The Unitarity-Limit Expansion for Two Nucleons with Perturbative Pions: Digest and Ideas

Theorists love nontrivial fixed points. In the Unitarity Limit, the NN 𝑆-wave binding energies are zero, the scattering lengths infinite, Physics is universal, i.e. insensitive to details of the interactions, and observables display richer symmetries, namely invariance under both scaling and Wigner’s combined SU(4) transformation of spin and isospin. In “Pionless” EFT, both are explicitly but weakly broken and hence perturbative in the Unitarity Window (phase shifts $45^\circ \lesssim \delta(𝑘) \lesssim 135^\circ$ , i.e. momenta $𝑘 \approx 𝑚 \pi $). This Unitarity Expansion provides strong hints that Nuclear Physics resides indeed in a sweet spot: bound weakly enough to be insensitive to the details of the nuclear interaction; and therefore interacting strongly enough that the NN scattering lengths are perturbatively close to the Unitarity Limit. In this paradigm change, NN details are less important than NNN interactions to explain the complexity and patterns of the nuclear chart. This presentation is a digest of the first quantitative exploration of corrections to this picture when pions are included [1] (see there for a more comprehensive list of references). Since the pion mass and decay constant introduce dimensionful scales in the NN system, they explicitly break the symmetries of the Unitarity fixed point. In $\chi EFT$, these symmetries must therefore be hidden and instead be classified as emergent. This text focuses on the $\chi EFT$ variant with Perturbative (“KSW”) Pions at next-to-next-to leading order ($N^2LO$). In the $^1S_0$ channel up to cm momenta $\lesssim 300 MeV$, the results are clearly converged order-by-order and agree very well with phase shift analyses. Apparent large discrepancies in the $^3 S_1 $channel even at 𝑘 ≈ 100 MeV are remedied by taking only the central part of the pion’s $N^2LO$ contribution. In contradistinction to the tensor part, it does not mix the different Wigner-SU(4) multiplets and hence is identical in $^1 S_0$ and $^3 S_1$ . With this formulation, pionic effects are small in both channels even for 𝑘 $\gtrsim 𝑚_\pi$ , i.e. where both Unitarity and pion effects are expected to be relevant. This leads to the Hypothesis that both scale invariance and Wigner-SU(4) symmetry in the Unitarity Expansion show persistence, i.e. the footprint of both combined dominates even for 𝑘 $\gtrsim 𝑚_\pi$ and is more relevant than chiral symmetry, so that the tensor/Wigner-SU(4) symmetry-breaking part of OPE is suppressed and does not enter before N3 LO. Included are also ideas about underlying mechanisms and LO results of $\chi EFT$ with Nonperturbative Pions in the expansion about Unitarity.