A Mathematica script for harmonic oscillator nuclear matrix elements arising in semileptonic electroweak interactions

Understand

Semi-leptonic electroweak interactions in nuclei - such as \beta decay, \mu 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 |\vec{p}|/M, where \vec{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 q^2, where q is the magnitude of the three-momentum transfer.