Comparison of Relativistic and Non-relativistic Faddeev calculations for Proton-Deuteron Elastic Scattering

TL;DR

The paper addresses the reliability of relativistic versus non-relativistic descriptions in proton-deuteron elastic scattering by comparing a fully relativistic Kharkiv potential with conventional NR NN potentials, and by constructing phase-preserving pseudo-relativistic potentials (PRPs) via CPS and KG transformations. It demonstrates that PRPs can reproduce key two-body observables while enabling relativistic calculations to be performed within a NR-like framework, and it uses the relativistic Faddeev equation to assess cross sections and polarization observables up to several hundred MeV. A major finding is that backward-angle cross sections exhibit notable relativistic enhancements with increasing energy (e.g., at $E=135$ MeV and beyond), while forward-angle differences are largely Kharkiv-specific. Differences between CPS and KG grow with energy but remain modest below about $300$ MeV, and polarization observables are generally well described by PRPs within this range, offering practical guidance for NR approaches up to ~$300$ MeV and informing future relativistic three-nucleon force implementations.

Abstract

This investigation compares non-relativistic and relativistic nucleon-nucleon (NN) potentials in the context of pd scattering. Conventional NN potentials (e.g., CDBonn, AV18) rely on the non-relativistic Schrödinger equation, whereas the Kharkiv potential is intrinsically relativistic. We employ the Coester-Pieper-Serduke (CPS) and Kamada-Glöckle (KG) conversion methods to construct a Pseudo-Relativistic Potential (PRP) from a realistic NN potential, preserving the deuteron binding energy and phase shifts. Calculations of the differential cross section using the relativistic Faddeev equation show that relativistic effects particularly the deviation at the backward angle become pronounced at 135 MeV. The differences in the forward angle were attributed to the characteristics of the Kharkiv potential itself. The reverse transformation of the Kharkiv potential into a Pseudo-Non-Relativistic Potential (PNRP) confirms that the backward-angle relativistic effect increases with energy in the range from 100 MeV to 400 MeV. Comparisons of the polarization observables indicate that relativistic effects, as well as the discrepancy between the CPS and KG transformations, become significant above 300 MeV. Nevertheless, non-relativistic calculations using the PRP remain generally reliable for polarization observables below 300 MeV.

Figures (11)

• Figure 1: Differential cross section of pd elastic scattering at 13 MeV. The nonrelativistic calculations are performed using CDBonn CDBonn, AV18 AV18, and Nijmegen Nijmegen potentials, and the relativistic calculations are performed using the Kharkiv potential Arslanaliev2022.
• Figure 2: Differential cross section of pd elastic scattering at 135 MeV. The line colors are the same as in Fig. \ref{['fig:1']}. Data sets 1 and 2 are from Sakai2000 and Sakamoto1996, respectively.
• Figure 3: Details of the forward and backward scattering angle regions in Fig. \ref{['fig:2']}.
• Figure 4: Details of the differential cross sections at forward and backward scattering angles at $E=135$ MeV. The realistic potentials (CDBonn, AV18, Nijmegen) are transformed into PRPs using the CPS method and substituted into the relativistic Faddeev equation. Data as in Fig. \ref{['fig:2']}.
• Figure 5: Differential cross sections at forward and backward scattering angles. This figure is identical to Fig. \ref{['fig:3']}, except that the Kharkiv potential has been replaced by its pseudo-nonrelativistic counterpart obtained via the inverse CPS transformation. The line colors match those in Fig. \ref{['fig:1']}. Experimental data as in Fig. \ref{['fig:1']}.