Short-range interactions in an effective field theory approach for nucleon-nucleon scattering

Abstract

We investigate in detail the effect of making the range of the "contact" interaction used in effective field theory (EFT) calculations of NN scattering finite. This is done in both an effective field theory with explicit pions, and one where the pions have been integrated out. In both cases we calculate NN scattering in the 1 S0 channel using potentials which are second-order in the EFT expansion. The contact interactions present in the EFT Lagrangian are made finite by use of a square-well regulator. We find that there is an optimal radius for this regulator, at which second-order corrections to the EFT are identically zero; for radii near optimal these second-order corrections are small. The cutoff EFTs which result from this procedure appear to be valid for momenta up to about 100 MeV/c. We also find that the radius of the square well cannot be reduced to zero if the theory is to reproduce both the experimental scattering length and effective range. Indeed, we show that, if the NN potential is the sum of a one-pion exchange piece and a short-range interaction, then the short-range piece must extend out beyond 1.1 fm, regardless of its particular form.

Low energy theorems for nucleon-nucleon scattering

“…However, if one is in regime where Q counting is valid, then two pion exchange and similar mechanisms only contribute to k cot(δ) at order Q 3 or higher and may be neglected, implying that the bound in ref. [8] is valid up to corrections of order Q/Λ.…”

“…One rather compelling way to see this is found in sect. IV of Scaldeferri et al [8]. There, using a generalization of conventional effective range theory, it is shown that for any energy independent potential model consisting of a one pion exchange plus a short distance potential (including a nonlocal potential), there is a rigorous lower bound on R. In particular, it is shown that in order to fit the scattering length and effective range, the short range potential must make a non-negligible contribution to the wave function out to at least 1.1 fm-i.e.…”

“…Moreover, it is clear from the derivation that this bound on R cannot be saturated, since the bound is saturated only if the wave function is strictly zero for r < R; the actual value of R beyond which the short distance potential ceases to make significant contributions is accordingly expected to be significantly larger suggesting a value of m π R ∼ 1. For example, if the short distance potential for the 1 S 0 channel is taken to be a square well whose depth and range is fit to the scattering length and effective range, then R = 2.3 fm [8], which corresponds to m π R ≈ 1.6. An obvious conclusion is that the scale of the short distance contribution to the nucleon-nucleon interaction is not well separated from m π making the chiral expansion highly problematic.…”

“…[8] includes only one pion exchange and short distance physics. It does not include higher order long distance effects such as two pion exchange which could alter the result.…”

Nucleon-nucleon optical model for energies up to 3 GeV

“…They also maintain color degrees of freedom. As well there are effective quantum field theories (EQF) that link the quark-gluon structure of the standard model to low energy nuclear physics [8,9]. Currently these latter approaches are very popular as they may give a foundation and interpretation of emergent structures.…”