Recent Applications of Self-Consistent Green's Function Theory to Nuclei

We discuss recent \emph{ab initio} calculations based on self-consistent Green's function theory. It is found that a simple extension of the formalism to account for two-nucleon scattering outside the model space allows to calculate non-soft interactions. With this, it is possible to make predictions for Lattice QCD potentials, obtained so far at pion masses of $m_\pi = 0.47$GeV/c$^2$. More traditional calculations that use saturating chiral EFT forces yield a good description of nuclear responses and nucleon knockout spectroscopy.

where the indices α, β, . . . label the states of a single-particle model space basis and g (0) (ω) is the uncorrelated propagator, which correspond to a mean field (MF) reference state. The central quantity is the irreducible self-energy, Σ (ω) ≡ Σ (∞) +Σ(ω), which acts as an optical potential for nucleons inside the correlated medium. One separates an enegy-independent term Σ (∞) that describes the average MF potential felt by all particles. The dynamic partΣ(ω) contains the effect of correlations beyond MF. We calculate this in the third-order algebraic diagrammatic construction [ADC (3)] approximation, which is a non perturbative resummation of 2p1h and 2h1p configurations [3]. We generally construct the selfenergy based on a reference MF state, which is derived to best approximate g(ω) and, therefore, solve Eq. (1) iteratively. The interested reader is referred to Refs. [3,4,5] for details of the formalism and of our implementation. Once the propagator is known, it is easy to extract information on the ground state energy, nucleonnucleus scattering, single-particle spectroscopy, response to external probes, and so on [3]. In the following we describe a few very recent applications based on nuclear forces obtained either directly from Lattice QCD calculations or from chiral effective field theory (EFT).
1. Taming the hard-core of forces from Lattice QCD The HAL QCD collaboration has devised an approach to extract n-nucleon forces from QCD simulation of 3n quarks in a space-time lattice [6,7,8]. This method generates systematically consistent two-, three-and many-nucleon interactions that are faithful to the few-body data and scattering phase shifts by construction. An earlier set of potentials was derived in the flavour SU(3) limit with different masses of the (pion) pseudo-scalar meson. Among these the force with lightest value of M PS = 469 MeV/c 2 shows saturation on nuclear matter [9]. We refer to this as the HAL469 S U(3) interaction. Note that more advanced interactions at near the physical pion mass are currently being computed [10]. For forces with a strong short-range repulsion, like the HAL QCD interactions, usual truncations of the oscillator space (of up to 12 shells in this case) are not sufficient and a resummation of ladder diagrams outside the model space is required. We do this by solving the Bethe-Goldstone Equation (BGE) in the excluded space according to Refs. [11,12] and add the corresponding diagrams to the mean field (MF) term of the self-energy, which becomes energy dependent [4]: where T BGE αγ, βδ (ω) are the elements of the scattering t-matrix in the excluded space. A static effective interaction is then extracted that we use to calculate the ADC(3) self-energy within the included model space. To do this, we solve the Hartree-Fock (HF) equations with the MF potential of Eq. (2): where latin indices label HF states. We then define a static interaction in this HF basis similarly to Refs. [4,13]: It should be noted that the BGE used to generate T BGE (ω) resums scattering states where at least one nucleon is outside the whole model space, while full ADC(3) correlations are computed for all nucleons inside the space. Hence, T BGE (ω) does not suffer from ambiguities with the choice of the single-particle spectrum at the Fermi surface encountered with the usual G-matrix used in Brueckner HF calculations. The combination of the ladder diagrams contained in the ADC(3) expansion and those in the T BGE (ω) accounts for the complete diagonalization of short-distance degrees of freedom. One can then study the infrared (IR) convergence of total binding energies to a complete basis set. Ref. [14] established that a harmonic oscillator (HO) model space, of frequency Ω and truncated to the first N max + 1 shells, behaves as a hard wall spherical box of radius Given an interaction that is independent of the model space, if the ultraviolet (UV) degrees of freedom are diagonalised exactly then the calculated ground state energies are expected to converge exponentially when increasing the effective radius L 2 : For the bare HAL469 interaction, if we use the SCGF without ladder diagrams from outside the model space, the extrapolation according to Eq. (6) will fail because the short-distance repulsion requires extremely large model spaces (N max >> 20) to reach UV convergence, see the left panel of Figure 1.
The results with T BGE (ω) and the interaction (4) included are displayed in the right panel and show good IR convergence. The total energy is extracted from a nonlinear least-squares fit to Eq. (6). The one-body propagators of 4 He, 16 O and 40 Ca have been calculated using T BGE (ω) to remove UV modes. We employed spherical harmonic oscillator spaces of different frequencies, Ω, and sizes up to N max =max{2n + } =11 (and N max ≤ 9 for 40 Ca) and we used the Koltun sum rule to extract the intrinsic ground state energies from g(ω). Fig. 2 shows that the complete resummation of ladder diagrams outside the model space results in a somewhat flat behaviour of the total energies for Ω ≈5-20 MeV. While some oscillations w.r.t. Ω are still present, the IR convergence pattern (shown in Fig. 1 for 4 He) is very clean for all three isotopes. This gives confidence that the short-range repulsion of HAL469 is relatively mild and that it is accounted for accurately. From calculations up to Ω=50 MeV and the IR extrapolation, we estimate a converged binding energy of 4.80(3) MeV for 4 He, where the error corresponds to the uncertainties in the extrapolation. This is to be benchmarked on the exact result of -5.09 MeV, which is known from Stochastic Variational calculations [15]. Since the method is size extensive, we assume a 10% error due to many-body truncations for all isotopes, as a conservative estimate.
We obtain -17.9(0.3)(1.8) MeV for the ground state energy of 16 O, where the first error is from the IR extrapolation. Hence, 16 O is unstable with respect to 4-α break up, by ≈2.5 MeV, although allowing an error in our binding energies of more than 10% could make it slightly bound. This is in contrast to the experiment, at the physical quark masses, where the 4-α breakup requires 14.4 MeV. On the other hand, 40 Ca we calculate -75.4(6.7)(7.5) MeV and this is stable with respect to α breakup by ≈24 MeV.   [16,17] and Coulomb excitation [18] experiments, respectively. The size of the HO basis in the calculation is N max = 13 (i.e., 14 major shells) and frequency Ω = 20 MeV. The interaction used is NNLO sat .

