# Course 2: Many-body methods for nuclear physics

The first edition of course 2 was given at GANIL in Caen, France from July 5 to July 25, 2015.

## Contents

- Motivation and background
- Course content
- Teaching
- Teachers and organizers
- Lecture notes and detailed program
- Admission and Application

## Motivation and background

The course aims at teaching basic theoretical approaches that are used in first-principle calculations of quantum mechanical observables of relevance for nuclear physics.

New radioactive beam experiments will continue to challenge nuclear theory. Properties of newly discovered isotopes and their observed decays at facilities like MSU/FRIB, RIKEN, or GANIL/SPIRAL2 will need to be qualitatively understood and quantitatively explained. New and disappearing shell closures appear to require a detailed understanding of the role of three-body interactions. Neutron radii as probed in PREX and CREX experiments will challenge theory. Extracting information related to correlations that change proton and neutron properties as a function of (N-Z)/A will require a better integration of nuclear reaction descriptions with nuclear structure approaches. While the course will focus on ab-initio methods and their applications, links with more empirical approaches like density functional theory (DFT) and its time-dependent implementation will be provided to present a balanced perspective allowing emerging new information to be put in a proper perspective. As nucleon-nucleon interactions are of prime importance for ab initio many-body calculations we will also devote one lecture to give an appropriate overview with emphasis on chiral interactions.

Clearly, conducting a research program in tune with new experimental developments requires extensive knowledge of various aspects of nuclear many- body theory, including open quantum systems. On the one hand, advances in ab- initio theory are pivotal to provide reliable predictions. On the other, it is the insight into the many-body structure of nuclei that allows understanding the physics and behavior of atomic nuclei, ideally guiding future experiments and discoveries. In this respect, a long-standing holy grail has been to construct microscopically motivated theories that handle both structure and reaction processes consistently, in order to reduce model dependence when drawing conclusions. It is therefore important to provide young scientists in the field of low-energy nuclear structure with adequate training in nuclear many-body theory.

High-quality ab-initio methods have been implemented in the last ten years to address nuclei up to masses of A~100. Methods such as coupled cluster (CC), Gorkov self-consistent Green’s function (SCGF) theory and in-medium similarity renormalization group (IMSRG) have recently achieved converged calculations with realistic chiral two- and three-nucleon interactions. Gorkov SCGF calculations for semi-magic open shells were beyond reach just a few years ago and are now a reality. These approaches are now addressing questions such as the origin of evolution of nuclear shells, the position of the drip lines, and the influence of the continuum at the drip lines. Some future key challenges for ab-initio theory will be to describe fully open-shell nuclei (beyond the semi-magic currently possible), predict scattering observables, and continue to develop strategies that can handle harder underlying nucleon-nucleon interactions in finite systems. CC methods and the SCGF method are applied with new vigor to the calculation of bulk properties of infinite matter including the treatment of three-body interactions.

In order to relate these calculations to measurements, to discuss questions on the nature of exotic isotopes and to motivate future experiments, it is becoming more and more important to evaluate the uncertainties in the data analysis due to reaction models. Various processes, such as transfer reactions (d,p), (p,d), (d,3He) or inverse kinematics knockout reactions like (p,2p), (p,pn), have been (and are) employed for this purpose. Unfortunately, experimental results are often inconsistent with each other due to the use of different kinematical regimes and different theoretical reaction models in the analysis of data. One possible path to resolve some of these discrepancies is given by the dispersive optical model (DOM), which relates elastic nucleon scattering to single-particle properties below the Fermi energy. Its link to SCGF many-body theory—that provides the DOM framework—is particularly helpful. Understanding these links is helpful to pursue theoretically guided extrapolations to the drip lines and also to determine overlap wave functions (which are not constrained in other approaches). Thus, covering the relations between the DOM and ab-initio approaches will help students to better understand some of the uncertainties discussed above.

All the above techniques are at the basis of our capabilities to understand and further investigate microscopically important open issues, including:

- Systematics of single-particle properties (that is both energies and spectral strength) along isotopic chains including the role of the nuclear tensor force.
- Charge-exchange reactions and collective modes in Z/N asymmetric nuclei.
- Response to electroweak probes
- Implications of collective response to stability and symmetry energy.
- Evolution of the coupling between single-particle and collective modes when approaching the drip lines

Further understanding of the extremes of stability come from studies of infinite (symmetric and neutron) nuclear matter, since these have important implications for the phase diagram of matter around normal nuclear density. The course will therefore cover some of the recent developments of CC and SCGF methods to calculate nuclear matter with NN and three-nucleon interactions. The role of fast nucleons—that can be probed in high-momentum experiments at Jefferson Lab—on the density dependence of the nuclear symmetry energy has also recently come into focus. The study of pairing in infinite nuclear systems is receiving a strong impetus from observational astrophysics of neutron stars and provides a link with the description of pairing in finite nuclei.

## Course content

The Nuclear Talent course on Many-body methods for Nuclear Physics was given at GANIL in Caen, Normandie, France, in collaboration with the University of Basse-Normadie, starting July 6 (arrival Sunday July 5) and ending July 24 (departure July 25) in 2015.

The course aimed at teaching modern theoretical approaches to the nuclear many-body problem that start from realistic nucleon-nucleon interactions. The goals were to (1) provide theoretical students with enough tools and background knowledge to comfortably perform basic nuclear structure and scattering calculations, and (2) develop a sufficient insight into modern many-body theory to guide experimental students into properly understanding the links with nuclear structure experiments.

A basic list of topics is given below, with further refinements by the lecturers.

