Indirect method for nuclear reactions and the role of the self energy

TL;DR

The work develops a unified framework tying nuclear structure to reaction observables through a complex, non-local, energy-dependent self-energy encoded as the optical potential. The Green's Function Transfer (GFT) formalism leverages this self-energy to compute inclusive and exclusive cross sections for indirect reactions, linking structure calculations (NFT and ab initio CC) directly to reaction kernels without ad hoc spectroscopic factors. It clarifies the relation between direct and indirect transfer via R-matrix concepts and shows how a consistent treatment of the $x$–$A$ system yields absolute cross sections that reflect underlying many-body dynamics. The approach is validated through NFT studies of $^{10}$Li and CC+GFT applications to Ca isotopes, and it provides a path to predictive, microscopic descriptions of reactions involving exotic nuclei, including explicit consideration of three-body dynamics beyond the spectator approximation.

Abstract

When a nuclear species (e.g., a nucleon or a deuteron nucleus) propagating freely is made to collide with a target nucleus, its trajectory is modified by exchanging variable amounts of energy, mass, linear and angular momentum with the target, according to its interaction with the nuclear medium. By addressing this perturbation away from the free path, one hopes to learn something about the nature of the medium through which our probe propagates. This is the essence of the experimental use of nuclear reactions for the purpose of gathering information about nuclear structure. In order to deal with the structure and the reaction aspects of a specific experiment on the same footing, it is therefore desirable to identify a theoretical construct that embodies the modification of the propagation of a particle in the medium with respect to the free case, and use it both for the determination of the nuclear spectrum (structure) and for the calculation of scattering observables (reaction). A candidate for such an object is the self energy, and we will try in the present lectures to put it at the center stage in the formulation of scattering theory.

Figures (6)

• Figure 1: Schematic representation of the inclusive reaction $a+ A\to b+(x+A)$, where the fragment $b$ is detected, and the nucleon $x$ is not. The interaction between $b$ and the rest of the system is approximated with an optical potential. The spectator approximation assumes that the interaction between $x$ and $A$ can be treated as a two-body problem, in which the fragment $b$ acts as a spectator.
• Figure 2: Schematic depiction of the distinctive features of the Distorted Wave Born approximation (DWBA) as compared with the Green's Function Transfer (GFT) formalism.
• Figure 3: Phase shifts (top) and spectral functions (bottom) for $^{10}$Li from the renormalized NFT calculation of barrancoMathrmLiReactionSpecific2020, compared with the unperturbed ("unp.") case. (a) Negative parity: the $2^-$ and $1^-$ channels develop a low--lying $\widetilde{1/2}^-$ resonance at $\sim$0.5 MeV. (b) Positive parity: the $s$--wave becomes virtual (large negative $a$) and a broad $\widetilde{5/2}^+$ resonance appears at a few MeV. (c,d) Differences of spectral functions highlight the redistribution of strength due to particle--vibration coupling.
• Figure 4: Absolute $(d,p)$ predictions from barrancoMathrmLiReactionSpecific2020: (a) strength function folded over the forward window $5.5^\circ$--$16.5^\circ$ (dominated by $p$--wave); (b) corresponding angular distributions in a representative energy interval; (c) strength integrated over backward angles $50^\circ$--$180^\circ$, where the $s$--wave becomes prominent; (d) low--energy angular distribution ($E{\lesssim}0.2$ MeV) emphasizing the virtual $s$--wave at $\theta_{\rm c.m.}\gtrsim 40^\circ$.
• Figure 5: Results obtained from the analysis of the $^9$Li($d,p$)$^{10}$Li reaction within the renormalized NFT + GFT framework: (i) parity inversion encoded in a virtual $\widetilde{1/2}^+$ (large negative scattering length) alongside a low--lying $\widetilde{1/2}^-$ resonance; (ii) dressing by the strong $2^+$ vibration of the $^{9}$Li core and associated two--phonon effects that redistribute $d_{5/2}$ strength; (iii) momentum matching in $(d,p)$ that suppresses the $s$--wave at forward angles yet enhances it at $\theta_{\rm c.m.}\gtrsim 40^\circ$.