Nuclear effective field theory: Status and perspectives

The nuclear-physics landscape has been redesigned as a sequence of effective field theories (EFTs) connected to the standard model through symmetries and lattice simulations of quantum chromodynamics (QCD). EFTs in this sequence are expansions around different low-energy limits of QCD, each with its own characteristics, scales, and ranges of applicability regarding energy and number of nucleons. The three main nuclear EFTs—chiral, pionless, and halo/cluster—are reviewed, highlighting their similarities, differences, and connections. The structural properties and reactions of nuclei that have been derived from the ab initio solution of the few- and many-body problem built upon EFT input are surveyed.

Figure 1

Landscape of nuclear EFTs in the plane of typical momentum $Q$ and pion mass $m_{\pi }$.

Figure 2

Reduction of pion exchange (dashed line) to a series of contact interactions between nucleons (solid lines) for $q^2\ll m_{\pi }^2$.

Figure 3

Bubble chain for a generic $S$-wave $NN$ scattering amplitude at LO from the $C_0$ interaction (solid circle).

Figure 4

NLO correction to the $NN$ scattering amplitude from the $C_2$ interaction (circled circle).

Figure 5

Diagrams representing the $AN$ nuclear potential in pionless EFT. The order of the contributions is indicated as $\mathcal{O}(Q^{\mu }/M_{\mathrm{hi}}^{\mu })$, $\mu \ge 0$, where $Q\sim M_{\mathrm{lo}}$ and $M_{\mathrm{hi}}\sim m_{\pi }$, so that the circles around the central solid circle denote inverse powers of $M_{\mathrm{hi}}$.

Figure 6

Bubble sum for the dressed dibaryon propagator obtained from the bare propagator (double dashed line).

Figure 7

The $NN$ scattering phase shift $\delta$ as a function of the nucleon laboratory energy $E_{\mathrm{lab}}$ in the ${{}^{3}S}_1$ partial wave for pionless EFT and chiral EFT at various orders ([525]) (from C.-J. Yang), and the chiral potential “EM500” ([246]). For comparison, we show the partial-wave analysis (PWA) of [579], with error bars smaller than the symbols.

Figure 8

Bubble diagram dressed with Coulomb exchange (wavy line).

Figure 9

Total cross section $\sigma$ for deuteron electrodisintegration as function of the photon energy $E_{\gamma }$. Good agreement is seen between measured data points and theoretical calculation (lines), which was achieved after the pionless calculation of [181] helped resolve a problem with the experimental analysis. Adapted from [678]. Courtesy of H. W. Grießhammer.

Figure 10

Integral equation for $nd$ scattering in the spin-quartet channel. A pair of similar but coupled equations describes the doublet channel.

Figure 11

Modification of the driving mechanism in Fig. 10, with the LO three-body force (solid circled) in the $nd$ doublet channel.

Figure 12

RG running of the three-body coupling $h_0$ at LO and NLO. Numerical data points are fitted using Eq. (44) at LO and an analogous expression ([425]) at NLO.

Figure 13

Correlation between the triton binding energy $B({}^3\mathrm{H})$ and the doublet $nd$ scattering length ${{}^{2}a}_{nd}$ (Phillips line) at LO and NLO, compared to results from various potential models and experiment. Bands indicate estimates of higher-order corrections: the larger band around the dotted line is LO, the smaller band around the dashed line is NLO.

Figure 14

Dibaryon kinetic-energy corrections for the $nd$ quartet-channel scattering amplitude at NLO. In the doublet channel, there are analogous diagrams with ${{}^{1}S}_0$ dibaryons.

Figure 15

Correlation between the alpha-particle binding energy $B({}^4\mathrm{He})$ and the triton binding energy $B({}^3\mathrm{H})$ (Tjon line) at LO in pionless EFT, compared to results from various chiral and phenomenological potentials, and experiment. The band indicates an estimate of higher-order corrections.

Figure 16

Illustration of the Efimov spectrum as a function of the three-body momentum $K$ [cf. Eq. (54)] and the inverse neutron-core scattering length $a_{cn}^{-1}$. The diagonal line in the fourth quadrant represents the neutron-core threshold. The solid lines indicate the Efimov states. The window of universality for bound states is represented by the shaded half circle, while the dashed and dot-dashed lines indicate a typical system with $a_{cn}>0$ and $a_{cn}<0$, respectively.

Figure 17

Boundary curves for the existence of excited Efimov states in two-neutron halo nuclei with different core masses $A$ as a function of the square roots of the neutron-core energy $E_{nc}$ and neutron-neutron energy $E_{nn}$ in units of the three-body ground-state energy $E_{gs}$. The sign of the square root is taken as positive (negative) if the corresponding two-body system $nc$ or $nn$ has a bound state (virtual state). Established halo nuclei are shown by the data points, while the shaded area gives the parameter range for the predicted halo nucleus ${}^{62}\mathrm{Ca}$. From [369].

