Unitarity for Two Nucleons with Pions

One can understand nuclei at the physical point by an expansion about the unitarity limit of infinite scattering length, with all other effective-range parameters zero. The $NN$ S-wave binding energies are then zero, and there is no scale at leading order. Nuclear Physics resides in a sweet spot: bound weakly enough to be insensitive to the details of the nuclear force; but dense enough that the $NN$ scattering lengths are perturbatively close to the unitarity limit. In this contribution, we study how new scales, namely the pion mass and decay constant change the picture in the $NN$ system. We find that when one imposes unitarity at zero energy, phase shifts do not significantly stray from unitarity at low energies in the $^3$S$_1$-$^3$D$_1$ and in the $^1$S$_0$ waves. Wigner's SU$(4)$ symmetry of combined spin and isospin transformations emerges then quite naturally. At a ``magic'' effective range $r_\chi\approx1.4\;$fm, the effects of these new scales are minimal in both channels. We observe that the physical values are close to it, provide further insight into unitarity with pions, and motivate a converging, perturbative expansion around the unitarity limit, with controlled corrections in the inverse scattering lengths, pion-nucleon interaction, ranges and isospin breaking.