Refs. [30] (x), [31] ($•$), and [32] (+).

Figure 6

Neutron analyzing power $A_y$ for neutron-deuteron elastic scattering at 10-MeV neutron energy as function of the center-of-mass scattering angle ${\Theta }_{\mathrm{c}}.\mathrm{m}.$. Results of CD Bonn (dotted curve), CD $\mathrm{Bonn}+\Delta$ (dashed curve), CD $\mathrm{Bonn}+\Delta +\text{U1}$ (solid curve), CD $\mathrm{Bonn}+\Delta +\text{U4}$ (dotted-dashed curve), and CD $\mathrm{Bonn}+\Delta +\text{U5}$ (double-dotted-dashed curve) are compared with the experimental data from Ref. [29].

Figure 7

Differential cross section and deuteron vector analyzing power for proton-deuteron elastic scattering at 135-MeV proton energy as function of the center-of-mass scattering angle ${\Theta }_{\mathrm{c}}.\mathrm{m}.$. The theoretical predictions of this figure are obtained omitting the Coulomb force for reasons explained in the caption of Fig. 5. Curves as in Fig. 6. The experimental data are quoted in Fig. 5.

Figure 8

Differential cross section for neutron-deuteron breakup at 13-MeV neutron laboratory energy in the space star configuration as function of the arclength $S$ along the kinematical curve. Results CD $\mathrm{Bonn}+\Delta$ (dashed curve), CD $\mathrm{Bonn}+\Delta +\text{U1}$ (solid curve), CD $\mathrm{Bonn}+\Delta +\text{U3}$ (dotted-dashed curve), and CD $\mathrm{Bonn}+\Delta +\text{U5}$ (double-dotted-dashed curve) are compared with the experimental data from Refs. [35, 36] (open and full squares). The proton-deuteron data from Ref. [37] (full circles) are also shown.

Figure 9

Differential cross section, proton analyzing power $A_y$, deuteron-to-proton spin-transfer coefficient $K_{yy}^y$, and deuteron analyzing power $A_{xx}$ for proton-deuteron elastic scattering at 135-MeV proton energy. Results based on the potentials CD Bonn (dotted curve), CD $\mathrm{Bonn}+\Delta$ (dashed curve), and CD $\mathrm{Bonn}+\Delta +\text{U1}$ (solid curve) are shown. The experimental data as in Fig. 5.

Figure 10

Differential cross section and nucleon-to-nucleon spin-transfer coefficients for proton-deuteron elastic scattering at 250-MeV proton energy. Note that this energy is slightly above the $\pi$-production threshold but remains well below the $\Delta$-production threshold. Curves as in Fig. 9. The experimental data are from Ref. [38].

Figure 11

Differential cross section and proton analyzing powers $A_x$ and $A_y$ for proton-deuteron breakup at 135-MeV proton energy. Results for three kinematical configurations $({28}^{\circ },{28}^{\circ },{20}^{\circ }),({48}^{\circ },{48}^{\circ },{20}^{\circ })$, and $({25}^{\circ },{25}^{\circ },{80}^{\circ })$ are given from left and to right, respectively. Curves as in Fig. 9. The experimental data are from Ref. [39].