Review Article Mean field and beyond description of nuclear structure with the Gogny force: A review L.M. Robledo1;2, T.R. Rodr´ıguez2, and R. R. Rodr´ıguez-Guzm´an3 1 Center for Computational Simulation, Universidad Polit´ecnicade Madrid, Campus de Montegancedo, Boadilla del Monte, 28660-Madrid, Spain 2 Departamento de F´ısicaTe´oricay Centro de Investigaci´onAvanzada en F´ısica Fundamental, Universidad Aut´onomade Madrid, E-28049 Madrid, Spain 3 Physics Department, Kuwait University, Kuwait 13060 E-mail: [email protected],[email protected],[email protected] May 2018 Abstract. Nowadays, the Gogny force is a referent in the theoretical description of nuclear structure phenomena. Its phenomenological character manifests in a simple analytical form that allows for implementations of techniques both at the mean field and beyond all over the nuclide chart. Over the years, multiple applications of the standard many-body techniques in an assorted set of nuclear structure applications have produced results which are in a rather good agreement with experimental data. The agreement allows for a simple interpretation of those intriguing phenomena in simple terms and gives confidence on the predictability of the interaction. The present status on the implementation of different many body techniques with the Gogny force is reviewed with a special emphasis on symmetry restoration and large amplitude collective motion. PACS numbers: 21.30.Fe, 21.60.Jz arXiv:1807.02518v1 [nucl-th] 6 Jul 2018 Mean field and beyond with the Gogny force 2 1. Introduction The Gogny force, named after the renowned French physicist Daniel Gogny [1], has been used to describe many different facets of nuclear structure since its inception back in the early seventies of the past century. It has been mainly used in a mean field framework including pairing correlations where the Hartree Fock Bogoliubov (HFB) mean field is the basic entity used to obtain the quantities appearing in the theory. In this category we include not only the calculation of potential energy surfaces (PES) using constraints on the relevant collective degrees of freedom so useful to describe the shape of the ground state or the path to fission, but also extensions like the collective Schr¨odingerequation (CSE), the Bohr Hamiltonian approach focused on quadrupole degrees for freedom and their coupling to rotations, the QRPA or the Interacting Boson Model (IBM) with parameters determined by HFB PES. As it stands, the mean field only provides wave functions in the intrinsic frame of reference where the mean field wave function is allowed to break symmetries of the Hamiltonian. In order to compute physical observables like mean values or transition probabilities it is very important to use wave functions in the laboratory frame which have the quantum numbers of the symmetries preserved by the nuclear interaction. Therefore, in addition to the mean field, mechanisms to restore the spontaneously broken symmetries have to be applied to perform the passage from the intrinsic to the laboratory frame. These mechanisms require the evaluation of overlaps between different HFB wave function that are subsequently used in integrals over the symmetry groups. This is referred to as "symmetry restoration" and its implementation with the Gogny force has been an active field of research since its first implementation in the early nineties. These ideas can also be used to deal with large amplitude collective motion in the spirit of the Generator Coordinate Method (GCM) often in combination with symmetry restoration. The purpose of this review is to describe the status of the implementation of all these techniques with the Gogny force putting special emphasis in those aspects related to symmetry restoration and large amplitude collective motion. There are already two long papers in the literature that overview some aspects of the applications of the Gogny force in nuclear structure but in our opinion they only offer partial views of the whole picture. In Ref [2] the authors review applications of the mean field, the 5D Collective Hamiltonian and the QRPA with the Gogny force. However, the coverage of the mean field is rather limited and fails to account for important applications like the study of high spin physics, finite temperature HFB or the applications to odd mass nuclei and multi-quasiparticle excitations as in the physics of high-K isomers. On the other hand, the review of [3] focuses only on aspects of symmetry restoration and large amplitude motion, not paying much attention to other applications beyond the mean field. Also the formal difficulties encountered in the implementation of symmetry restoration techniques with density dependent forces are scarcely discussed and also relevant technical details are overlooked. In our review we have tried to give a complete, albeit not very deep, description of all different techniques used in nuclear structure paying some attention to some relevant technical details like Mean field and beyond with the Gogny force 3 evaluation of matrix elements, Pfaffian techniques to evaluate overlaps