Single-particle spectroscopic strength from nucleon transfer reactions with a three-nucleon force contribution

The direct reaction theory widely used to study single-particle spectroscopic strength in nucleon transfer experiments is based on a Hamiltonian with two-nucleon interactions only. We point out that in reactions with a loosely-bound projectile, where clustering and breakup effects are important, an additional three-body force arises due to three-nucleon ($3N$) interaction between two nucleons belonging to different clusters in the projectile and a target nucleon. We study the effects of this force on nucleon transfer in $(d,p)$ and $(d,n)$ reactions on $^{56}$Ni, $^{48}$Ca, $^{26m}$Al and $^{24}$O targets at deuteron incident energies between 4 and 40 MeV/nucleon. Deuteron breakup is treated exactly within a continuum discretized coupled-channel approach. It was found that an additional three-body force can noticeably alter the angular distributions at forward angles, with consequences for spectroscopic factors' studies. Additional study of transfer to $2p$ continuum in the $^{25}$F$(p,2p)^{24}$O reaction, involving the same overlap function as in the $^{24}$O($d,n)^{25}$F case, revealed that $3N$ force affects the $(d,n)$ and $(p,2p)$ reactions in a similar way, increasing the cross sections and decreasing spectroscopic factors, although its influence at the main peak of $(p,2p)$ is weaker. The angle-integrated cross sections are found to be less sensitive to the 3N force contribution, they increase by less than 20$\%$. Including 3N interactions in nucleon removal reactions makes an essential step towards bringing together nuclear structure theory, where 3N force is routinely used, and nuclear direct reaction theory, based on two-nucleon interactions only.