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Low-momentum effective theory for nucleons
E. Epelbaoum, W. Glöckle, Ulf-G. Meißner
TL;DR
The paper addresses how to construct a low-momentum nucleon theory while retaining the relevant physics of a precise two-nucleon interaction. It uses unitary transformations to decouple low- and high-momentum components in an S-wave Malfliet–Tjon model, and finds that the effective theory reproduces the original bound-state and scattering results below the cutoff.
Problem
Chiral approaches systematically treat low-momentum nuclear-force matrix elements, but shallow bound states and Lippmann–Schwinger equations require handling higher momenta.
Source: p. 2Method
Unitary transformations decouple low- and high-momentum components of an S-wave two-nucleon potential, producing an effective interaction projected below a chosen cutoff.
Source: p. 3Results
The effective theory reproduces the original low-energy phase shifts and bound-state energy exactly below the chosen cutoff.
Source: pp. 4–5Takeaways & Limitations
The construction may support effective chiral few-nucleon forces and consistent treatments of relativistic effects in few- and many-nucleon systems.
Source: p. 5The demonstrated construction uses a simple S-wave potential and a low/high momentum separation defined by a cutoff.
Source: pp. 2–3
Abstract
from arXiv · showhide
Starting from a precise two-nucleon potential, we use the method of unitary transformations to construct an effective potential that involves only momenta less than a given maximal value. We describe this method for an S-wave potential of the Malfliet-Tjon type. It is demonstrated that the bound and scattering state spectrum calculated within the effective theory agrees exactly with the one based on the original potential. This might open an avenue for the construction of effective chiral few-nucleon forces and for a systematic treatment of relativistic effects in few-body systems.
Figures
The figures compare effective and original two-nucleon quantities across momentum cutoffs, showing how the low-momentum construction changes potentials while preserving observables and transformed bound-state information.
- For momenta below 400 MeV, the effective and original two-nucleon potentials are compared directly, with the effective potential shown by solid lines and the original by dashed lines.Source: Figure 1 · p. 7
- For momenta below 200 MeV, the same potential comparison probes the stronger low-cutoff regime.Source: Figure 2 · p. 8
- Phase shifts from the effective and original potentials are compared against laboratory kinetic energy for cutoffs Λ = 400 and 200 MeV.Source: Figure 3 · p. 9
- The deuteron momentum-space wave functions from the effective and original potentials are compared at Λ = 200 MeV.Source: Figure 4 · p. 10