Generalized gauge-space rotations in atomic nuclei: A critical insight

We critically reexamine the concepts of pairing rotations and moments of inertia in gauge space extracted from experimental binding energies. Our analysis focuses on pairing correlations among like nucleons, neutron-proton pairing, and $α$-type correlations. By investigating $α$ separation energies and binding-energy differences along chains of fixed isospin projection and subtracting macroscopic contributions, we reveal a remarkably smooth and nearly universal behavior in the residual $α$ correlation energy. These results exhibit the parabolic trends characteristic of collective rotations in gauge space. We demonstrate that the standard definition of the gauge-space moment of inertia for like-nucleon pairing is dominated by macroscopic contributions from Coulomb and symmetry energies. Once these are removed, the remaining moment of inertia becomes negative. This suggests that the observed behavior reflects the loss of correlation energy due to Pauli-blocking effect. Our results indicate that $α$ correlations constitute a genuine collective mode associated with quartetting dynamics arising from the coherent coupling of two superfluid components.

💡 Research Summary

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The paper presents a critical reassessment of the concepts of pairing rotations and gauge‑space moments of inertia that are traditionally extracted from experimental nuclear binding energies. The authors focus on three types of correlations: like‑nucleon (neutron‑neutron and proton‑proton) pairing, neutron‑proton (np) pairing, and α‑type (quartetting) correlations.

First, they define an α‑correlation energy
(E_{\alpha}(N,Z)=B(N,Z)-B(N-2,Z-2)-B_{\alpha}),
where (B) denotes the positive binding energy and (B_{\alpha}) is the binding of a free α particle. By plotting (E_{\alpha}) for nuclei above (^{100})Sn they observe a large spread that follows the expected Coulomb trend: heavier nuclei have more negative values. To isolate the genuine nuclear interaction contribution, they subtract a macroscopic liquid‑drop term (B_{M}) that contains Coulomb, symmetry, surface and volume energies. The residual quantity

(E’{\alpha}=B-B{M}-(B_{N-2,Z-2}-B_{M,N-2,Z-2})-B_{\alpha})

exhibits an astonishingly smooth, nearly universal behavior. When nuclei are grouped by fixed isospin projection (T_{z}) (α‑decay chains) and particle‑hole symmetry is enforced around the major shells (Z=82, N=82, 126), the curves become almost perfect parabolas. This indicates that after removing macroscopic effects the α‑correlation energy depends only on the number of valence α particles (or holes) and follows a collective rotational pattern in gauge space. The authors argue that this is a clear signature of a genuine quartet mode – a collective degree of freedom built from the coherent coupling of the two superfluid components (neutron and proton pairs).

Next, the paper revisits the standard expression for the pairing‑rotational energy of a nucleus with mass number (A):

(E(A)=E(A_{0})+\lambda_{A_{0}}(A-A_{0})+\frac{(A-A_{0})^{2}}{2J_{A_{0}}}).

Here (J_{A_{0}}) is the gauge‑space moment of inertia, (\lambda_{A_{0}}) the chemical potential, and (A_{0}) a reference (often the mid‑shell) nucleus. Historically, fitting this quadratic form to raw binding energies yields a positive (J), which has been interpreted as a positive pairing moment of inertia. The authors demonstrate that the dominant quadratic curvature in the raw data originates from macroscopic symmetry and Coulomb terms. After subtracting these contributions, the curvature is dramatically reduced and, more importantly, its sign reverses: the extracted microscopic (J) becomes negative.

To explain the sign change, they invoke the exact solution of the pairing Hamiltonian. For an even‑even system with (n) pairs, the ground‑state energy can be approximated as

(E(n)\simeq nE_{2}+n(n-1)G),

and the two‑pair separation energy as

(S_{2}(n)\simeq -E_{2}-2(n-1)G).

The linear term ((nE_{2})) reflects the binding contributed by each added pair, while the quadratic term ((n(n-1)G)) embodies the Pauli‑blocking effect: as more pairs occupy the valence space, the phase space for additional pairing shrinks, leading to a loss of correlation energy per added pair. This Pauli‑blocking mechanism is precisely what produces a negative microscopic moment of inertia once the macroscopic background is removed.

The authors then turn to neutron‑proton pairing. The np pairing rotational energy is often written as

(E_{np}=\frac{(Z-Z_{0})(N-N_{0})}{2J_{np}}),

with the associated moment of inertia extracted from a double difference of binding energies

(\delta V_{pn}= \frac{1}{4}