QQ-onia package: a numerical solution to the Schrödinger radial equation for heavy quarkonium

Abstract

This paper presents the basics of the QQ-onia package, a software based upon the Numerov O ( h 6 ) method which can be used to solve the Schrödinger radial equation using a suitable potential V ( r ) for the heavy quarkonium system. This package also allows the analysis of relevant properties of those resonances such as the square of the wave functions at the origin, their corresponding derivatives for l ≠ 0 states, typical heavy-quark velocities, and mean square radii. Besides, it includes a tool to analyze the spin dependent contributions to the heavy quarkonia spectrum, providing the splitting of n 3 S 1 − n 1 S 0 , as well as the n 3 P J − n 1 P 1 energy levels. Finally a simple software implemented in QQ-onia to compute E1 transition rates is presented. Program summary Program title: QQ-onia package Catalogue identifier: AECQ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECQ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 17 706 No. of bytes in distributed program, including test data, etc.: 2 334 506 Distribution format: tar.gz Programming language: PAW, Physics Analysis Workstation ( http://wwwasd.web.cern.ch/wwwasd/paw/) Computer: PC/Workstation Operating system: Windows-XX and Unix (Linux) Classification: 11.1, 11.6 Nature of problem: Software to solve the Schrödinger radial equation using a suitable potential V ( r ) for the heavy quarkonium system, allowing to perform spectroscopy. It also allows the analysis of relevant quantities of those resonances such as the square of the wave functions at the origin, their corresponding derivatives for l ≠ 0 states, typical heavy-quark velocities, and mean square radii. The package is a (userfriendly) multipurpose tool for dealing with different heavy quarkonium systems, providing a way to study the influence of a given potential on a series of relevant physical quantities, by either varying parameterized values of a well-known potential form, or by including new terms. Solution method: Based upon the Numerov O ( h 6 ) method, we perform a matching procedure to the reduced wave function at the cut point. We also perform a normalization technique for these wave functions taking into account the different domains when we use a Numerov backward–forward technique. In the case of l ⩾ 2 we present a way to find the corresponding derivatives at the origin by only calculating the reduced radial wave function and first derivative. When estimating the heavy quark velocity, we introduce an additional way to compute this quantity from the virial theorem. The calculated reduced wave functions and radial wave functions at the origin are later used to obtain the heavy quarkonia nL splitting and E1 transition rates. Additional comments: Using Windows, to optimize the edition of the files, please, open it with MFC-WORDPAD. Running time: It depends on the choice of the r range, and the number of energy steps.