Modern theory of nuclear forces

Effective field theory allows for a systematic and model-independent derivation of the forces between nucleons in harmony with the symmetries of quantum chromodynamics. The foundations of this approach are reviewed and its application for light nuclei at various resolution scales is discussed. The extension of this approach to many-body systems is sketched.

Figure 1

(Color online) Schematic plot of the central nucleon-nucleon potential. The longest range contribution is the one-pion exchange; the intermediate range attraction is described by two-pion exchanges and other shorter ranged contributions. At even shorter distances, the $NN$ interaction is strongly repulsive.

Figure 2

(Color online) Scales in the two-nucleon problem and the range of validity of the corresponding EFTs as explained in the text. Here ${\Lambda }_{\chi }$ is the hard scale related to spontaneous chiral symmetry breaking, with ${\Lambda }_{\chi }\simeq M_{\rho }$, with $M_{\rho }=770\mathrm{MeV}$ the mass of the rho meson.

Figure 3

Expansion of a heavy-particle exchange diagram in terms of local light-particle operators. The solid and dashed lines denote light and heavy particles, respectively. The filled circle and square denote insertions with zero and two derivatives in order. The ellipsis stands for operators with more derivatives.

Figure 4

(Color online) Effective potential in the ${}^{1}S_0$ channel for three different quark masses in quenched LQCD. The dashed line is the asymptotic OPE for $M_{\pi }=380\mathrm{MeV}$, $m=1.2\mathrm{GeV}$, and $g_{\pi N}^2∕\left(4\pi \right)=14.0$. From 257.

Figure 5

The bubble diagrams with the contact interaction $C_0^t$ or $C_0^s$ contributing to the two-nucleon scattering amplitude.

Figure 6

Diagrams for the inclusion of higher-order contact interactions.

Figure 7

(Color online) Double-differential cross sections of the ${}^2\mathrm{H}(e,e^{\prime })$ reaction with errors (hatched bands) extracted from the experiment. The gray bands and dashed lines are calculations in pionless EFT and a potential model (413). Figure is courtesy of H. W. Grießhammer.

Figure 8

The integral equation for the boson-dimeron scattering amplitude. The single (double) line indicates the boson (dimeron) propagator.

Figure 9

(Color online) Phase shifts for neutron-deuteron scattering below the deuteron breakup at LO (dash-dotted line), NLO (dashed line), and next-to-next-to-leading order ${\mathrm{N}}^2\mathrm{LO}$ (solid line). The filled squares and circles give the results of a phase shift analysis and a calculation using AV18 and the Urbana IX three-body force, respectively. Figure is courtesy of L. Platter.

Figure 10

(Color online) The Tjon line correlation as predicted by the pionless theory. The gray circles and triangles show various calculations using phenomenological potentials (357). The squares show the results of chiral EFT at NLO for different cutoffs, while the diamond gives the ${\mathrm{N}}^2\mathrm{LO}$ result (143; 154). The cross shows the experimental point.

Figure 11

Representation of the two-pion exchange Feynman diagram in terms of time-ordered graphs. Solid and dashed lines represent nucleons and pions, respectively.

Figure 12

Chiral expansion of the two-nucleon force up to next-to-next-to-next-to-leading order $\left({\mathrm{N}}^3\mathrm{LO}\right)$. Solid dots, filled circles, squares, and diamonds denote vertices with ${\Delta }_i=0$, 1, 2, and 3, respectively. Only irreducible contributions of the diagrams are taken in to account as explained in the text.

Figure 13

(Color online) Chiral expansion of the three-nucleon force up to ${\mathrm{N}}^3\mathrm{LO}$. Diagrams in the first line (NLO) yield vanishing contributions to the 3NF if one uses energy-independent formulations as explained in the text. The five topologies at ${\mathrm{N}}^3\mathrm{LO}$ involve the two-pion exchange, one-pion-two-pion-exchange, ring, contact-one-pion exchange, and contact-two-pion-exchange diagrams in order. Shaded blobs represent the corresponding amplitudes. For remaining notation see Fig. 12.

