# Course 3: Few-body methods and nuclear reactions

Few-body methods and nuclear reactions was first held at the ECT* in Trento, Italy from July 20 to August 7, 2015.

## Contents

- Motivation and background
- Course objectives and learning outcomes
- Format and teaching
- Prerequisites
- Course outline
- Teachers and organizers

## Motivation and background

Few-body theories and phenomenology represent an important frontier between nuclear physics and QCD, the underlying theory of strong interactions. Great efforts are done by the few-body nuclear community to analyze the capability of the different interactions, that, based on various symmetry considerations, have been suggested over many decades, to describe the few-body nuclear dynamics. To this aim one has to minimize the error in the solution of the quantum mechanical problem, which could taint the comparison with experimental data.

The subject of this course was different methods used to describe few-nucleon systems. Though during the course the theory of the interaction will not be discussed in detail, a brief introduction to this subject was given. The choice of protons and neutrons as effective explicit degrees of freedom in the Hamiltonian implies the presence of forces of many-body (particularly three-body) nature, besides the usual nucleon- nucleon (NN) interaction. This fact gives particular importance to the A=3,4 systems which was discussed in detail.

The ultimate motivation in a program devoted to solve the quantum mechanical problem of interacting nucleons is to evaluate the capability of the underlying theory to describe observables. The ground state energy and the mean square radius are the first observables to be analyzed. Methods dealing with this kind of observables enter in the category of bound state methods. Very different techniques have to be applied to describe scattering observables as cross sections, asymmetries, recombination rates, response functions, etc. Specific differences dealing with bound state or scattering state techniques were discussed in detail during the course. Possibilities of extensions of the methods to larger systems and/or non-nuclear systems were also discussed.

## Course objectives and learning outcomes

The Nuclear Talent course on Few-body methods and nuclear reactions ran at the ECT* in Trento, in collaboration with the University of Trento, starting July 20 (arrival July 19) and ending August 7 (departure August 8) in 2015.

The objectives and learning outcomes of the course were to provide the students tools in order to:

- Understand the basic ingredients of the methods used to solve the few-nucleon problem
- Make contact with the numerical techniques and programming languages used by experts in solving the few-nucleon problem.
- Understand the main characteristics of the NN interaction and the origin of the three-nucleon force
- Be able to solve numerically the two-body Schrödinger equation for bound and scattering states
- From those results, be able to construct observables and compare to experimental data
- Understand the general properties of basis functions for a general number of fermions including spin-isospin quantum numbers
- Be able to solve selected problems in systems with A=3,4
- Know the low energy behavior of the main scattering observables
- Know and verify correlations between bound state and scattering observables
- Be aware of the possibility to extend the methods to non-nuclear systems

## Format and teaching

There were lectures in the mornings for a total of 50 hours and exercises in the afternoons for a total of additional 50 hours. The intention was to immediately apply, in the afternoons, the explanations given in the mornings. To this end exercises were organized in such a way that the students got the necessary tools to solve selected problems by performing numerical applications by themselves. The lecturer team made a strong effort to merge the different subjects into a unitary presentation, uniforming the formalism, stressing cross referencing. While each week had only two lecturers in charge, all members of the team were present and ready to make comments and clarifications if needed. The whole team was involved in the exercises, supervising and discussing with the students. The latter were divided in small working groups, but collective discussions of the problems encountered, at the end of each (or every two) exercise sessions were encouraged. Weekly comprehensive discussions about the contents of the lectures in relations to the achievements of the groups were organized as well.

On request a final assignment, to be completed after the end the course, was be given to students who need 7 certified ECTS credits. These correspond to 3.5 credits in the US. The final assignment will be graded with marks A (4.0), B (3.5), C (3.0), D (2.5), E (2.0) and failed for Master students and passed/not passed for PhD students.

