Standard Big Bang Nucleosynthesis (BBN) predicts the abundances of the light elements in the early universe. Even if the overall agreement with the experimental data is good, still some discrepancies exist on the relic abundances of ${}^7$Li and ${}^6$Li. In order to exclude or confirm these scenarios, the BBN model needs precise input parameters, in particular the cross-sections of the BBN nuclear reaction network. However, the suppression of the cross-sections due to the Coulomb barrier makes the measurement very difficult and so affected by large systematic errors. Therefore, reliable theoretical calculations result fundamental in order to reduce the uncertainties. In this work we present a theoretical study of two nuclear reactions connected to ${}^6$Li abundance and recently the $α$+d$\rightarrow$ ${}^6$Li + $γ$ and the p+${}^6$Li$\rightarrow$${}^7$Be+$γ$ radiative captures. For the first reaction we use a so-called ab-initio approach in which we solve the full six-body problem by using realistic nuclear potentials to describe the nucleon interactions. In particular we concentrate on the calculation and characterization of the final state of the reaction, the ${}^6$Li ground state, focusing on the electromagnetic static structure and the quantities relevant from the astrophysical point of view such as the asymptotic normalization coefficient. For doing this we use the Hyperspherical Harmonic approach developed by the Pisa group providing for the first time the possibility of using this approach beyond A = 4 nuclear systems. The second reaction is instead studied by using a two-body cluster approach where the proton and ${}^6$Li are considered as structureless particles. The angular distribution of the emitted photon obtained in this work were used by the LUNA Collaboration to determine the efficiency of the detector used in the measurement of the reaction.
