Wigner SU(4) symmetry, clustering, and the spectrum of $$^{12}$$C
Shihang Shen, Timo A. Lähde, Dean Lee, Ulf-G. Meißner
Abstract
Abstract We present lattice calculations of the low-lying spectrum of $$^{12}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>12</mml:mn> </mml:msup> </mml:math> 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 $$\sim 15$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:mn>15</mml:mn> </mml:mrow> </mml:math> MeV above the ground state, and explore the structure of each state using a large variety of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> cluster and harmonic oscillator trial states, projected onto given irreducible representations of the cubic group. We are able to verify earlier findings for the $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> clustering in the Hoyle state and the second $$2^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:math> state of $$^{12}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>12</mml:mn> </mml:msup> </mml:math> 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 $$^{12}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>12</mml:mn> </mml:msup> </mml:math> C system, or the effects of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> clustering are diminishing their influence. This is in agreement with previous findings from ab initio shell model calculations.