Test of the Charge Symmetry Hypothesis of NN Interaction from the Coulomb-Free p–p Scattering Cross Section and Its Relation to Universality

We examine the results on the determination of the Coulomb-free ¹ \(S_0\) $S_0$ proton–proton (p–p) scattering length by analyzing the cross section of the quasi-free p + d \(\rightarrow \) $\to$ p + p + n reaction at center-of-mass energies below 1 MeV. This was achieved using a Bayesian data-fitting approach, yielding a p–p scattering length \(a_{pp} = -18.17^{+0.52}_{-0.58}|_{stat}\pm 0.01_{syst}\) $a_{\mathit{pp}}=-18.{17}_{-0.58}^{+0.52}{|}_{\mathit{stat}}\pm 0.{01}_{\mathit{syst}}$  fm and effective range \(r_0 = 2.80\pm 0.05_{stat}\pm 0.001_{syst}\) $r_0=2.80\pm 0.{05}_{\mathit{stat}}\pm 0.{001}_{\mathit{syst}}$  fm. We test the stability of the results against the upper energy cutoff and fitting data sets separately. A model based on the Eckart potential is introduced for an effective description in the universal window. In this model, the short-range interaction is considered as a whole, similar to how the s-wave phase-shift \(\delta \) $\delta$ functions in describing low-energy nucleon–nucleon scattering data. Based on our analysis, we confirm that the obtained parameters accurately represent the characteristics of the short-range physics and the influence of the up-down quark mass difference on the charge symmetry breaking is less significant as initially anticipated. Additionally, we suggest evaluating the charge symmetry breaking of the short-range interaction rather than solely focusing on the nuclear interaction.