Colloquium: Three-body forces: From cold atoms to nuclei

It is often assumed that few- and many-body systems can be accurately described by considering only pairwise two-body interactions of the constituents. We illustrate that three- and higher-body forces enter naturally in effective field theories and are especially prominent in strongly interacting quantum systems. We focus on three-body forces and discuss examples from atomic and nuclear physics. In particular, the importance and the challenges of three-nucleon forces for nuclear structure and reactions, including applications to astrophysics and fundamental symmetries, are highlighted.

Figure 1

Three-nucleon force arising from virtual excitation of a $\Delta (1232)$ degree of freedom. Solid (dashed) lines indicate nucleons (pions).

Figure 2

Illustration of the field redefinition in Eq. (5) that trades an off-shell two-body interaction for a three-body interaction.

Figure 3

Illustration for the resolution dependence of two- and three-body interactions.

Figure 4

The bubble diagrams with the contact interaction $g_2$ contributing to the two-body scattering amplitude.

Figure 5

The integral equation for the boson-dimer scattering amplitude. The single (double) lines indicate the boson (dimer) propagators.

Figure 6

Unrenormalized three-body energies $B_3$ as a function of the momentum cutoff $\Lambda$ (solid lines). The dotted line indicates the cutoff where a new three-body state appears at the boson-dimer threshold (dash-dotted line). The dashed line shows a hypothetical renormalized energy. The inset shows the running of the three-body force $g_3(\Lambda )\sim -H(\Lambda )$ with $\Lambda$.

Figure 7

Illustration of the Efimov spectrum: The energy variable $K=\mathrm{sgn}(E)\sqrt{m|E|}$ is shown as a function of the inverse scattering length $1/a$. The solid lines indicate the Efimov states while the hashed areas give the scattering thresholds. The dashed vertical line indicates a system with fixed scattering length.

Figure 8

Three-body loss coefficient $K_3$ in a gas of ultracold ${}^7\mathrm{Li}$ atoms as a function of scattering length (in units of the Bohr radius $a_0$) for the $|m_F=1⟩$ state (solid circles) and the $|m_F=0⟩$ state (open diamonds). The solid lines represent fits to the universal EFT prediction. The dashed lines represent the $K_3\sim a^4$ upper (lower) limit for $a>0$ ($a<0$). From 73.

Figure 9

The Tjon line correlation between $B({}^3\mathrm{H})$ and $B({}^4\mathrm{He})$. The experimental value is shown by the cross. The shaded band gives the LO universal result (153), while the outer solid lines include NLO corrections obtained using the resonating group method (108). The pluses and diamonds show calculations using phenomenological NN and $\mathrm{NN}+3\mathrm{N}$ potentials, respectively (134); the circle gives a chiral EFT result at ${\mathrm{N}}^2\mathrm{LO}$ (54); and the triangles are based on an SRG-evolved ${\mathrm{N}}^3\mathrm{LO}$ NN potential (132; 87).

Figure 10

Topology of the two-pion-exchange 3NF. Solid (dashed) lines indicate nucleons (pions).

Figure 11

Topology of the leading midrange (left) and short-range (right) 3NFs. Solid (dashed) lines indicate nucleons (pions).

Figure 12

Left panel: Elastic nucleon-deuteron cross section at 10 MeV. The almost indistinguishable bands correspond to chiral $Q^2$ and $Q^3$ calculations. Data are from 166, 97, 155, and 161. Right panel: Nucleon-deuteron breakup cross section at 19 MeV for the space-star configuration at $\alpha =56°$ [see 116 for the definition of the kinematics]. The bands are the same as in the left panel. Data are from 116.

Figure 13

The nucleon vector analyzing power $A_y$ for elastic nucleon-deuteron scattering at 10 MeV. The shaded band covers predictions based on chiral NN and 3N forces at order $Q^3$ for various cutoffs. The solid line is the result of the CD-Bonn NN combined with the TM99 3N potential. Data are as in the left panel of Fig. 12. From 105.

Figure 14

Ground-state energy of ${}^4\mathrm{He}$ as a function of the SRG flow parameter $\lambda$ starting from chiral NN and 3N interactions at ${\mathrm{N}}^3\mathrm{LO}$ and ${\mathrm{N}}^2\mathrm{LO}$, respectively. For details, see 102.

