Nuclear Physics Around the Unitarity Limit

We argue that many features of the structure of nuclei emerge from a strictly perturbative expansion around the unitarity limit, where the two-nucleon $S$ waves have bound states at zero energy. In this limit, the gross features of states in the nuclear chart are correlated to only one dimensionful parameter, which is related to the breaking of scale invariance to a discrete scaling symmetry and set by the triton binding energy. Observables are moved to their physical values by small perturbative corrections, much like in descriptions of the fine structure of atomic spectra. We provide evidence in favor of the conjecture that light, and possibly heavier, nuclei are bound weakly enough to be insensitive to the details of the interactions but strongly enough to be insensitive to the exact size of the two-nucleon system.

Figure 1

Three-nucleon binding energies at NLO as a function of the $3N$ sharp cutoff. (Red) dashed curves: results of Ref. [12], keeping the physical $a_t$ at LO. (Green) solid curves: effect of taking both $S$ waves to the unitarity limit at LO and then including the physical values perturbatively at NLO, along with one-photon exchange and its counterterm. Horizontal lines: experimental values. Top (bottom) lines: ${}^3\mathrm{He}$ (${}^3\mathrm{H}$, fitted).

Figure 2

${}^4\mathrm{He}$ binding energy as function of the Gaussian cutoff. (Blue) solid and (green) dashed lines: standard Pionless EFT and full unitarity at LO, respectively, with partial waves up to $j_{\max }=3/2$. (Blue) dot-dashed and (green) dotted line: same for $S$ waves only ($l_{\max }=0$). (Green) diamonds ($l_{\max }=0$) and circles ($j_{\max }=3/2$): first-order corrections in $1/a_{s,t}$ are added. Results for $j_{\max }=2$ are almost identical to $j_{\max }=3/2$ and not shown. Large symbols on right edge: $\mathrm{\Lambda }\to \infty$ extrapolation (see text).

Figure 3

Tjon line: correlation between the ${}^4\mathrm{He}$ and ${}^3\mathrm{H}$ binding energies. (Blue) solid curve: standard pionless LO result; (green) dashed upper curve: unitarity limit at LO. Additional points nearly on top of the blue curve: inverse scattering lengths as first-order perturbation. Star: experimental point.