A three-dimensional momentum-space calculation of three-body bound state in a relativistic Faddeev scheme

M R Hadizadeh,M Radin,K Mohseni

doi:10.1038/s41598-020-58577-4

Abstract

In this paper, we study the relativistic effects in a three-body bound state. For this purpose, the relativistic form of the Faddeev equations is solved in momentum space as a function of the Jacobi momentum vectors without using a partial wave decomposition. The inputs for the three-dimensional Faddeev integral equation are the off-shell boost two-body t–matrices, which are calculated directly from the boost two-body interactions by solving the Lippmann-Schwinger equation. The matrix elements of the boost interactions are obtained from the nonrelativistic interactions by solving a nonlinear integral equation using an iterative scheme. The relativistic effects on three-body binding energy are calculated for the Malfliet-Tjon potential. Our calculations show that the relativistic effects lead to a roughly 2% reduction in the three-body binding energy. The contribution of different Faddeev components in the normalization of the relativistic three-body wave function is studied in detail. The accuracy of our numerical solutions is tested by calculation of the expectation value of the three-body mass operator, which shows an excellent agreement with the relativistic energy eigenvalue.

Highlights

In Table in the matrix 1 we show an example of the convergence of the matrix elements of the boost potential Vk(p, p′, x′) as a function of the iteration number calculated for three different values of the Jacobi momentum k = 1,5,10 fm−1 for the Malfliet-Tjon-V (MT-V) bare potential in the fixed points (p = 1.05 fm−1, p′ = 2.60 fm−1, x′ = 0.50)
The direct solution of the relativistic Lippmann-Schwinger equation using relativistic interactions is one of the methods which has been successfully implemented in traditional partial wave decomposition calculations
One of the novel techniques for the calculation of the relativistic and boost interactions from the nonrelativistic interactions is solving a quadratic equation using an iterative scheme proposed by Kamada and Glöckle

Summary

Introduction

In the first step we solve the three-dimensional integral Eq (16) to calculate the matrix elements of the boost interaction Vk(p, p′, x′) from the nonrelativistic interaction Vnr(p, p′, x′) by an iterative scheme. By having the matrix elements of the boost interactions Vk(p, p′, x′) we solve the Lippmann-Schwinger integral Eq (11) to calculate the off-shell boost t–matrices Tk(p, p′, x′; ε) and symmetrize it on the angle variable to get Tksym(p, p′, x′; ε).

Results

Conclusion