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We investigate three models for the charge symmetry breaking (CSB) of thenucleon-nucleon ($NN$) interaction (based upon $\rho$-$\omega$ mixing, nucleonmass splitting, and phenomenology) that all reproduce the empirical value forthe CSB of the $^1S_0$ scattering length ($\Delta a_{CSB}$) accurately. Wereveal that these models make very different predictions for CSB in $^3P_J$waves and examine the impact of this on some observable quantities of $A\geq 3$nuclear systems. It turns out that the $^3$H-$^3$He binding energy differenceis essentially ruled by $\Delta a_{CSB}$ and not very sensitive to CSB from $P$waves. However, the Coulomb displacement energies (which are the subject of theNolen-Schiffer anomaly) receive about 50% of their CSB contribution from $NN$partial waves beyond $^1S_0$. Consequently, the predictions by the various CSBmodels differ here substantially (10-20%). Unfortunately, the evaluation of theleading Coulomb contributions carry a large uncertainty such that nodiscrimination between the competing CSB models can presently be made. Todecide the issue we suggest to look into nuclear few-body reactions that aresensitive to CSB of the nuclear force.Comment: 21 pages including 4 figures, Revte

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