# Analytical approach for the Quartet Condensation Model

## TL;DR

This paper derives explicit analytical formulas for the Quartet Condensation Model, simplifying calculations of isovector pairing correlations in nuclei by replacing complex recurrence relations with polynomial expressions.

## Contribution

It provides the first complete analytical expressions for key quantities in the QCM, enabling straightforward numerical implementation.

## Key findings

- Analytical formulas for the norm of quartet states
- Explicit expressions for the mean value of the Hamiltonian
- Simplified computational approach for isovector pairing calculations

## Abstract

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. We present the final analytical expressions for the above mentioned quantities, for all cases up to four quartets in the valence shell. The analytical QCM expressions were obtained by a straightforward implementation of the SO(5) algebra in the symbolic computer algebra system Cadabra2. 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 matter of copying and pasting the presented formulas. We introduce in this work the 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.