- Quick review of Wick’s theorem and many-body perturbation theory.
- Many-body propagators and concept of spectral and response functions.
- Introduction to approaches for finite nuclei: SCGF, CCM, IM-SRG.
- Advanced ab-initio methods implementations for finite systems: ADC(3), Λ-CCSD(T), etc…
- Correlations in nuclei and link between structure and reactions.
- Theory of optical potentials, empirical DOM and elastic scattering.
- Methods for infinite matter: SCGF, CCM. The problem of nuclear saturation.
- Asymmetric nucleonic matter: thermodynamics and symmetry energy.
- BCS gap equations and extensions in infinite matter.
- Particle symmetry breaking and pairing in finite systems.

## Teaching

The course was held from July 6 to July 24 in 2015 at GANIL in Caen, Normandie, France. It was organized in collaboration with the University of Caen Basse-Normadie. The course took the form of an intensive program of three weeks, with a total time of approximately 45 h of lectures and directed exercises, about 60 h devoted to exercises and possible computational projects and a final assignment worth approximately 2 weeks of work. The total workload amounted to 150-170 hours, corresponding to 7 ECTS in Europe (3.5 credits in the US). The final assignment was graded with marks A, B, C, D, E and failed for Master students and passed/not passed for PhD students.

The organization of the day is follows:

Time Activity 9am-12pm Lectures, directed exercises 12pm-2pm Lunch 2pm-6pm Hands-on sessions, computational projects 6pm-7pm Wrap-up of the day

## Teachers and organizers

The teachers were

- Carlo Barbieri (CB), University of Surrey, c.barbieri@surrey.ac.uk
- Wim Dickhoff (WD), Washington University, St. Louis, wimd@wuphys.wustl.edu
- Gaute Hagen (GH), Oak Ridge National Laboratory, hageng@ornl.gov
- Morten Hjorth-Jensen (MHJ), Michigan State University and University of Oslo, hjensen@nscl.msu.edu
- Arturo Polls (AP), University of Barcelona, artur@ecm.ub.es

The organizers were

- Carlo Barbieri (University of Surrey), c.barbieri@surrey.ac.uk
- Wim Dickhoff (Washington University, St. Louis), wimd@wuphys.wustl.edu
- Francesca Gulminelli (University of Caen Basse Normadie) gulminelli@lpccaen.in2p3.fr
- Morten Hjorth-Jensen (Michigan State University and University of Oslo), hjensen@nscl.msu.edu
- Marek Płoszajczak (GANIL) ploszajczak@ganil.fr, and contact person

## Lecture notes and detailed program

All teaching material for the course can be found at its github address. The source files are at the github repository and a simple git clone will allow you to fetch the most recent version of all lecture notes and project/exercise files. If you have never used version control software, we recommend strongly that you get used to it. We require that when working with the different projects that all groups establish their own github repositories.

We provide also at the same repository some background material on second quantization and nuclear physics. We recommend strongly that you study this material before attending the course, and if you are not too familiar with second quantization we would advice you to study the material at the above links to the github repository. Parts of the material is based on the first 7 chapters of the textbook "Many-body Theory Exposed!" by Dickhoff and Van Neck. These chapters deal with antisymmetric states, second quantization, independent-particle model, two-body states, nuclear matter, single-particle propagator in a one-body system, and single-particle propagator in a many- fermion system. The slides can be downloaded from the following the second quantization link. Additional material introductory material and more material on second quantization can be found at the github address github address of the course.

We recommend also that you study the project part at the github link as that defines the topics we will work on during the course. An essential element of the Talent courses is to develop a large project(s) which allows you to study and understand theoretical concepts in nuclear physics. These concepts will in turn allow you to interpret results from experiments and understand the pertinent physics in terms of the underlying forces and laws of motion. Together with the regular lectures in the morning, the hope is that during these three weeks you will be able to write and run a program which implements at least one of the methods discussed during the lectures. The lectures will also cover additional material which aims at giving you a broader view on what can be achieved with the methods to be discussed. Combined with the 'hands-on' afternoon sessions, the hope is that the lectures and the computational projects will together allow you to achieve these goals. For those of you who would like to get credits to be transferred to your home university, the project(s) can be extended upon allowing you to include further elements to the many-body methods. The load of the final project is estimated to be 80 hours. In total, attendance at the course and doing the final project amounts to seven ECTS. There is no credit transfer for Northern-American students.

Monday 2015-07-06 | Review second quantization; Hartree-Fock; time-independent perturbation theory (MHJ) | Many-body PT, FCI (MHJ) | Lecture slides | Projects and exercises |
---|---|---|---|---|

Tuesday 2015-07-07 | Theory of Finite Nuclei (FN) 1: CCSD (MHJ) | Time-dependent perturbation theory; Propagators and Dyson equation (CB) | Lecture slides | Projects and exercises |

Wednesday 2015-07-08 | Coupled cluster theory continues (MHJ) | FN 2: SCGF (CB) | Lecture slides | Projects and Exercises |

Thursday 2015-07-09 | Infinite nuclear matter (INM) 1: Brueckner-Hartree-Fock approach to infinite nuclear matter (AP) | Correlations and experiment: (e,e'p) and other data (WD) | Lecture slides | Projects and Exercises |

Friday 2015-07-10 | INM 2: Theory of Infinite nuclear matter. SCGF (AP) | Dispersive Optical Model (DOM) 1 (WD) | Lecture slides | Projects and Exercises |

Prospective student participants will be expected to already have some experience in programming and be familiar with data manipulation and graphical utilities, etc. For the projects we would recommend using either Fortran or C++ as programming languages, but Python can be used as well. A good foundation in both standard mathematical methods for scientists and a practitioners knowledge of intermediate level (advanced undergraduate/postgraduate) quantum mechanics is important. Students who have not already studied the above formally will be expected to study suggested pre-course materials in advance of the course.