SevenOperators, a Mathematica script for harmonic oscillator nuclear matrix elements arising in semileptonic electroweak interactions

Semi-leptonic electroweak interactions in nuclei-such as β decay, μ capture, charged- and neutral-current neutrino reactions, and electron scattering-are described by a set of multipole operators carrying definite parity and angular momentum, obtained by projection from the underlying nuclear charge and three-current operators. If these nuclear operators are approximated by their one-body forms and expanded in the nucleon velocity through order | over(p, →) | / M, where over(p, →) and M are the nucleon momentum and mass, a set of seven multipole operators is obtained. Nuclear structure calculations are often performed in a basis of Slater determinants formed from harmonic oscillator orbitals, a choice that allows translational invariance to be preserved. Harmonic-oscillator single-particle matrix elements of the multipole operators can be evaluated analytically and expressed in terms of finite polynomials in q², where q is the magnitude of the three-momentum transfer. While results for such matrix elements are available in tabular form, with certain restriction on quantum numbers, the task of determining the analytic form of a response function can still be quite tedious, requiring the folding of the tabulated matrix elements with the nuclear density matrix, and subsequent algebra to evaluate products of operators. Here we provide a Mathematica script for generating these matrix elements, which will allow users to carry out all such calculations by symbolic manipulation. This will eliminate the errors that may accompany hand calculations and speed the calculation of electroweak nuclear cross sections and rates. We illustrate the use of the new script by calculating the cross sections for charged- and neutral-current neutrino scattering in ¹²C. Program summary: Program title: SevenOperators. Catalogue identifier: AEAY_v1_0. Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAY_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.: 2227. No. of bytes in distributed program, including test data, etc.: 19 382. Distribution format: tar.gz. Programming language: Mathematica. Computer: Any computer running Mathematica; tested on Mac OS X PowerPC (32-bit) running Mathematica 6.0.0. Operating system: Any running Mathematica. RAM: Memory requirements determined by Mathematica; 512 MB or greater RAM and hard drive space of at least 3.0 GB recommended. Classification: 17.16, 17.19. Nature of problem: Algebraic evaluation of harmonic oscillator nuclear matrix elements for the one-body multipole operators governing semi-leptonic weak interactions, such as charged- or neutral-current neutrino scattering off nuclei. Solution method: Mathematica evaluation of associated angular momentum algebra and spherical Bessel function radial integrals. Running time: Depends on the complexity of the one-body density matrix employed, but times of a few seconds are typical.