Figure 14

Diagrams contributing to the four-nucleon force at ${\mathrm{N}}^2\mathrm{LO}$. For notation see Fig. 12.

Figure 15

(Color online) Neutron-proton phase shifts in $S$ and P waves at ${\mathrm{N}}^3\mathrm{LO}$ in comparison with the Nijmegen (437; 404) (filled circles) and Virginia Tech (15) (open triangles) PWA. Shaded bands (dashed lines) refer to the calculations by EGM (141) [EM (128)].

Figure 16

(Color online) Neutron-proton phase shifts in $D$ waves and the mixing angles ${ϵ}_{1,2}$ at ${\mathrm{N}}^3\mathrm{LO}$. For notation see Fig. 15.

Figure 17

(Color online) Neutron-proton ${}^{1}S_0$ partial wave at NLO (dashed band), ${\mathrm{N}}^2\mathrm{LO}$ (light-shaded band), and ${\mathrm{N}}^3\mathrm{LO}$ (dark-shaded band) in comparison with the Nijmegen (437; 404) (filled circles) and Virginia Tech (15) (open triangles) PWA.

Figure 18

(Color online) Differential cross section for elastic $nd$ scattering at $E_{\mathrm{lab}}=10\mathrm{MeV}$ (left panel) and $65\mathrm{MeV}$ (right panel). Light (dark) shaded bands depict the results at NLO $\left({\mathrm{N}}^2\mathrm{LO}\right)$. The neutron-deuteron data at $10\mathrm{MeV}$ are from 252. The remaining data at $10\mathrm{MeV}$ are the Coulomb and IB-corrected proton-deuteron data from 429, 400, and 416. The data at $65\mathrm{MeV}$ are proton-deuteron data from 469.

Figure 19

(Color online) The proton-to-proton (left panel) and proton-to-deuteron (right panel) polarization transfer coefficients in $d(\vec{p},\vec{p})d$ and $d(\vec{p},\vec{d})p$ reactions at $E_p^{\mathrm{lab}}=22.7$. Light (dark) shaded bands depict the results at NLO $\left({\mathrm{N}}^2\mathrm{LO}\right)$. Data are from 218; 309.

Figure 20

(Color online) Chiral EFT predictions for neutron-deuteron breakup cross section (in $\mathrm{mb}{\mathrm{MeV}}^{-1}{\mathrm{sr}}^{-2}$) along the kinematical locus $S$. Light shaded (dark shaded) bands refer to the results at NLO $\left({\mathrm{N}}^2\mathrm{LO}\right)$. Left panel: The SST at $E_N=13\mathrm{MeV}$. Neutron-deuteron data (open triangles) are from 425; 439; proton-deuteron data (filled circles) are from 401. Right panel: The symmetric constant relative-energy (SCRE) configuration with $\alpha =56°$ at $E_N=19\mathrm{MeV}$ (321). Dashed and dash-dotted lines are results based on the CD Bonn 2000 2NF (329) combined with the TM99 3NF (100) and the coupled channel calculation including the explicit $\Delta$ and the Coulomb interaction (Deltuva et al., 2005b), respectively.

Figure 21

(Color online) States dominated by $P$-shell configurations for ${}^{10}\mathrm{B}$, ${}^{11}\mathrm{B}$, ${}^{12}\mathrm{C}$, and ${}^{13}\mathrm{C}$. The excitation energy scales are in MeV. The calculation is carried out in the framework of NCSM based on chiral ${\mathrm{N}}^3\mathrm{LO}$ 2NF from 128 and ${\mathrm{N}}^2\mathrm{LO}$ 3NF. For more details on the calculation see 350. Figure is courtesy of Petr Navratil.

