• bound state wave function
• few-nucleon system
• hyperspherical harmonics method
• integral relations
• Kohn variational principle
• phase shifts and mixing parameters
• scattering states

The study of the scattering states in few-nucleon systems is of great interest because it allows to perform a strong test for the models of the nuclear interaction and it provides the necessary inputs to study nuclear reactions. In this thesis we intend to use a method based on integral relations (IR) [1] to calculate scattering parameters, such as phase shifts and mixing angles, using bound-state-like wave functions. This is essentially a feasibility test of great interest for the following reasons: A≤4 scattering states are currently described with accurate methods, which however cannot work with any potential model. The method based on the IR should overcome this problem (see below). Furthermore, accurate methods for A>4 bound systems exist and they could be used in conjunction with the IR in order to study also A>4 scattering systems.

In this work, we use the hyperspherical harmonics (HH) method [2,3] for bound states to perform our feasibility test and we concentrate on the A=2,3 nuclear systems. Therefore we start introducing the HH method in the general case of a A-nucleon system. To be noticed that the HH method is one of the most accurate techniques used to solve the nuclear bound state problem for A≤4, with two- or three-body, local or non-local potentials, and its extension to a greater number of nucleons is promising [4]. Within the HH method, the bound state problem is solved variationally, applying the Rayleigh-Ritz (RR) variational principle. The eigenvalue equation is solved finding the eigenvectors of coefficients, which allow us to construct the bound state wave functions to be used in the IR. We discuss then the application of the HH method to the A-nucleon scattering state problem, reviewing first its implementation in conjunction with the Kohn variational principle (KVP). Then we derive the IR at first and second order.

We proceed applying the formalism to the case of a two-nucleon system and we calculate phase shifts and mixing angles for the following two-nucleon realistic potential models: the local Argonne AV18 [5] and the non-local Idaho chiral potential derived at next-to-next-to-next-to-leading order in the chiral expansion (N3LO) [6]. We choose these two potential models to make contact with the results available in the literature and to test the method with two very different potentials. With the IR, we are able to calculate directly the R-matrix, from which phase shifts and mixing angles are easily extracted. The AV18 and N3LO phase shifts and mixing parameters obtained with this method, for different states and for different values of the energy, are in very good agreement with those found in the literature [5,6].

We then study the three-nucleon system. Also in this case the bound state problem is solved variationally, using the RR principle. In particular, the form of the wave function of the system is more involved, due to the possible permutations of the three particles and the consequent presence of transformation coefficients, as well as the necessary expansion over a maximum of N=18 combinations of angular, spin and isospin momenta of the particles. These combinations are traditionally called ''channels'' in the HH method [2,3]. The IR are the same in form as those for the two-body problem but the calculation is computationally more involved. This is the main reason why we have considered only the AV14 potential model [7] in this feasibility test. In fact, the AV14 allows us to make contact with the results found in the literature [1], but it also shares with the AV18 similar features, as for instance a short-range hard repulsion, which makes the test with these potentials probably among the most difficult cases under the computational point of view. We have obtained results that are in agreement with those found in the literature [1].
Therefore we can conclude that our feasibility study is positive and the method of the IR is very promising.

In a near future, we expect to apply the method of the IR (i.e. use of bound-state-like wave functions to describe the scattering states) for A=3 also with other potential models, as the N3LO [6] and the CD-Bonn potential [8]. This last one is a widely used non-local potential model, which, due to its peculiar structure, cannot be applied within the HH method based on the KVP. Furthermore, we plan to extend the method of the IR to nuclear scattering problems with A≥4.

References:
[1] A. Kievsky, M. Viviani, and L. E. Marcucci, Phys. Rev. C 85, 014001 (2012)
[2] A. Kievsky et al, J. Phys. G: Nucl. Part. Phys. 35, 063101 (2008)
[3] L. E. Marcucci, J. Dohet-Eraly, L. Girlanda, A. Gnech, A. Kievsky, and M. Viviani, Front. Phys. 8, 69 (2020)
[4] A. Gnech, M. Viviani, and L. E. Marcucci, arXiv 2004.05814 (2020), submitted to Phys. Rev. C
[5] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995)
[6] R. Machleidt and D. R. Entem, Phys. Rep. 503, 1 (2011)
[7] R. B. Wiringa, R. A. Smith, and T. L. Ainsworth, Phys. Rev. C 29, 1207 (1984)
[8] R. Machleidt, Phys. Rev. C 63, 024001 (2001)