Isovector Dipole Nuclear Response
In this Section we focus on the 16 O, 22 O, 40 Ca and 48 Ca nuclear response produced by an isovector dipole electric field, E1. The corresponding operator, corrected for the center-of-mass displacement, is: which probes the excitation spectrum with multipolarity and parity J π =1 − . We calculate the response function R(E) of Eq. (7) by solving the usual random phase approximation (RPA) in the particle-hole channel and starting from the same MF propagator that is used as reference state to obtain the selfenergy in previous sections. As mentioned in the introduction, this includes effects of correlations that go beyond the usual HF mean field and, in particular, those responsible for reproducing the correct centroid of giant resonances. We then compute the total photoabsorption cross section as and the dipole polarizability, which is the accumulated E1 strength weighted by the inverse of the energy: In the above relations, α denotes the fine-structure constant.
Our results for the total cross sections for 16 40 Ca and 48 Ca, computed in the SCGF approach (second column), the CC-LIT approach (third column) and obtained by integrating the experimental data (fourth column).  NNLO sat chiral interaction that reproduces radii in this region and has good saturation properties [19]. For all isotopes considered, the position of the giant dipole resonance is fairly well reproduced. By moving from the N=Z nucleus 16 O to the neutron-rich 22 O, the strength of the response is distributed towards lower energies, with the first RPA peak shifted to energy below 10 MeV. In the corresponding experimental curve, there appears a soft dipole mode of excitation, with a weaker strength compared to the giant resonance at higher energies.
In Table 1, dipole polarizabilities α D of Eq. (9) are compared with values obtained from the integrated experimental spectrum and from a Coupled Cluster combined with the Lorentz Integral Transform (CC-LIT) approach calculation [20,21]. For the 16 O, where the experimental dipole polarizability is fully integrated over the entire spectrum, our computed α D lacks 15% of the strength with respect to both the experimental value and result of the CC-LIT approach. The total strength in the SCGF calculation is better recovered for higher mass 40 Ca and 48 Ca, and even within the experimental error for 48 Ca. Correcting the discrepancies with CC-LIT may require improvements of the many-body truncation to go beyond the RPA and we are currently working to implement such extensions.

Spectroscopic factors
The fragmentation of the single-particle spectral function is mostly determined by long-range correlations (LRC) and it is constrained by general properties at the energy surface, such as the density of states and gaps [22]. As an example, Fig. 5 demonstrates the dependence of spectroscopic factors (SFs) for the dominant quasiparticle peaks in 56 Ni on the particle-hole gap ∆E ph (which has an experimental value of 6.1 MeV). This is calculated from a chiral N3LO two-nucleon interaction only but with a modification of its monopole strength in the p f shell. The monopole correction controls the separation among the p 3/2 and the f 7/2 orbits. Hence, by varying its strength one can control the predictions for ∆E ph . Calculations for spaces of different sizes and HO frequencies are not converged w.r.t. the model space but all lines lay on top of each other, showing that there exists a clear correlation and that SFs can be strongly constrained by observable quantities at low energy.
Since the correlation of SFs with ∆E ph is strong, we calculated the oxygen chain using NNLOsat Hamiltonian of Ref. [19], which has the advantage of predicting gaps accurately. The spectroscopic factors obtained in the ADC(3) scheme are shown in the right panel of Fig. 5 and are sensibly smaller than previous results from older Hamiltonians that had a too dilute spectrum at the Fermi surface [23]. Remarkably, the NNLOsat results are very close (almost equal) to past FRPA calculations of 16 O where the quasiparticle energies were phenomenologically constrained to their experimental values [24]. These are also in good agreement with (p,2p) measurements from R 3 B at GSI [25].   Figure 5. Left. Dependence of spectroscopic factors of the first quasiparticle and quasihole peaks in 56 Ni as a function of the particle-hole gap ∆E ph [4]. The calculated quenching is strongly correlated to the gap even when full convergence with respect to the model space is still not achieved. Note that ∆E ph ≡ E A+1 0 − E A−1 0 is an experimentally observable quantity. Here, it is varied over a range of values by tuning the monopoles of the NN interaction. Right. Calculated spectroscopic factors for protons and neutrons around Oxygen isotopes obtained from NNLOsat. Each point refer to the separation of a nucleon from the isotope indicated nearby to the ground state of the daughter nucleus. Likewise, stronger quenching is found for those isotopes with smaller ∆E ph gap.