The $nnn$ and $ppp$ correlation functions

Scattering experiments with three free nucleons in the ingoing channel are extremely challenging in terrestrial laboratories. Recently, the ALICE Collaboration has successfully measured the scattering of three protons indirectly, by using the femtoscopy method in high-energy proton-proton collisions at the Large Hadron Collider. In order to establish a connection with current and future measurements of femtoscopic three-particle correlation functions, we analyse the scenarios involving $nnn$ and $ppp$ systems using the hyperspherical adiabatic basis. The correlation function is a convolution of the source function and the corresponding scattering wave function. The finite size of the source allows for the use of the free scattering wave function in most of the adiabatic channels except the lowest ones. The scattering wave function has been computed using two different potential models: $(i)$ a spin-dependent Gaussian potential with parameters fixed to reproduce the scattering length and effective range and $(ii)$ the Argonne $v_{18}$ nucleon-nucleon interaction. Moreover, in the case of three protons, the Coulomb interaction has been considered in its hypercentral form. The results presented here have to be considered as a first step in the description of three-particle correlation functions using the hyperspherical adiabatic basis, opening the door to the investigation of other systems, such as the $pp\Lambda$ system. For completeness, the comparison with the measurement by the ALICE Collaboration is shown assuming different values of the source radius.