Practical approximation scheme for the pion dynamics in the three-nucleon system

Abstract

We discuss a working approximation scheme to a recently developed formulation of the coupled πNNN-NNN problem. The approximation scheme is based on the physical assumption that, at low energies, the 2N-subsystem dynamics in the elastic channel is conveniently described by the usual 2N-potential approach, while the explicit pion dynamics describes small, correction-type effects. Using the standard separable-expansion method, we obtain a dynamical equation of the Alt-Grassberger-Sandhas (AGS) type. This is an important result, because the computational techniques used for solving the normal AGS equation can also be used to describe the pion dynamics in the 3N system once the matrix dimension is increased by one component. We have also shown that this approximation scheme treats the conventional 3N problem once the pion degrees of freedom are projected out. Then the 3N system is described with an extended AGS-type equation where the spin-off of the pion dynamics (beyond the 2N potential) is taken into account in additional contributions to the driving term. These new terms are shown to reproduce the diagrams leading to modern 3N-force models. We also recover two sets of irreducible diagrams that are commonly neglected in 3N-force discussions, and conclude that these sets should be further investigated, because a claimed cancellation is questionable.PACS numbers: 25.10.+s, 21.45.+v, 25.80.Hp, and 21.30.Fe Typeset using REVT E X 1 Preprint DFPD 00/TH/22

Three-Body Dynamics in One Dimension

“…In table 1 we show the effect of one-pion (OPE3) and two-pion exchange (TPE3) three-nucleon force (3NF) type corrections. For the details of the explicit form of these correction terms we refer to the literature [7,5]. It is seen that unlike the TPE, the OPE corrections have only a small effect on the 'triton' binding energies.…”

“…In particular, we investigate numerically the effects of irreducible "pionic" corrections [6,7] in the one-dimensional three-body bound state by solving the homogeneous version of the AGS equation. Three two-body potentials that give the same 2N binding at −2.225MeV but with different "stiffness" of the repulsion terms have been considered.…”

Irreducible pionic effects in nucleon-deuteron scattering below 20 MeV

“…[6,12,14,15], we have incorporated directly into the Faddeev-AGS threenucleon equation the irreducible effects generated by the one-pion-exchange mechanism in presence of a nucleon-nucleon correlation. The detailed expression has been discussed previously in Ref.…”

“…[12]. Clearly, all three types of diagrams are related to specific aspects of the pion dynamics which cannot be incorporated in the description of a purely nucleonic system interacting through a pair-wise potential.…”

“…Previously, this problem has been analyzed with few-body techniques for quite a few years [9][10][11]. The solution of the resulting set of 21×21 coupled equations represents a formidable task, and for practical reasons an approximation scheme to reduce the complexity of the original equations to an approximated but more tractable form has been developed [12]. The approximation scheme builds up the complete dynamical solution starting from the zeroth-order solution, given by the standard quantum-mechanical Alt-GrassbergerSandhas (AGS) equation [13].…”