A complete characterization of the structure of nuclei can be obtained by combining information arising from inelastic scattering, Coulomb excitation and γ−decay, together with oneand 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. ∗ andrea.idini@gmail.com † gregory.potel@gmail.com ‡ barranco@us.es § vigezzi@mi.infn.it ¶ broglia@mi.infn.it 1 ar X iv :1 50 4. 05 33 5v 2 [ nu cl -t h] 2 9 A pr 2 01 5 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 Sn involved in the characterization of the spherical, superfluid 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 v14( S0)(≡ v p ) Argonne pairing potential. HFB provides an embodiment of the quasiparticle spectrum while QRPA a realization of density (J = 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 p were calculated (see [7]; see also [8–15]). Adding v p to the bare interaction v bare p , the total pairing interaction v p was determined. With these elements, the Nambu-Gor’kov (NG) equation was solved selfconsistently using Green’s function techniques [16–20], and the parameters characterizing the renormalized quasiparticle states obtained. Within this framework, the quasiparticle energies Ẽν are given by Ẽν = √ (̃ν − F )2 + ∆̃ν . The renormalised single-particle energy ̃ν − F = Zν [( ν − F ) + Σ ν ], is written in terms of the HF energy ν and of the even part of the normal self-energy, the quantity Zν providing a measure of the single-particle character of the orbital ν. The state dependent pairing gap ∆̃ν = ∆̃ bare ν + ∆̃ ind ν obeys the generalized gap equation ∆̃ν = −Zν ∑ ν′>0 〈ν ′ν̄ ′|vbare p + v p |νν̄〉Nν′ ∆̃ν′ 2Ẽν′ , (1)