Investigation of the charge symmetry of the nuclear interaction using quasi-free scattering
Authors: Tumino Aurora , Rapisarda Giuseppe G. , La Cognata Marco , Oliva Alessandro , Kievsky Alejandro , Bertulani Carlos A. , D'Agata Giuseppe , Gattobigio Mario , Guardo Giovanni L. , Lamia Livio , Lattuada Dario , Pizzone Rosario G. , Romano Stefano , Sergi Maria L. , Spartà Roberta , Viviani Michele 
Published: January 2025

The study of nuclear forces, in particular the interaction between nucleons in the spin singlet state \(^1S_0\), allows us to examine the concepts of charge independence and charge symmetry. This latter property implies that the interaction between two protons is identical to that between two neutrons, except for small effects due to the different mass and charge of the particles. These effects leave their fingerprints on the scattering lengths that characterize low-energy nucleon-nucleon interactions. Precise measurements of the neutron-proton (\(n\)-\(p\)), proton-proton (\(p\)-\(p\)), and neutron-neutron (\(n\)-\(n\)) \(s\)-wave scattering lengths are essential for understanding charge symmetry breaking and refining nuclear force models. \(^1S_0\) proton-proton (\(p\)-\(p\)) scattering length requires Coulomb effects to be theoretically removed; yet, the Coulomb-free \(p\)-\(p\) data strongly depends on various theoretical techniques to subtract the Coulomb contribution. In this study, Tumino et al. (2023) aimed at measuring the Coulomb-free \(p\)-\(p\) scattering length directly, namely, minimizing the model dependence uncertainty. It has been extracted from the \(p\)-\(p\) scattering cross section measured at center-of-mass energies below 1 MeV via the quasi-free \(p+d \to p+p+n\) reaction, applying the Trojan Horse Method. A Bayesian data-fitting approach employing the expression of the s-wave nucleon-nucleon scattering cross section yielded a \(p\)-\(p\) scattering length \(a_{pp} = -18.17^{+0.52}_{-0.58}|_{\mathrm{stat}} \pm 0.01|_{\mathrm{syst}} \, \mathrm{fm}\) and an effective range \(r_0 = 2.80 \pm 0.05_{\mathrm{stat}} \pm 0.001_{\mathrm{syst}} \, \mathrm{fm}\), to be compared with the values in the literature, \(a_{pp} = -17.3 \pm 0.4 \, \mathrm{fm}\) and \(r_0 = 2.85 \pm 0.04 \, \mathrm{fm}\) (Machleidt and Slaus, 2001). It is important to underscore that the Coulomb-free \(p\)-\(p\) scattering length in the literature is devoid of short-range physics, which should be incorporated for a meaningful comparison. Therefore, a model founded on universality principles was developed to interpret these findings. It incorporates the short-range interaction as a whole, including nuclear and residual electromagnetic effects, similar to how the s-wave phase shift \(\delta\) operates in describing low-energy nucleon-nucleon scattering data. The calculated scattering length including short-range physics, \(a_{pp} = -17.6 \pm 0.4 \, \mathrm{fm}\), is in very good agreement with the Trojan Horse data, while model dependence is negligible as, in the universality framework, results are almost insensitive to the details of the interaction. The comparison with the current accepted short-range \(a_{pp} \) and \( a_{nn}\) values suggests that differences in the masses of up and down quarks and their electromagnetic interactions have a smaller-than-expected impact within the context of charge symmetry breaking, while other contributions such as partial waves beyond \(^1S_0\), three-nucleon interactions and the difference between quark scalar densities in proton and neutrons may play a role, especially at higher energies.
Abstract of the lecture delivered at Palazzo Grimaldi in Modica on January 10, 2025, on the occasion of the ceremony for the award of the Grimaldi Prize 2024.