Figure 22

(Color online) Isoscalar (left panel) and isovector (right panel) components of the $2\pi$-exchange potential in coordinate space for $\tilde{\Lambda }=700\mathrm{MeV}$. Dashed and solid (dotted and dash-dotted) lines refer to the NLO and ${\mathrm{N}}^2\mathrm{LO}$ results in the deltafull (deltaless) theory, respectively. There are no contributions to ${\tilde{V}}_C$ and ${\tilde{W}}_{T,S}$ (${\tilde{V}}_{T,S}$ and ${\tilde{W}}_C$) at NLO $\left({\mathrm{N}}^2\mathrm{LO}\right)$ in the deltaless theory.

Figure 23

(Color online) $F$-wave $NN$ phase shifts for $\tilde{\Lambda }=700\mathrm{MeV}$. The dotted curve is the LO prediction, long-dashed (short-dashed) and solid (dashed-dotted) lines show the NLO and ${\mathrm{N}}^2\mathrm{LO}$ results with (without) the explicit $\Delta$ contributions. The filled circles depict the results from the Nijmegen PWA (437).

Figure 24

(Color online) Class-II (left panel) and class-III (right panel) $2\pi$-exchange potentials at ${\mathrm{N}}^3\mathrm{L}\stackrel{̸}{\mathrm{O}}$ in the $\Delta$-full theory (shaded bands) compared to the results in the $\Delta$-less theory at ${\mathrm{N}}^3\mathrm{L}\stackrel{̸}{\mathrm{O}}$ (dashed lines) and ${\mathrm{N}}^4\mathrm{L}\stackrel{̸}{\mathrm{O}}$ (dash-dotted lines). The bands arise from the variation in $\delta m_{\Delta }^1$ and $\delta m_{\Delta }^2$ according to Eq. (2.50). Notice further that the leading (i.e., ${\mathrm{N}}^3\mathrm{L}\stackrel{̸}{\mathrm{O}}$) contributions to ${\tilde{V}}_{T,S}^{\mathrm{II}}\left(r\right)$ and subleading (i.e., ${\mathrm{N}}^4\mathrm{L}\stackrel{̸}{\mathrm{O}}$) contributions to ${\tilde{V}}_C^{\mathrm{II}}\left(r\right)$ vanish in the $\Delta$-less theory. In all cases, the spectral function cutoff $\tilde{\Lambda }=700\mathrm{MeV}$ is used.

Figure 25

(Color online) LO (dashed line) and NLO (solid line) results for the total cross section for the reaction $pp\to d{\pi }^{+}$ in comparison with the data from 256 (open circles), 242 (filled circles), and 116 (filled squares). The hatched area gives the estimated uncertainty at NLO. Figure is courtesy of C. Hanhart.

Figure 26

One-pseudoscalar-meson-exchange diagrams at LO for the hyperon-nucleon interaction.

Figure 27

(Color online) Total cross sections as a function of $p_{\mathrm{lab}}$ for $\Lambda p\to \Lambda p$ and ${\Sigma }^{\pm }p\to {\Sigma }^{\pm }p$. The shaded band is the LO EFT result for $\Lambda =550,\dots 700,\mathrm{MeV}$, the dashed curve is the Jülich 04 model (226), and the solid curve is the Nijmegen NSC97f model (405). Note that the total cross sections for ${\Sigma }^{\pm }p\to {\Sigma }^{\pm }p$ are obtained by integrating the differential data in a limited angular range (see 125).

Figure 28

(Color online) Total cross sections as a function of $p_{\mathrm{lab}}$ for ${\Xi }^{-}p\to {\Xi }^{-}p$ (left panel) and ${\Xi }^{-}p\to \Lambda \Lambda$ (right panel). The shaded band is the LO EFT result for $\Lambda =600\mathrm{MeV}$ and varying the LEC $C_{{}^{1}S_0}^{\Lambda \Lambda }$ of the additional singlet contact term within natural bounds given a mildly attractive interaction in the ${}^{1}S_0$ channel of the $\Lambda \Lambda$ interaction.

Figure 29

Overview of the various pieces of the transfer matrix calculation.

Figure 30

(Color online) $NN$ $S$-wave phase shifts and mixing angles vs center-of-mass momentum with actions ${\mathrm{LO}}_1$ and ${\mathrm{LO}}_2$.

