Improved convergence in the three-nucleon system at very low energies

Abstract

Neutron-deuteron scattering in the context of "pion-less" Effective Field Theory at very low energies is investigated to next-to-next-to-leading order. Convergence is improved by fitting the two-nucleon contact interactions to the tail of the deuteron wave-function, a procedure known as Z-parameterisation and extended here to the three-nucleon system. The improvement is particularly striking in the doublet-S wave (triton) channel, where better agreement to potential-model calculations and better convergence from order to order in the power counting is achieved for momenta as high as ∼ 120 MeV. Investigating the cut-off dependence of the phase-shifts, one confirms numerically the analytical finding that the first momentum-dependent threebody force enters at N 2 LO. The other partial waves converge also substantially faster. Effective-range parameters of the nd-system are determined, e.g. for the quartet-Swave scattering length a q = [6.35 ± 0.02] fm, which compares favourably both in magnitude and uncertainty with recent high-precision potential-model determinations. Differential cross-sections up to E lab ≈ 15 MeV agree with data.

with effective field theory

“…One can write the Faddeev integral equation in the kinematics defined by the two cluster-configurations exist in the three-nucleon system.We follow the notation suggested by Grießhammer in [21]. The Nd t -cluster with total spin S = 3 2 or S = 1 2 , depending on whether the deuteron and nucleon spins are parallel or anti-parallel; and the Nd s -cluster which has total spin S = 1 2 , as d A s is a scalar.…”

“…In the doublet channel, the Faddeev equation is two-dimensional in cluster-configuration space as both Nd t -and Nd s -configurations contribute [21]:…”

“…One included the specific three-body force term in the doublet-channel Faddeev equation (2.3) at a given order n of expansion as suggested by Bedaque et al [4,5,11,21] if and only if that term is needed to cancel cut-off dependences in the observables which are stronger than cut-off dependence from the suppressed terms of order (QR) n+1 where Q is a typical external momentum and R is the short distance scale in position space beyond which the EFT breaks down. This argument suggests that from the point of view of the EFT, cutoff dependences cancel order by order in the expansion.…”

“…In that case, a unique solution exists in the 2 S 1/2 -channel for each Λ and H = 0, but no unique limit as Λ → ∞. As long-distance phenomena must however be insensitive to details of the shortdistance physics (and in particular of the regulator chosen), Bedaque et al [4,5,11,21] showed that the system must be stabilized by a three-body force…”

“…As one needs a three-body force at LO, H 0 ∼ Q −2 , all three-body forces obtained by expanding H in powers of E are also enhanced, with the interactions proportional to H 2 entering at N 2 LO [11,21]. The power-counting for the three-body forces is hence…”