On two nucleons near unitarity with perturbative pions

Abstract

We explore the impact of perturbative pions on the Unitarity Expansion in the two-nucleon S-waves of Chiral Effective Field Theory at next-to-next-to leading order (\(\textrm{N}^{2}\)LO). Pion exchange explicitly breaks the nontrivial fixed point’s universality, i.e.  invariance of S waves under both conformal and Wigner’s combined \(\textrm{SU}(4)\) spin–isospin transformations. On the other hand, Unitarity explicitly breaks chiral symmetry. The two seem incompatible in their respective exact-symmetry limits. \({\upchi }\textrm{EFT}\) with Perturbative Pions in the Unitarity Expansion resolves the apparent conflict in the Unitarity Window (phase shifts \(45^\circ \lesssim \delta (k)\lesssim 135^\circ \)), i.e.  around momenta \(k\approx m_\pi \) most relevant for low-energy nuclear systems. Its only LO scale is the scattering momentum; NLO adds only scattering length, effective range and non-iterated one-pion exchange (OPE); and \(\textrm{N}^{2}\)LO only once-iterated OPE. Agreement in the \({\phantom {0}}^{1}\textrm{S}_{0}\) channel is very good. Apparently large discrepancies in the \({\phantom {0}}^{3}\textrm{S}_{1}\) channel even at \(k\approx 100\;\textrm{MeV}\) are remedied by taking at \(\textrm{N}^{2}\)LO only the central part of OPE. In contradistinction to the tensor part, it is identical in the \({\phantom {0}}^{1}\textrm{S}_{0}\) and \({\phantom {0}}^{3}\textrm{S}_{1}\) channels. Both channels then match empirical phase shifts and pole parameters well within mutually consistent quantitative theory uncertainty estimates. Pionic effects are small, even for \(k\gtrsim m_\pi \). Empirical breakdown scales are consistent with \(\overline{\Lambda }_{\textrm{NN}}=\frac{16\pi f_\pi ^2}{g_A^2M}\approx 300\;\textrm{MeV}\), where iterated OPE is not suppressed. We therefore conjecture: Both conformal and Wigner 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-\(\textrm{SU}(4)\) symmetry-breaking part of OPE does not enter before \(\textrm{N}^{3}\)LO. We also discuss the potential relevance of entanglement and possible resolution of a conflict with the strength of the tensor interaction in the large-\(N_C\) expansion.