or computer code implementations. Although not belonging to the family of Gogny forces we have decided to include the description of translational invariance restoration with the Brink-Boecker interaction as an illustration of this important, and often overlooked, aspect of symmetry restoration. The present review and the one of [2] can be considered as complementary as we do not cover in too much details the subjects treated by Peru and Martini. The theoretical description of nuclear reactions with the Gogny force is not described in this review. In the last years and with the help of increasing computing power, more and more of the nuclear structure microscopic input required for reactions can be obtained from sound theoretical models with the Gogny force [4,5,6]. All these models are described in this review but their connection with reaction theory is scarcely discussed in the text. The review focuses almost exclusively on applications with the Gogny force. The many body techniques used to describe nuclear structure have also been used with other interactions/functionals like the non-relativistic family of Skyrme energy density functionals or the relativistic models with great success. Although those calculations are similar and in many cases complementary to the ones presented here we are not going to discuss them and we refer the interested reader to the vast literature already available in the form of reviews. The review is divided in six sections including the Introduction, the second section is devoted to the description of the different parametrizations of the Gogny force and several recent improvements/departures from it. In Section 3 the mean field method, adapted to deal with density dependent interactions is discussed and several examples of application with the Gogny force are presented. In Section 4 two methods beyond the mean field but not requiring Hamiltonian overlaps are described: namely the QRPA and the IBM mapping procedure. In Section 5, the issue of symmetry restoration is discussed in general and later applications to the most common types of symmetry restoration (parity, particle number projection, angular momentum projection, linear momentum projection) are presented along with several applications with the Gogny force. The difficulties encountered in the application of the symmetry restoration techniques to the case of phenomenological density dependent interactions is also addressed. Finally, in Section 6 the standard method to deal with large amplitude collective motion is discussed. Among the applications, we discuss the application of the method with symmetry restored wave functions as well as the mixing of multi particle-hole excitations intimately connected with the Configuration Interaction method. Also approximate methods based on the Gaussian overlap approximation are discussed. We conclude with a summary and perspectives section. Several appendixes with a more technical information are also included. Mean field and beyond with the Gogny force 4 2. The Gogny force: its origins, motivation and present implementations In this section a historical overview of the origins and motivation of the Gogny force is presented with special emphasis in the fitting protocols used in each of the different main parametrizations considered (D1, D1', D1S, D1N, D1M). A few paragraphs will also be devoted to the newly proposed D2 Gogny force with a finite range density dependent interaction. Finally, we also discuss less known parametrizations including some specific terms and some forces inspired by the Gogny interaction. 2.1. The Gogny force: guiding principles The Gogny force was conceived in a period of time when the Skyrme interaction had started to become fashionable mostly because of its ability to describe nuclear properties at the simple Hartree Fock (HF) mean field level [7,8]. At that time, the fact that in Skyrme forces different interactions had to be used in the pairing and particle-hole (ph) channel was considered as a drawback. Also the necessity to consider a window around the Fermi level where the pairing interaction was active, was often considered as an annoying characteristic. In order to have a pairing force derived from the same central potential than the particle-hole (ph) channel, a finite range interaction, with its natural ultraviolet cutoff, had to be implemented. This is the main reason why the Gogny force was created: it had a finite range central potential that could also be used to obtain the pairing interaction. The central potential was inspired by early attempts by Brink and Boecker [9] to derive a finite range central potential with a Gaussian form for nuclear structure calculations. Going finite range was a technical challenge for the computers available at that time. However, combining together the simplicity of Gaussian shape for the central potential and a nice property of the harmonic oscillator wave functions [10], to be discussed below, gave the opportunity to get a reasonable implementation of the HF or the Hartree Fock Bogoliubov (HFB) mean fields on those days computers [11].