Well funneled nuclear structure landscape: renormalization

A complete characterization of the structure of nuclei can be obtained by combining information arising from inelastic scattering, Coulomb excitation and $\gamma-$decay, together with one- and two-particle transfer reactions. In this way it is possible to probe the single-particle and collective components of the nuclear many-body wavefunction resulting from their mutual coupling and diagonalising the low-energy Hamiltonian. We address the question of how accurately such a description can account for experimental observations. It is concluded that renormalizing empirically and on equal footing bare single-particle and collective motion in terms of self-energy (mass) and vertex corrections (screening), as well as particle-hole and pairing interactions through particle-vibration coupling allows theory to provide an overall, quantitative account of the data.

Nuclear structure is both a mature [1-3] and a very active field of research [4,5] , and time seems ripe to attempt a balance of our present, quantitative understanding of it. Here we take up an aspect of this challenge and try to answer to the question: how accurately can theory predict structure observables in terms of single-particle and collective degrees of freedom and of their couplings?
In pursuing this quest one of two paths can be taken: 1) select one nuclear property, for example the single-particle spectrum, and study it throughout the mass table [6]; 2) select a target nucleus A which has been fully characterized through inelastic scattering and Coulomb excitation (A(α, α )A * ), together with one-(A(d, p)A + 1, A(p, d)A − 1) and two-((A + 2(p, t)A, A(p, t)A − 2) particle transfer processes and study the associated, complete nuclear structure information involving the island of nuclei A, A±1 and A±2 in terms of the corresponding absolute differential cross sections and decay transition probabilities. Here we have chosen the second way and selected the group of nuclei 118,119,120,121,122 Sn involved in the characterization of the spherical, superfluid 120 Sn target nucleus.
Single-particle and collective vibrations constitute the basis states. The calculations are implemented in terms of a SLy4 effective interaction and a v 14 ( 1 S 0 )(≡ v bare p ) Argonne pairing potential. HFB provides an embodiment of the quasiparticle spectrum while QRPA a realization of density (J π = 2 + , 3 − , 4 + , 5 − ) and spin (2 ± , 3 ± , 4 ± , 5 ± ) modes. Taking into account renormalisation processes (self-energy, vertex corrections, phonon renormalization and phonon exchange) in terms of the particle-vibration coupling (PVC) mechanism, the dressed particles as well as the induced pairing interaction v ind p were calculated (see [7]; see also [8][9][10][11][12][13][14][15] and v ind to∆ ν are about equal, density modes leading to attractive contributions which are partially cancelled out by spin modes, as expected from general transformation properties of the associated operators entering the particlevibration coupling vertices [21][22][23]. Theory (SLY4 +QRPA+ (PVC) REN+NG) provides a quantitative account of the experimental value (∆ exp ≈ 1.45 MeV). It is to be noted that in carrying out the above calculations use has been made of empirically renormalized collective modes. This is because SLy4 leads to little collective density vibrations (cf. Table   1, where, for concreteness, the bare QRPA results characterizing the low-lying 2 + of 120 Sn are collected [7], see also [24]), in keeping with the associated value of the effective mass 0.7 m. In fact, collectivity is closely associated with a density of levels (∼ m * ) consistent with an effective mass m * = m ω m k /m ≈ m. This is achieved by coupling the two-quasiparticle QRPA SLy4 solutions to 4qp doorway states made out of a 2qp uncorrelated component and an empirically tuned QRPA collective mode [25] (see Fig. 2 of [7], cf. also [26]), an example of the fact that in a consistent PVC renormalised description of the nuclear structure, one has to treat (dress), on equal footing, all degrees of freedom (i.e. single-particle and collective modes). In this way, not only self-energy but also vertex corrections are consistently included (sum rules conserving processes), and thus the "bare" QRPA mode is properly clothed, bringing theory in overall agreement with experiment (see Table 1, second and third lines; for details cf. [27] as well as [7]).
