Quantitative relativistic effects in the three-nucleon problem

Abstract

The quantitative impact of the requirement of relativistic invariance in the three-nucleon problem is examined within the framework of Poincaré invariant quantum mechanics. In the case of the bound state, and for a wide variety of model implementations and reasonable interactions, most of the quantitative effects come from kinematic factors that can easily be incorporated within a non-relativistic momentum-space three-body code.

Poincaré invariant three-body scattering at intermediate energies

“…It has the advantage that the Faddeev equation provides a mathematically well-defined method for exactly solving the strong interaction dynamics. The Faddeev equation in this framework is more complicated than the corresponding non-relativistic equation, due to the non-linear relation between the mass and energy in relativistic theories, but these difficulties can be overcome [4,7,14]. An important advance that allows these calculations to be extended to energies in the GeV range is the use of numerical methods based on direct integrations, rather than partial wave expansions.…”

“…In this CPS method the relativistic interaction can not be analytically calculated from the non-relativistic one. However, there is a simple analytic connection between the relativistic and non-relativistic two-body t-matrices rel p ) is scattering equivalent to the non-relativistic one at the same relative momentum p [7].…”

First order relativistic three-body scattering

“…This two-body t-matrix, t(k, k ′ ; ε) is the starting point for all calculations which will be presented in the following. In principle there are other methods to obtain a phase-shift equivalent relativistic potential [20], however in this work we want to focus on the relativistic effects visible in three-body scattering observables, and thus use only one fixed scheme.…”

“…This, procedure has internal inconsistencies which show up if these models are used as input in larger systems, but they clearly indicate that the problem of identifying relativistic effects is more subtle than simply computing non-relativistic limits. In this paper we focus on differences between relativistic and non-relativistic calculations with two-body input that have the same cross section and use the same two-body wave functions [18,19,20].…”

Relativistic Hamiltonian dynamics and few-nucleon systems

“…In order to have a strong and immediate comparison with the experiments we have focused our efforts on the development of Poincaré covariant calculations of the electromagnetic ff's, extending our approach from the Deuteron [5] to the Trinucleon. The adopted Bakamjian-Thomas (BT) procedure (see, e.g., [6]) allows us to exploit realistic wave functions for Few-Nucleon Systems (for A=3, see, e.g., [7]) in order to evaluate matrix elements of a Poincaré covariant current operator [8] for an interacting system. The relativistic effects imposed by Poincaré covariance materialize in the relativistic kinematics and in the presence of the so-called Melosh rotations (see, e.g., [6]), that allows one to use the standard Clebsh-Gordan machinery to obtain many-nucleon wave functions with the correct angular coupling.…”