Figure 31

(Color online) Neutron-deuteron scattering $S$-wave phase shifts in the spin-$3∕2$ quartet channel vs the square of relative momentum. The data for proton-deuteron and neutron-deuteron scatterings are taken from 454.

Figure 32

(Color online) Deuteron binding energy as a function of the pion mass $M_{\pi }$. The shaded areas correspond to the allowed values. The light shaded band gives the uncertainty due to the unknown value of the LECs ${\overline{D}}_{S,T}$ using the central value ${\overline{d}}_{16}-1.23{\mathrm{GeV}}^{-2}$. The dark shaded band gives the uncertainty if, in addition to the variation in ${\overline{D}}_{S,T}$, the LEC ${\overline{d}}_{16}$ is varied in the range from ${\overline{d}}_{16}-0.91{\mathrm{GeV}}^{-2}\text{to}{\overline{d}}_{16}-1.76{\mathrm{GeV}}^{-2}$ given in 161. The heavy dot shows the binding energy for the physical value of the pion mass.

Figure 33

(Color online) Inverse of the $S$-wave scattering lengths in the spin-triplet and spin-singlet nucleon-nucleon channels as a function of the pion mass $M_{\pi }$. Filled triangles and rectangles show the lattice calculations from 189, 190 and 25, respectively. For remaining notation see Fig. 32.

Figure 34

(Color online) Binding energies $B_3$ of the triton ground and first two excited states as a function of $M_{\pi }$. The circles, squares, and diamonds give the chiral EFT result, while the solid lines are calculations in the pionless theory. The vertical dotted line indicates the critical pion mass $M_{\pi }^{\mathrm{crit}}$ and the dashed lines are the bound state thresholds.

Figure 35

(Color online) Doublet neutron-deuteron scattering length $a_{nd}^{1∕2}$ in the critical region computed in the pionless EFT. The solid line gives the LO result, while the crosses and circles show the NLO and ${\mathrm{N}}^2\mathrm{LO}$ results. The dotted lines indicate the pion masses at which $a_{nd}^{1∕2}$ diverges.

Figure 36

(Color online) Generalized in-medium vertices of lowest order. The thick solid lines correspond to insertions of a Fermi sea and each circle to the insertion of the operator $\Gamma$ as defined in the text.

Figure 37

Diagrams contributing to $E/N$ in nuclear matter. Upper panel: One- and two-pion exchange diagrams contributing to the energy per particle at two and three loops. Solid and dashed lines denote nucleons and pions, respectively. Lower panel: Three-body diagrams related to $2\pi$ exchange with single delta excitations (double lines). These represent interactions between three nucleons in the Fermi sea.

Figure 38

(Color online) The various one- and two-body matter density form factors with leading-order error bands for the ground state of ${}^{20}\mathrm{C}$ in the low-energy region: ${\mathcal{F}}_{nn}\left(k^2\right)$, solid line; ${\mathcal{F}}_{nc}\left(k^2\right)$, dotted line; ${\mathcal{F}}_n\left(k^2\right)$, lighter dashed line; and ${\mathcal{F}}_c\left(k^2\right)$, lighter dot-dashed line.

Figure 39

(Color online) The diagonal matrix elements of the nucleon-nucleon potential in the ${}^{1}S_0$ partial wave. Left panel: Chiral nucleon-nucleon potentials at ${\mathrm{N}}^3\mathrm{LO}$ by EGM (141) and EM (128) for various cutoffs. Right panel: Same potentials evolved down to a cutoff $\Lambda =2.2{\mathrm{cm}}^{-1}$. Figure courtesy of A. Schwenk.

Figure 40

(Color online) Results for $E_0∕E_0^{\text{free}}$ vs the Fermi momentum $k_F$ at LO (open triangles) and NLO (filled triangles). For comparison, we also display the results of FP 1981 (182), APR 1998 (5), CMPR $v6$ and $v{8}^{\prime }$ 2003 (83), SP 2005 (422), and GC 2007 (211).