Figure 18

Correlation between the ${}^{61}\mathrm{Ca}\text{−}n$ scattering length and the two-neutron separation energy $S_{2n}$ of ${}^{62}\mathrm{Ca}$. The emergence of an excited Efimov state around $S_{2n}=230\mathrm{keV}$ is indicated by the vertical dashed line. The shaded area indicates the region where halo EFT breaks down. From [368].

Figure 19

The LO contribution to the irreducible vertex for an $A_0$ photon coupling to the ${}^{10}\mathrm{Be}$-neutron $S$-wave bound state. The thick solid line indicates the ${}^{11}\mathrm{Be}$ ground state, while the solid, dashed, and curly lines represent the neutron, ${}^{10}\mathrm{Be}$ core, and photon, respectively.

Figure 20

The two diagrams contributing to the irreducible vertex for the $S$-to-$P$-state transition ${\mathrm{\Gamma }}_{j\mu }$ at LO. The double line indicates the ${}^{11}\mathrm{Be}$ excited state; the other lines are as in Fig. 19.

Figure 21

Diagrams contributing to photodissociation of the ${}^{11}\mathrm{Be}$ ground ($S$-wave) state. The notation is as in Fig. 20.

Figure 22

Differential $B(E1)$ strength for Coulomb dissociation of ${}^{11}\mathrm{Be}$ into ${}^{10}\mathrm{Be}+n$ as a function of excess energy of the neutron $E^{*}$. The dashed (solid) lines show the halo EFT result of [378] at LO (NLO) folded with the detector resolution. The experimental data are from [600], while the dotted line (almost on top of the solid line) gives the effective range model of [739]. From [378].

Figure 23

(Right panel) NLO $S$ factor as function of energy (solid blue curve). Shading indicates a 68% interval. The dashed line gives the LO result. (Left panel) One-dimensional probability distribution functions for $S(0)$ (blue line and histogram) and $S(20\mathrm{keV})$ (red dashed line). From [796].

Figure 24

Halo EFT results for the $\alpha \alpha S$-wave phase shift by [396] as a function of the laboratory energy $E_{\mathrm{lab}}$. LO and NLO phase shifts are given by the blue dashed and red solid lines, respectively. The experimental data are from [13], while the lattice EFT results are from [241].

Figure 25

Sample of diagrams representing the pion-range components of the $AN$ nuclear potential in chiral EFT according to Eq. (74) with $n=2$. The order of the contributions is indicated as $\mathcal{O}(Q^{\mu }/M_{\mathrm{hi}}^{\mu })$, $\mu \ge 0$, where $Q\sim m_{\pi }$ and $M_{\mathrm{hi}}\sim M_{\mathrm{QCD}}$. Arrows in the $3N$ and $4N$ columns indicate the changes for $n=1$. A solid (double) line stands for a nucleon (nucleon excitation), while a dashed line stands for a pion. A circle around the central solid circle denotes an inverse power of $M_{\mathrm{hi}}$.

Figure 26

(Top panel) Ground-state energy (negative of the binding energy) per nucleon, and (bottom panel) residuals (differences between computed and experimental values) of charge radii for selected nuclei computed with a variety of chiral potentials (labeled a–i and ${\mathrm{NNLO}}_{\mathrm{sat}}$). From [238].

Figure 27

The $NN$ ${{}^{3}P}_0$ phase shift $\delta$ as a function of the center-of-mass momentum $k_{\mathrm{c}.\mathrm{m}.}$ at different chiral orders: $\mathcal{O}(1)$ (green), $\mathcal{O}(Q^2/M_{\mathrm{QCD}}^2)$ (red), and $\mathcal{O}(Q^3/M_{\mathrm{QCD}}^3)$ (blue). Bands represent the variation of a coordinate-space cutoff in the range 0.6–0.9 fm, with the dashed line showing the $\mathcal{O}(Q^3/M_{\mathrm{QCD}}^3)$ results with a 0.3 fm cutoff. The cyan band shows the $\mathcal{O}(Q^3/M_{\mathrm{QCD}}^3)$ potential in Weinberg’s prescription with the regularization procedure of [263], where the momentum-space cutoff in pion loops (the Lippmann-Schwinger equation) is varied between 500 and 700 (450 and 650) MeV. The solid black line is the Nijmegen phase-shift analysis ([723]). From [614].

Figure 28

Nucleon electric (${\alpha }_{E1}$) and magnetic (${\beta }_{M1}$) polarizabilities extracted with chiral EFT. Ellipses show the $1\sigma$-error fits of [553] for proton (blue, on lower band) and neutron (red, on upper band), where statistic, systematic, and theory errors are added in quadrature. For comparison, Particle Data Group averages before (large green ellipses) and after (small black ellipses) these fits are also given. The bands are the constraints from the Baldin sum rule. From [349].

Figure 29

Triton spectrum in the vicinity of a limit cycle as function of the pion mass. The circles, squares, and diamonds give the chiral-potential result of [266], while the solid lines are ${\mathrm{N}}^2\mathrm{LO}$ calculations in pionless EFT ([379]). The vertical dotted line indicates the critical pion mass. The thresholds for stable three-nucleon states are given by the dashed lines. From [379].