Wigner SU(4) symmetry, clustering, and the spectrum of $^{12}$C

TL;DR

This paper uses lattice calculations with a spin- and isospin-independent nucleon interaction to explore the low-energy spectrum and clustering phenomena in carbon-12, confirming alpha clustering and suggesting weak spin-orbit effects.

Contribution

It demonstrates that a simple SU(4)-symmetric nucleon interaction can accurately reproduce the low-lying spectrum and clustering features of $^{12}$C, highlighting the potential dominance of alpha clustering effects.

Findings

Successful reproduction of low-lying spectrum up to 15 MeV

Verification of alpha clustering in the Hoyle state

Indication of weak spin-orbit interactions in $^{12}$C

Abstract

We present lattice calculations of the low-lying spectrum of $^{12}$C using a simple nucleon-nucleon interaction that is independent of spin and isospin and therefore invariant under Wigner's SU(4) symmetry. We find strong signals for all excited states up to $\sim 15$~MeV above the ground state, and explore the structure of each state using a large variety of $\alpha$ cluster and harmonic oscillator trial states, projected onto given irreducible representations of the cubic group. We are able to verify earlier findings for the $\alpha$ clustering in the Hoyle state and the second $2^+$ state of $^{12}$C. The success of these calculations to describe the full low-lying energy spectrum using spin-independent interactions suggest that either the spin-orbit interactions are somewhat weak in the $^{12}$C system, or the effects of $\alpha$ clustering are diminishing their influence. This is in agreement with previous findings from {\it ab initio} shell model calculations.