Charge symmetry breaking of the nucleon-nucleon interaction: ρ-ω mixing versus nucleon mass splitting

We investigate three models for the charge symmetry breaking (CSB) of the nucleon-nucleon $\mathrm{}\left(\mathrm{NN}\right)\mathrm{}$ interaction (based upon $\rho -\omega$ mixing, nucleon mass splitting, and phenomenology) that all reproduce the empirical value for the CSB of the ${}^1{S}_0$ scattering length $\left(\Delta a_{\mathrm{CSB}}\right)$ accurately. We reveal that these models make very different predictions for CSB in ${}^3{P}_J$ waves and examine the impact of this on some observable quantities of $A>~3$ nuclear systems. It turns out that the ${}^3\mathrm{H}{-}^3\mathrm{He}$ binding energy difference is essentially ruled by $\Delta a_{\mathrm{CSB}}$ and not very sensitive to CSB from P waves. However, the Coulomb displacement energies (which are the subject of the Nolen-Schiffer anomaly) receive about 50% of their CSB contribution from $\mathrm{NN}$ partial waves beyond ${}^1{S}_0.$ Consequently, the predictions by the various CSB models differ here substantially (10%–20%). Unfortunately, the evaluation of the leading Coulomb contributions carries a large uncertainty such that no discrimination between the competing CSB models can presently be made. To decide the issue we suggest looking into nuclear few-body reactions that are sensitive to CSB of the nuclear force.