The organization of the day was as 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

## Prerequisites

Students should have working proficiency in quantum mechanics. This includes knowledge of the following basic physics concepts and mathematical tools: concept of wave-function, Schroedinger equation, spin, Hilbert spaces, linear operators, differential operators, group theory. Students are also expected to have strong operating programming skills in Fortran, C/C++ and/or Python. Other languages are allowed but may not be supported by the teaching staff.

## Course Outline

### Introduction

Since the course covers a wide range of topics, it will be necessary to unify the different backgrounds of the students and to introduce the notation. Moreover an outline of the main ingredients appearing in the construction of modern NN interaction and 3-nucleon forces (3NFs) will be given. Specific topics are

- The NN interaction, from phenomenological potentials to chiral perturbation theory
- The two-nucleon problem: bound states
- The two-nucleon problem: scattering states
- Some properties of scattering states: low energy behavior and integral relations Exercises will include
- Brief introduction to programming languages and the use of computing resources
- Writing a code to solve the two-nucleon system using partial wave decomposition
- Calculation of observables and comparison to the experimental data

### The Faddeev and Faddeev-Yakubowsky Equations

Here the Faddeev and Faddeev-Yakobowsky equations will be discussed in configuration as well as in momentum space. The different numerical techniques used to solve the equations will be illustrated. The discrete and the continuum spectrum will be analyzed. The connection between the solution of the equations and the description of three- and four-nucleon reactions will be discussed. Important aspects of this chapter will be the discussion of the three-nucleon force and the treatment of the Coulomb interaction. Complex scaling will be introduced as a method to compute the breakup amplitude in configuration space Exercises will include:

- Derivation of the Faddeev and Faddeev-Yakobowsky equations in the case of separable and s-wave interactions. Applications to the calculation of the Efimov spectrum.
- Use of the Faddeev and Faddeev-Yakobowsky equations to describe simple reactions as n-d and n-3H.

### Methods based on Basis expansions

The construction of a general basis for fermions will be addressed. The specific case of the HH basis will be discussed in detail, however the relation with the harmonic oscillator (HO) will be given. The particular case of the adiabatic expansion will be illustrated. The direct applications of large basis to a bound state problem will be discussed in the context of the Rayleigh-Ritz variational principle. Application to scattering states using the Kohn variational principle will be discussed as well. The description of few-nucleon reactions using expansion basis will be illustrated. In this context the Resonating Group Method (RGM) will be introduced. Exercises will include:

- the study of the convergence properties of the HH/HO basis using different types of potentials
- numerical implementation of large matrix algebra

### Few-nucleon reactions with external probes

The study of few-nucleon systems using external probes will be discussed. The main ingredients for the calculation of electro-weak reactions will be introduced. Bound state type techniques to calculate reactions (Lorentz Integral Transform, Complex Scaling and other methods) will be explained and put in practice in physical cases. Exercises will include:

- calculation of the dipole response function to the deuteron, 3He and 4He for simplified potentials with different techniques
- use of different routines for inversion of integral transforms

## Teachers and organizers

The teachers were

- Nir Barnea (The Hebrew University of Jerusalem), nir@phys.huji.ac.il
- Mario Gattobigio (INLN - University of Nice), mario.gattobigio@inln.cnrs.fr
- Alejandro Kievsky (INFN Pisa), alejandro.kievsky@pi.infn.it
- Rimantas Lazauskas (IPHC Strasbourg), Rimantas.Lazauskas@iphc.cnrs.fr
- Andreas Nogga (FZ Jülich), a.nogga@fz-juelich.de
- Giuseppina Orlandini (University of Trento, Italy), orlandin@science.unitn.it

The organizers were

- Alejandro Kievsky (INFN Pisa), alejandro.kievsky@pi.infn.it
- Giuseppina Orlandini (University of Trento, Italy) and contact person, orlandin@science.unitn.it

Student coordinator and advisor

- Morten Hjorth-Jensen (Michigan State University and University of Oslo, Norway), hjensen@nscl.msu.edu