The exact discontinuity of a partial wave along the left-hand cut and the exact $N/D$ method in non-relativistic scattering

Abstract

We first deduce the analytical continuation in the complex planes of the initial and final three-momenta of the Lippmann-Schwinger equation in coupled or uncoupled partial-wave amplitudes. This result allows us to deduce a master equation whose solution is the exact discontinuity of the on-shell partial-wave amplitudes along the left-hand cut. This equation is always a linear non-singular integral equation whose solution is fixed exclusively by the knowledge of the potential, applicable to either regular or singular potentials. The capability of calculating exactly this discontinuity allows one to settle the exact $N/D$ method in two-body non-relativistic scattering for coupled and uncoupled waves. We exemplify this new advance in scattering theory by explicitly checking the agreement between the Lippmann-Schwinger equation with the corresponding solutions of the exact $N/D$ method for some examples that involve regular and singular potentials, either attractive or repulsive.

103 pages, 21 figures

References in corpus (5)

• Chiral effective field theory and nuclear forces
• Semilocal momentum-space regularized chiral two-nucleon potentials up to fifth order
• Renormalization of chiral two-pion exchange NN interactions. Momentum vs. coordinate space
• Renormalization of Singular Potentials and Power Counting
• New results from a number operator interpretation of the compositeness of bound and resonant states

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• Power counting in chiral effective field theory and nuclear binding
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• Decay properties of $N^{\ast }(1895)$
• Ladder resummation of spin 1/2 fermion many-body systems with arbitrary partial-wave content
• Non-perturbative methods for NN singular interactions
• Nuclear matter from the ladder resummation in terms of the experimental nucleon-nucleon scattering amplitudes
• Effective range expansion with a long-range force
• Renormalization of NN Chiral EFT, my personal mixed feelings