Figure 15

Excitation energies in MeV of light nuclei, ${}^{10}\mathrm{B}$ and ${}^{13}\mathrm{C}$, obtained in the ab initio no-core shell model (NCSM) with chiral EFT interactions (NN to ${\mathrm{N}}^3\mathrm{LO}$ and 3N to ${\mathrm{N}}^2\mathrm{LO}$). From 130.

Figure 16

Ground-state energies of the neutron-rich oxygen isotopes relative to ${}^{16}\mathrm{O}$, including experimental values of the bound isotopes ${}^{16–24}\mathrm{O}$. Energies obtained from (a) phenomenological forces SDPF-M and USD-B, (b) a $G$-matrix interaction and including Fujita-Miyazawa 3NFs due to $\Delta$ excitations, and (c) from low-momentum interactions $V_{\mathrm{low}k}$ and including chiral 3NFs at ${\mathrm{N}}^2\mathrm{LO}$ as well as only due to $\Delta$ excitations. The changes due to 3NFs based on $\Delta$ excitations are highlighted by the shaded areas. (d) Schematic illustration of a two-valence-neutron interaction generated by 3NFs with a nucleon in the ${}^{16}\mathrm{O}$ core. For details see 141.

Figure 17

Excitation energies of bound excited states in ${}^{23}\mathrm{O}$ compared with experiment (42; 163). The NN-only results are calculated in the $sd$ and $sd{f}_{7/2}{p}_{3/2}$ shells with empirical single-particle energies (SPEs). The $\mathrm{NN}+3\mathrm{N}$ energies are obtained in the same spaces, but with calculated SPEs including 3NFs at ${\mathrm{N}}^2\mathrm{LO}$. For details see 91. The dashed lines give the one-neutron separation energy $S_n$.

Figure 18

Nuclear matter energy per particle vs Fermi momentum $k_{\mathrm{F}}$ at the Hartree-Fock level (left) and including second-order (middle) and third-order particle-particle and hole-hole contributions (right), based on evolved ${\mathrm{N}}^3\mathrm{LO}$ NN potentials and ${\mathrm{N}}^2\mathrm{LO}$ 3NFs fit to the ${}^3\mathrm{H}$ binding energy and the ${}^4\mathrm{He}$ charge radius (87). Theoretical uncertainties are estimated by the NN (lines) and 3N (band) cutoff variations.

Figure 19

Mass-radius range for neutron stars based on chiral EFT $\mathrm{NN}+3\mathrm{N}$ interactions, combined with a general extrapolation to high densities (88); the light gray region labeled NN+3N includes an update for the $1.97M_{\odot }$ neutron star discovered recently. The predicted range is consistent with astrophysical modeling of x-ray burst sources [see, e.g., the inner gray shaded region from the 167 analysis]. For comparison, we also show equations of state commonly used in supernova simulations [lines from 139].

Figure 20

Leading two-body axial currents and the corresponding 3NF contributions in chiral EFT. Solid (dashed) lines indicate nucleons (pions), and the wavy lines represent the axial currents.

Figure 21

The ratio $⟨E_1^A{⟩}_{\mathrm{theo}}/⟨E_1^A{⟩}_{\mathrm{emp}}$ that determines the ${}^3\mathrm{H}$ half-life as a function of the low-energy coupling $c_D$, which relates the leading two-body axial currents and 3NFs (see Fig. 20). The empirical range is given by the horizontal band. Results are shown based on different ${\mathrm{N}}^3\mathrm{LO}$ NN potentials and including ${\mathrm{N}}^2\mathrm{LO}$ 3NFs and consistent two-body axial currents. For comparison, the result without 3NFs and without two-body currents (no MEC, no 3NF) is given. For details, see 69.

Figure 22

Nuclear matrix elements $M^{0\nu \beta \beta }$ for neutrinoless double-beta decay of different nuclei. Results are shown based on chiral EFT currents at successive orders, including one-body currents at orders $Q^0$ and $Q^2$, and the predicted long-range parts of two-body currents at order $Q^3$ [for a discussion of the short-range contributions, see 123]. For comparison, we also show shell-model results (SM09) of 124 based on phenomenological one-body currents only.

Figure 23

Order of 3NF contributions in pionless and chiral EFT and in EFT with explicit $\Delta$ degrees of freedom ($\mathrm{\text{chiral}}+\Delta$). Open vertices in the last column indicate the differences of the low-energy constants in chiral and $\mathrm{\text{chiral}}+\Delta$ EFT.