Correlations among 3N observables

We use a correlation matrix of two-nucleon (2N) potential parameters to investigate correlations among three-nucleon (3N) observables for the elastic nucleon-deuteron (Nd) scattering process. We employ the up-to-date 2N semilocal momentum-space regularized chiral interactions and the phenomenological OPE-Gaussian potential and using the Faddeev formalism we calculate 3N observables and their correlations. As an example, we present the correlation between the differential cross section and the deuteron analyzing power T 20 .


Introduction
The question about the existence of correlations among nuclear physics observables is closely linked to constructing models of nuclear forces. Namely, the knowledge of the correlations among the various 3N observables will allow the determination of the set of observables, which should be used to fit the parameters of three-body force models. Such information will also be a valuable guide for planning future experiments in low-energy nuclear physics. To the best of our knowledge, the correlations between two-and three-nucleon observables (2N and 3N respectively) have not been investigated yet. The aim of this paper is to start such a study focusing on 3N observables.
Recently R. Navarro Peréz and his collaborators from the Granada group carefully revised the existing 2N data and prepared a new database [1], removing from the data these points for which uncertainties were unknown or poorly defined. They excluded also data sets inconsistent with other data, which led to a self-consistent database at the 3σ level. Currently, this database (Granada-2013) is a standard set of data used for fixing parameters of the 2N forces. In addition, R. Navarro Pérez et al. have derived new models of the 2N interaction, for example the One-Pion-Exchange-Gaussian (OPE-Gaussian) potential [2,3]. In 2018 the Bochum-Bonn group introduced sophisticated 2N potentials up to the fifth-order in the chiral 055.1 expansion with a semilocal regularization in momentum space for the pion exchange contributions [4]. They used the self-consistent Granada-2013 database to fix the free parameters of the model. For both potentials [2] and [4] a detailed error analysis was performed. In particular, the authors quantified for the first time the statistical uncertainties of the potential parameters and obtained not only the values of 2N potential parameters but also their covariance matrices. The knowledge of a correlation matrix of the 2N potential parameters allowed us to perform a statistical analysis aimed at finding theoretical uncertainties for the elastic nucleon-deuteron observables (3N observables). As a result in [5,6], we successfully analyzed the propagation of uncertainties of 2N potential parameters to a given 3N observable. It was found that this uncertainty is much smaller than the uncertainty arising from using various models of the nuclear interaction.

Methods and theoretical formalism
Given the mean values of the 2N potential parameters and their correlation matrix, we sampled 50 sets of the potential parameters. Next, for each set of potential parameters, we apply the Faddeev framework to calculate the elastic Nd scattering observables [7]. This means that we first calculate the deuteron wave function by solving the Schrödinger equation and obtain the t-matrix elements from the Lippman-Schwinger equation for a given 2N interaction in momentum space. Next, we solve the Faddeev equation in momentum space to compute the transition amplitude from which any 3N observable can be obtained. In this study, we neglect 3N interactions and use only the 2N force, which enters the Faddeev equation via the precalculated t-matrix operator. The Faddeev equation for the transition amplitude T |φ〉 reads: where the initial state |φ〉 is composed of a deuteron and a relative momentum eigenstate of the projectile nucleon, P is a permutation operator, G 0 is the free 3N propagator and t is a solution of the Lippmann-Schwinger equation with an interaction V (given by the models of 2N interaction from the Granada or Bochum-Bonn groups). We solve Eq. 1 numerically, in momentum space, using partial wave decomposition and by generating the Neumann series and summing it by the Padé method. More information can be found in Refs. [7] and [8].

Results
We analyzed many pairs of 3N observables for which we computed correlation coefficients at selected incoming nucleon energies and in the whole range of scattering angles. As an example in Fig. 1, we present the angular dependence of the correlation coefficient for a chosen pair of 3N observables as functions of the center-of-mass scattering angle in the range θ c.m. ∈ [0 • , 180 • ] at two laboratory energies of the incident neutron 13 and 65 MeV. Correlations between two selected 3N observables, i.e. the differential cross section and the deuteron tensor analyzing power T 20 were investigated with the semilocal momentum-space regularized (SMS) chiral N 2 LO, N 4 LO and N 4 LO+ potentials [4] with the value of the regulator parameter Λ = 450 MeV and with the OPE-Gaussian potential [2]. In general, as shown in Fig. 1, the differential cross section, dσ/dθ c.m. , appears weakly correlated with the deuteron tensor analyzing power, T 20 , for the two employed potentials and for both energies. However, it is observed that for some intervals of the scattering angle, a relatively strong correlation for this particular pair of 3N observables exists. In the case of the SMS chiral N 2 LO potential at scattering energy E = 13 MeV the absolute value of the correlation coefficient does not exceed With a growing order of the chiral expansion, one reaches a more accurate version of the chiral 2N force. One example is the SMS chiral N 4 LO. Using it we observe the same behavior, but the magnitude of the correlation coefficient at E=13 MeV is different. For the SMS chiral N 4 LO+ predictions one can see again more or less the same behavior as for the SMS chiral N 2 LO, especially for small scattering angles for both energies. For the phenomenological OPE-Gaussian potential, we see that the behavior of predictions is similar to the one for results based on the SMS chiral N 4 LO force for both energies.

Conclusion
We showed that it is possible to analyze the correlation among various 3N observables resulting from the correlation matrices provided with the models of 2N interactions from the Granada and Bochum-Bonn groups. It was also demonstrated that when using the chiral interaction the angular dependence of the correlation coefficients for the differential cross section and the deuteron analyzing power depends on the order of chiral expansion as well as on the scattering energy. It is interesting that a strong correlation appears at specific intervals of scattering angles. This could indicate a dependence of the transition amplitude on specific partial waves.