To test how robust the results displayed in Fig. 1  Because of the PVC mechanism, the different valence quasiparticle states undergo renormalization and fragmentation, phenomena which can be specifically probed with one-particle transfer reactions. In Fig. 4(a) we display the absolute differential cross sections associated with the reaction 120 Sn(d, p) 121 Sn(lj), calculated making use of the spectroscopic amplitudes associated with the strongest populated fragments of the valence orbitals h 11/2 , d 3/2 , s 1/2 and In discussing the 120 Sn(p, d) 119 Sn reaction we concentrate on the d 5/2 orbital, the most theoretically challenging of all of the valence single-particle strength functions. This is because this state, being further away from the Fermi energy ( d 5/2 = −11.3 MeV, F ≈ −8MeV) than the other four valence orbitals (see Table 2), is embedded in a denser set of doorway states (of type s 1/2 ⊗ 2 + , d 3/2 ⊗ 2 + , g 7/2 ⊗ 2 + , h 11/2 ⊗ 3 − , etc.), as compared to the other ones. Consequently, it can undergo accidental degeneracy and thus conspicuous fragmentation. As seen from Table 3, although the calculated summed cross sections (σ = 6.15 mb) agree, within experimental errors, with observation (7.93 ± 2 mb), theory predicts an essentially uniform fragmentation of the strength over an energy interval of ≈ 760 keV, while the data [32] is consistent with a concentration of the strength at an energy close to that of the lowest theoretical 5/2 + level (1090 keV).
In keeping with the above scenario we have shifted the bare single-particle energy d 5/2 by 600 keV ((-11.3 + 0.6) MeV= -10.7 MeV), amounting to a 6% change in the k−mass (i.e. from 0.7m to 0.74m, Table 2 ), and recalculated all the quantities discussed above.
Making use of the corresponding nuclear structure results and of global optical parameters, the absolute differential cross sections associated with the 5/2 + states populated in the reaction 120 Sn(p, d) 119 Sn and lying below 2 MeV have been calculated. They are displayed in Fig. 4(b) in comparison with the experimental data. Theory provides now a quantitative account of the experimental findings. In particular, of the fact that the strength function is dominated by a single peak. With the 600 keV shift, it is predicted at an energy of 1050 keV carrying 4.4 mb and it is observed at 1090 keV with a cross section of 5.35 ±1.3 mb.
The resulting overall agreement between theory and experiment is further confirmed by Fig.   4(c) where the absolute value of the one-particle transfer strength function associated with the population of 5/2 + states predicted by the calculation is compared with experiment. As observed in Fig. 4(d), also this two-dimensional projection of the multidimensional nuclear structure landscape is funnelled, testifying to the physical robustness of the findings.
An alternative approach to the one discussed above which leads to almost identical findings regarding the d 5/2 fragmentation, can be obtained by treating the energy of the five valence orbitals as parameters to be optimized selfconsistently within the framework of the full NG calculations, so as to best reproduce the quasiparticle spectrum (Opt. Table 2).
The results can be expressed in terms of a state-dependent k−mass is displayed in the inset to Fig. 1(a). Again, this two-dimensional section of the nuclear structure landscape is of a well funnelled character. Within this context, it is noted that a measure of the reliability with which theory can describe the nuclear structure is provided by the relative dimensionless standard deviations σ rel (equal to e.g. σ(E qp )/ E qp in the case of the quasiparticle spectrum) associated with each of the different observables, and taken at the minimum of the nuclear structure landscape, as shown in Table 4.
We conclude that a theoretical description of nuclear structure based on single-particle (mean field with m k ≈ 0.7m) and collective motion (QRPA) and on their interweaving controlled by the particle-vibration coupling mechanism and leading to renormalization of both types of nuclear excitations through mass (self-energy) and screening (vertex) corrections and induced pairing, can provide an overall quantitative account of the nuclear structure representative of a mass zone (group of nuclei displaying homogeneous properties like e.g. sphericity and superfluidity, likely circumscribed by phase transition domains). Allowing for a weak state dependence of the k-mass, determined by optimising the energy of the valence single-particle orbitals to reproduce the quasiparticle spectrum, the theoretical description of the nuclear structure probed in terms of direct reaction absolute differential cross sections and based on renormalized single-particle and collective degrees of freedom, becomes accurate within a 10% error level. The PVC mechanism is found to play a central role in   Fig. 2(a)).
[29] In carrying out the calculations reported in Figs  Orbital      Table II  (h) Mean square deviation between the experimental and theoretical energies of the members of the h 11/2 ⊗ 2 + multiplet shown in (g).