[Paper Review] Renormalized QRPA and double beta decay: a critical analysis

Jorge G. Hirsch, Peter O. Hess+1 more|arXiv (Cornell University)|Feb 6, 1996

Nuclear physics research studies5 citations

TL;DR

This paper critically evaluates the renormalized Quasiparticle Random Phase Approximation (RQRPA) for double beta decay by exactly solving the proton-neutron monopole Lipkin model. It demonstrates that RQRPA violates the Ikeda Sum Rule—even with exact wave functions—due to neglect of scattering terms in both the Hamiltonian and one-body operators, undermining its reliability for ββ decay calculations.

ABSTRACT

The proton-neutron monopole Lipkin model, which exhibites the properties which are relevant for the description of the double beta decay (\\beta \\beta) transitions, is solved exactly. The exact results are compared with the ones obtained by using the Quasiparticle Random Phase (QRPA) and renormalized QRPA (RQRPA) approaches. It is shown that the RQRPA violates the Ikeda Sum Rule and that this violation may be common to any extension of the QRPA which neglects scattering terms in the participant one-body operators as well as in the Hamiltonian. This finding remains valid even when exact wave function are used to compute two-quasiparticle leading order terms of the transition operators. It underlines the need of additional developments before the RQRPA could be adopted as a reliable tool to compute \\beta \\beta processes.

Motivation & Objective

• To assess the reliability of the renormalized QRPA (RQRPA) approach in describing double beta decay transitions.
• To investigate whether the violation of the Ikeda Sum Rule in RQRPA is inherent to its formalism or an artifact of approximations.
• To compare exact solutions of the proton-neutron monopole Lipkin model with QRPA and RQRPA results for ββ decay matrix elements.
• To determine the impact of neglecting scattering terms in one-body operators and the Hamiltonian on the validity of RQRPA.
• To evaluate whether RQRPA can be trusted as a computational tool for double beta decay without further theoretical refinement.

Proposed method

• Exact diagonalization of the proton-neutron monopole Lipkin model Hamiltonian to obtain reference solutions for double beta decay matrix elements.
• Application of the standard QRPA and RQRPA approaches to the same model Hamiltonian for comparative analysis.
• Use of exact wave functions in RQRPA calculations to isolate the effect of formal approximations from numerical errors.
• Evaluation of the Ikeda Sum Rule violation in RQRPA by comparing predicted transition strengths with sum rule constraints.
• Systematic comparison of RQRPA results with exact solutions to identify systematic deviations due to formal approximations.
• Analysis of the role of scattering terms in one-body operators and the Hamiltonian in preserving sum rules and physical consistency.

Experimental results

Research questions

• RQ1 Does the RQRPA approach violate the Ikeda Sum Rule when applied to the proton-neutron monopole Lipkin model?
• RQ2 Is the violation of the Ikeda Sum Rule in RQRPA due to the neglect of scattering terms in the one-body operators and Hamiltonian?
• RQ3 Can the use of exact wave functions in RQRPA calculations eliminate the violation of the Ikeda Sum Rule?
• RQ4 To what extent is the RQRPA formalism inconsistent with fundamental sum rules in the context of double beta decay?
• RQ5 What are the implications of these inconsistencies for the reliability of RQRPA in predicting double beta decay matrix elements?

Key findings

• The RQRPA approach violates the Ikeda Sum Rule even when exact wave functions are used for the two-quasiparticle matrix elements.
• The violation arises specifically from the neglect of scattering terms in both the one-body transition operators and the Hamiltonian within the RQRPA formalism.
• This violation is not an artifact of approximate wave functions but a fundamental flaw in the RQRPA framework.
• The same inconsistency is likely to occur in any extension of QRPA that omits scattering terms in the relevant operators.
• The findings indicate that RQRPA cannot be considered a reliable method for double beta decay calculations without further theoretical development.
• The exact solution of the Lipkin model provides a benchmark that exposes the limitations of RQRPA in preserving fundamental sum rules.