Nonobservable nature of the nuclear shell structure: Meaning, illustrations, and consequences
Abstract
Background: The concept of singlenucleon shells constitutes a basic pillar of our understanding of nuclear structure. Effective singleparticle energies (ESPEs) introduced by French [Proceedings of the International School of Physics "Enrico Fermi," Course XXXVI, Varenna 1965, edited by C. Bloch (Academic Press, New York, 1966)] and Baranger [Nucl. Phys. A 149, 225 (1970), 10.1016/03759474(70)906925] represent the most appropriate tool to relate manybody observables to a singlenucleon shell structure. As briefly discussed in Duguet and Hagen [Phys. Rev. C 85, 034330 (2012), 10.1103/PhysRevC.85.034330], the dependence of ESPEs on onenucleon transfer probability matrices makes them purely theoretical quantities that "run" with the nonobservable resolution scale λ employed in the calculation.
Purpose: Given that ESPEs provide a way to interpret the manybody problem in terms of simpler theoretical ingredients, the goal is to specify the terms, i.e., the exact sense and conditions, in which this interpretation can be conducted meaningfully.
Methods: While the nuclear shell structure is both scale and scheme dependent, the present study focuses on the former. A detailed discussion is provided to illustrate the scale (in)dependence of observables and nonobservables and the reasons why ESPEs, i.e., the shell structure, belong to the latter category. Stateoftheart multireference inmedium similarity renormalization group and selfconsistent Gorkov Green's function manybody calculations are employed to corroborate the formal analysis. This is done by comparing the behavior of several observables and of nonobservable ESPEs (and spectroscopic factors) under (quasi) unitary similarity renormalization group transformations of the Hamiltonian parametrized by the resolution scale λ .
Results: The formal proofs are confirmed by the results of ab initio manybody calculations in their current stage of implementation. In practice, the unitarity of the similarity transformations is broken owing to the omission of induced manybody interactions beyond threebody operators and to the nonexact treatment of the manybody Schrödinger equation. The impact of this breaking is first characterized by quantifying the artificial running of observables over a (necessarily) finite interval of λ values. Then the genuine running of ESPEs is characterized and shown to be convincingly larger than the one of observables (which would be zero in an exact calculation).
Conclusions: The nonobservable nature of the nuclear shell structure, i.e., the fact that it constitutes an intrinsically theoretical object with no counterpart in the empirical world, must be recognized and assimilated. Indeed, the shell structure cannot be determined uniquely from experimental data and cannot be talked about in an absolute sense as it depends on the nonobservable resolution scale employed in the theoretical calculation. It is only at the price of fixing arbitrarily (but conveniently) such a scale that one can establish correlations between observables and the shell structure. To some extent, fixing the resolution scale provides ESPEs (and spectroscopic factors) with a quasiobservable character. Eventually, practitioners can refer to nuclear shells and spectroscopic factors in their analyses of nuclear phenomena if, and only if, they use consistent structure and reaction theoretical schemes based on a fixed resolution scale they have agreed on prior to performing their analysis and comparisons.
 Publication:

Physical Review C
 Pub Date:
 September 2015
 DOI:
 10.1103/PhysRevC.92.034313
 arXiv:
 arXiv:1411.1237
 Bibcode:
 2015PhRvC..92c4313D
 Keywords:

 21.60.Cs;
 21.10.Jx;
 21.10.Pc;
 Shell model;
 Spectroscopic factors and asymptotic normalization coefficients;
 Singleparticle levels and strength functions;
 Nuclear Theory;
 Nuclear Experiment
 EPrint:
 14 pages, 9 figures, accepted for publication in Physical Review C