Analytical approach for the quartet condensation model

Background: Within the quartet condensation model (QCM), the isovector pairing correlations for $N=Z$ nuclei are described with a very high accuracy by a “condensate” of $\alpha$-like quartets. The usual approach involves cumbersome recurrence relations in order to compute numerically the relevant quantities of the model: the norm of the quartet states and the mean value of the isovector pairing Hamiltonian as functions of the pair mixing amplitudes.

Purpose: We present the final analytical expressions for the above-mentioned quantities, for all cases up to four quartets in the valence shell.

Method: The analytical QCM expressions were obtained by a straightforward implementation of the SO(5) algebra in the symbolic computer algebra system cadabra2.

Results: The norm of the quartet states and the mean value of the Hamiltonian are polynomial functions of the mixing amplitudes. The numerical implementation of the QCM model is thus made trivial as a matter of copying and pasting the presented formulas.

Conclusions: We introduce in this work a method of computer-aided analytical calculus for a many-body setting. In particular, we provide precise and easy-to-use tools for the description of isovector pairing correlations.